Dendritations of Surfaces
Alfonso Artigue
math.DS


Strong laser alignment of solvent-solute aggregates in the gas-phase
    
    December 30, 2023
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In this paper we develop a generalization of foliated manifolds in the context of metric spaces. 
In particular we study dendritations of surfaces that are defined as maximal atlases of compatible upper 
semicontinuous local decompositions into dendrites.
Applications are given in modeling stable and unstable sets of topological dynamical systems. 
For this purpose new forms of expansivity are defined.







§ INTRODUCTION

In this paper we develop a Theory of Foliations from a viewpoint 
of Continuum Theory. We define and study what we call continuumwise foliations 
or simply cw-foliations and as a special case dendritations. 
These are generalizations of foliations of smooth manifolds, laminations, singular pseudo-Anosov foliations and 
the generalized foliations used by Hiraide to study expansive homeomorphisms. 
The idea is to consider monotone upper semicontinuous decompositions as local charts.
We will not assume that the plaques are distributed as a product structure as in standard foliation theory.
Moreover, two plaques in a common local chart have not even to be homeomorphic.

As we will show in Theorem <ref>, cw-foliations are a conceptual framework to understand the 
distribution of local stable and unstable continua in the dynamical systems that we will consider. 
A motivation is to classify cw-expansive surface homeomorphisms. We say that f is cw-expansive (continuum-wise expansive) <cit.> if 
there is δ>0 such that if (f^n(A))<δ for all n∈ and A is connected then 
(A)=1, i.e., A is a singleton.
Important examples of cw-expansive dynamics are Anosov diffeomorphisms of a compact manifold of arbitrary dimension and 
pseudo-Anosov maps of compact surfaces with singular points and 1-prongs. 

For an Anosov diffeomorphism

local stable sets form foliated charts (at least C^0, see <cit.>). 
If we consider an expansive homeomorphism of a compact surface, via L,Hi, we know that local stable sets form 
a singular foliation. 
Recall that a homeomorphism f is expansive if there is δ>0 such that if (f^n(x),f^n(y))<δ for all n∈ then y=x.
In fact, Hiraide and Lewowicz independently proved that expansive homeomorphisms of compact surfaces are conjugate to pseudo-Anosov diffeomorphisms. 
If we consider cw-expansivity on a compact surface we have that local stable sets may be far 
from determining foliations in the standard sense. 
Moreover, a local stable set may not even be a finite union of arcs.
For example, in <cit.>, it is constructed a cw-expansive homeomorphism on a compact surface with a fixed point 
whose local stable set is connected but not locally connected, see <ref>. 
This example suggests that the goal of classifying all the cw-expansive surface homeomorphisms requires new technology. 

In <ref>, we introduce some definitions located between expansivity and cw-expansivity 
that will be called cwN-expansivity. 
In a sense, a version of N-expansivity from a viewpoint of continuum theory. 
Recall that, given N≥ 1, we say that f is N-expansive <cit.> if there is δ>0 such that 
if A⊂ X (an arbitrary subset) and (f^n(A))≤δ for all n∈ then (A)≤ N (i.e., A has at most N points).
We will say, see Definition <ref>, that a homeomorphism is cwN-expansive if there 
is δ>0 such that if A,B⊂ X are continua, (f^n(A))<δ for all n≥ 0 and 
(f^n(B))<δ for all n≤ 0 then (A∩ B)≤ N.
Also, a homeomorphism is -expansive (where the F means finite) if there 
is δ>0 such that if A,B⊂ X are continua, (f^n(A))<δ for all n≥ 0 and 
(f^n(B))<δ for all n≤ 0 then A∩ B is a finite set.
These new forms of expansivity allow us to prove the local connection of stable sets and to conclude that 
they are dendrites. See Theorem <ref>.
Recall that a dendrite is a Peano continuum containing no simple closed curve, a continuum is a compact connected metric space 
and a Peano continuum is a locally connected continuum. 

It is known, see Theorem <ref>, that such dendrites (the stable and 
unstable local sets mentioned above) have a uniform size, i.e., there is δ>0 such that 
for all x in the surface the stable and the unstable dendrites of x meet the boundary of the disc 
centered at x with radius δ. 
In this way we arrive naturally to the concept of dendritic decomposition, see <ref>. 


The topic of topological decompositions has a long literature, see for example Sm,Ro,Da,Nadler2,ChaCha. 
For applications in dynamical systems the reader is referred to ArAnomalous,CaPa,PX,BM,KTT. 
A celebrated result proved by Moore in <cit.> states that 
the quotient space of an upper semicontinuous decomposition of the two-dimensional sphere 
in non-separating continua is again the sphere (assuming that the decomposition has at least two elements, in other case the quotient is a point). 
In some sense, Moore's decompositions are a generalization of zero-dimensional foliations, i.e., the decomposition in singletons. 
A standard foliated chart of [0,1]×[0,1] by horizontal lines has two properties: 
each plaque is an arc and the quotient space is an arc. 
By Proposition <ref>, a local chart of a dendritation has the following properties: each plaque is a dendrite and 
the quotient space is a dendrite. 
The mentioned properties of a standard foliated chart do not characterize the foliated chart, 
see Example <ref>. 
In <ref> we give necessary and sufficient conditions for a decomposition to be 
a standard foliated chart.
In Corollary <ref> we conclude that a continuous and -smooth dendritation is a standard foliation.
The terms continuous and -smooth have a special meaning in this paper, see Definitions 
<ref> and <ref>.

In <cit.> Hiraide considered generalized foliations 


for the study of expansivity from a topological viewpoint (see <ref>). 
However, this definition seems to be useful jointly with the pseudo orbit tracing property, 
and not in our case. 
For example, pseudo-Anosov singular foliations are not generalized foliations in the sense of Hiraide 
and they are our main examples of dendritations.
Our generalization of a foliation is designed to accompany cw-expansivity.

The concept of cwN-expansivity appears naturally in the deep study of the articles L,Hi. 
In these papers expansive homeomorphisms of surfaces are classified. 
A careful reading reveals that several arguments can be done 
assuming cw1-expansivity instead of expansivity. 
The concept of cw1-expansivity was previously considered in <cit.> 
where some topological properties of stable sets were proved.
In Theorem <ref> we show that every cw1-expansive homeomorphism of a compact surface is expansive. 
The key of this proof is Theorem <ref> where we give sufficient conditions for a cw1-expansive homeomorphism 
of a compact metric space to be expansive.
In Example <ref> we show that cw1-expansivity does not imply expansivity on arbitrary compact metric spaces 
(this space is not locally connected). 
In <ref> we show that there is a cw2-expansive homeomorphism of the two-dimensional sphere that is not 2-expansive.
It is a pseudo-Anosov with 1-prongs.

We obtained some general results on surface dendritations.
In Theorem <ref> we show that 
for every dendritation of a closed surface there is a residual set 
of points without ramifications.
In particular, generic leaves are one-dimensional submanifolds. 
This is a consequence of another result by Moore <cit.>, 
saying that at most a countable number of disjoint triods can be embedded in a plane.
In Theorem <ref> we consider a -expansive homeomorphism of a compact surface. 
We show that stable and unstable continua form dendritations. 
We also prove that: no leaf is a Peano-continuum, generic leaves are non-compact one-dimensional manifolds 
and in a dense subset of the surface stable and unstable leaves are topologically transverse.
One can feel that this result gives a nice generic picture of what is a -expansive homeomorphism of a compact surface. 
However, we think that the goal of classifying surface -expansive homeomorphism is far from the present paper, not to mention cw-expansivity.

A brief sketch of the paper is as follows.
In <ref> we introduce new forms of expansivity and we study the topology of stable and unstable sets. 
In <ref> we recall the main results from continuum theory that will be needed and 
study decompositions, the local charts of our foliations.
In <ref> we define and study cw-foliations on metric spaces. 
They are defined via atlases of upper semicontinuous decompositions. 
We study the induced partition of the space into leaves.
The contents of <ref> and <ref> are independent.
In <ref> the results of the previous sections are joined to study the stable and the unstable 
cw-foliations defined by a cw-expansive homeomorphism of a metric space.
In <ref> we study some special cases of dendritations of surfaces. 
We give sufficient conditions to prove that they are (singular) foliations. 
Also, some properties of the stable and the unstable dendritations of a homeomorphism with 
some kind of expansivity are derived.
Throughout the paper several open problems are given.

The author thanks the referee for the careful reading and several suggestions that helped to improve the article.



§ VARIATIONS OF EXPANSIVITY



We start presenting general results of cw-expansive homeomorphisms on Peano continua. 
In <ref> we introduce cwN-expansivity and we summarize the main variations of 
expansivity that will be considered in this paper.
In <ref> we present some examples 
that will be used to illustrate our results in subsequent sections.
In particular, in <ref> we give an example of a cw2-expansive homeomorphism on the two-dimensional sphere that is not 
2-expansive.
In <ref> basic topological properties of stable sets are stated. 
We recall the Invariant Continuum Theorem for such dynamics.
In <ref> we introduce capacitors as a tool to understand 
what happens with unstable continua between two close stable plates. 
In <ref> we give another form of expansivity, called partial expansivity, 
that generalizes partial hyperbolicity of diffeomorphisms. We do not develop this concept in the 
paper, however we think that it could be of interest. 
In <ref> we consider the relationship between stable sets and stable continua. 
Also, we give sufficient conditions for a cwN-expansive homeomorphism to be N-expansive. 
For these purposes we introduce expansivity modulo an equivalence relation.



 §.§ CwN-expansivity



Let f X→ X be a homeomorphism of a compact metric space (X,). 
We say that A⊂ X is a subcontinuum if A is compact and connected.
Denote by (X) the space of subcontinua of X 

    (X)={A⊂ X: A is a non-empty continuum}.

In (X) we consider the Hausdorff metric. 
It is usually called hyperspace of X and has the remarkable properties of being compact (if X is compact) 
and arc-connected (provided that X is connected), see Nadler,Nadler2.
For δ>0 define the sets

    _δ={A∈(X):(A)≤δ},


    ^s_δ={A∈(X):f^n(A)∈_δ for all n≥ 0},


    ^u_δ={A∈(X):f^n(A)∈_δ for all n≤ 0}.

The continua in ^s_δ are called δ-stable 
and those in ^u_δ are δ-unstable.
We also consider the sets

    [ ^s={A^s∈(X):lim_n→ +∞(f^n(A^s))= 0},; ^u={A^u∈(X):lim_n→ -∞(f^n(A^u))= 0}. ]

The continua in ^s are called stable 
and those in ^u are unstable.


 The sets _δ,^s_δ and ^u_δ are closed in (X).




  Given N≥ 1 we say that f is cwN-expansive if there is δ>0 such that 
  if A^s∈^s_δ and A^u∈^u_δ then 
  (A^s∩ A^u)≤ N.
  In this case δ is a cwN-expansivity constant.

  We say that f is -expansive if there is δ>0 such that 
  if A^s∈^s_δ and A^u∈^u_δ then 
  A^s∩ A^u is a finite set.



In Table <ref> the main variations of expansivity considered in this paper are summarized.





 §.§ Examples



In this section we explain the examples that we had in mind while developing this paper. 
We start with three classical families for which there are well established theories 
for modeling stable and unstable sets:


  * Anosov diffeomorphisms. These diffeomorphisms are defined on smooth manifolds 
and are characterized by the uniform hyperbolicity on the tangent bundle.
Considering the distribution of stable and unstable sets,
they are the most regular kind of expansive homeomorphisms 


because they form continuous foliations (see <cit.>). 

  * Smale spaces. These are expansive homeomorphisms of compact metric spaces with local product structure (equivalently, canonical coordinates 
 or pseudo-orbit tracing property). 
 The category includes Smale's basic sets of Axiom A diffeomorphisms. 
 Stable and unstable sets can be modeled with Hiraide's generalized foliations (see <ref>).
 
  * Pseudo-Anosov diffeomorphisms. These diffeomorphisms are defined on compact surfaces. 
 They have local product structure except for a finite number of points called singular. 
 Stable and unstable sets form singular foliations (see for example <cit.>). 

Next we describe other examples that will be essential in the development of the paper. 
Except for the first one, we assume that the reader is familiar with Smale's derived from Anosov diffeomorphisms 
(which in particular is an interesting example of a Smale space).



  §.§.§ A pseudo-Anosov with 1-prongs


The dynamics of pseudo-Anosov diffeomorphisms is not simple, at least from author's viewpoint. 
In this section we wish to discuss in detail some properties of a special example, 
a pseudo-Anosov with 1-prongs of the sphere. 
This example has been considered several times in the literature. 
It has some properties that may not be easy to predict at first sight.
In <cit.>*Example 1, p. 140 Walters considered it to show that 
a factor of an expansive homeomorphism may not be expansive.
In <cit.>*2.4 it is proved that 
the local stable set of some points is not locally connected.
In <cit.> it is shown that it is not entropy expansive, in fact they show that 
there are arbitrarily small horseshoes.
We will show that it is cw2-expansive but not N-expansive, for all N≥ 1.

The example is as follows. Let T^2=S^1× S^1 be the two dimensional torus where S^1=/. 
 Consider the equivalence relation p∼ -p for p∈ T^2. 
 The quotient space is a two-dimensional sphere S^2=T^2/∼. 
 Denote by π T^2→ S^2 the canonical projection.
 On the torus consider the Anosov diffeomorphism f̃ T^2→ T^2 defined by f̃(x,y)=(2x+y,x+y). 
 Define f S^2→ S^2 by f(π(p))=π(f̃(p)) for all p∈ T^2.
For a more detailed construction the reader is referred to the works mentioned above.

The homeomorphism f is cw2-expansive.



Denote by ^s and ^u the stable and the unstable singular foliations of f, respectively. 
These are transverse foliations except at the singularities. 
Singular points are 1-prongs and the foliations looks as in Figure <ref>. 
Then, a small arc of the stable foliation intersects in at most two points 
an unstable arc. 
Thus, the proof is reduced to show that every stable continuum is contained in a stable leaf.
Arguing by contradiction, let C⊂ S^2 be a stable continuum that it is not contained in a stable leaf. 
Then there is a hyperbolic periodic point p∈ S^2 such that ^s(p)∩ C≠∅, where ^s(p) denotes the stable leaf trough p. 
Since p is hyperbolic we see that (f^n(C)) cannot go to zero as n→+∞ because 
C is not contained in ^s(p). This contradiction proves that every stable continuum is contained in a stable leaf. 


As usual we define local stable and unstable sets as 

    W^s_ϵ(p)={x∈ S^2:(f^n(p),f^n(x))≤ϵ for all n≥ 0}
    
    W^u_ϵ(p)={x∈ S^2:(f^n(p),f^n(x))≤ϵ for all n≤ 0}

respectively. 
The next result can be derived from <cit.>, 
however, since we think that more details can be given, a proof is included. 
The author learned this proof from J. Vieitez and J. Lewowicz. 
We will use the notation A for the closure of A.



For all ϵ>0 there is a Cantor set C such that (f^n(C))≤ϵ for all n∈,
in particular 
f is not N-expansive for all N≥ 1. 



Let q∈ S^2 be a 1-prong of f. 
Take p∈ S^2 such that the orbit {f^n(p):n≥ 0} is dense in S^2. 
A point with this property will be called transitive.
Consider ϵ>0. 
We will show that W^s_ϵ(p) contains a Cantor set contained in the unstable arc of p. 
Consider n≥ 0 such that there is x_1≠ f^n(p) with x_1∈ W^s_ϵ/2(f^n(p))∩ W^u_ϵ/2(f^n(p)). 
We have that p_1=f^-n(x_1)∈ W^s_ϵ/2(p)∩ W^u_ϵ/2(p). 
Also we can assume that p_1 is in the unstable arc of p, see Figure <ref>. 
Since f^n(p) and f^n(p_1) are in a stable arc and 
p is transitive we have that p_1 is transitive too. 


Denote by C_1={p,p_1}. 
For each x∈ C_1 take a transitive point y in the unstable arc of p with 
y∈ W^s_ϵ/4(x). 
Define C_2 as the set containing C_1 and these two new points. 
Again, for each x∈ C_2 take a transitive point y in the unstable arc of p with 
y∈ W^s_ϵ/8(x). Inductively we define a sequence of sets {C_n:n≥ 1} 
with |C_n|=2^n+1. By construction each C_n is contained in W^s_ϵ(p). 
Denote by C=∪_n≥ 1C_n. 
Since W^s_ϵ(p) is closed we have that C⊂ W^s_ϵ(p). 
Also, C has no isolated point and it cannot contain an arc because f is cw-expansive (in fact cw2-expansive as we proved). 
Then, C is a Cantor set contained in the unstable arc of p and in W^s_ϵ(p).




  §.§.§ Quasi-Anosov diffeomorphisms



By definition, a quasi-Anosov diffeomorphism is a C^1 diffeomorphism f M→ M 
of a compact manifold such that {df^n(v):n∈} is unbounded for every non-vanishing tangent vector v, 
where df^n is the differential of f^n and · is the norm induced by a Riemannian metric on M.
They were characterized by Mañé <cit.> as Axiom A diffeomorphisms with quasi-transverse stable and unstable spaces.
Recall that Axiom A means that the non-wandering set is hyperbolic and periodic points are dense in the non-wandering set. 
For an Axiom A diffeomorphism, at every point x∈ M the tangent space contains a contracting subspace E^s_x and an expanding subspace E^u_x. 
The quasi-transversality condition means that E^s_x∩ E^u_x=0 for all x∈ M.

We proceed to sketch the construction of a particular quasi-Anosov diffeomorphism 
that is not Anosov <cit.>.
Consider two derived from Anosov diffeomorphisms 
f_i M_i→ M_i, i=1,2, where M_i is an n-torus,
f_1 is conjugate to f_2^-1 and
f_1 presents a codimension one shrinking repeller and a sink fixed point p_1.
Consequently, f_2 presents a codimension one expanding attractor and a source fixed point p_2.
Let B_i, i=1,2, be an open ball around p_i such that B_1⊂ f_1^-1(B_1) and 
B_2⊂ f_2(B_2). 
Consider the manifolds with boundary N_i=M_i∖ B_i
and a diffeomorphism φf_1^-1(B_1)∖ B_1→f_2(B_2)∖ B_2
such that M=N_1∪ N_2/x∼φ(x) is a closed manifold and 
there is a diffeomorphism f M→ M extending the dynamics of f_1 and f_2.

In <cit.> it is proved that for n=3, there is a diffeomorphism φ making f quasi-Anosov.
As we said, the non-wandering set is hyperbolic. 
At wandering points stable and unstable manifolds are one-dimensional and its tangent lines are quasi-transverse.
Since M is 3-dimensional, they are not transverse and f is not Anosov.



  §.§.§ Q^r-Anosov diffeomorphisms


In <cit.> the construction of <ref>
was considered for the simpler case n=2. 
On a surface there is not enough space to construct a quasi-Anosov diffeomorphism (not being Anosov). 
However, the construction can be performed.
In this case the map φ will introduce tangencies between stable and unstable manifolds at wandering points. 
Assuming that f is of class C^r, if these tangencies are of order at most r≥ 2 we say that f is Q^r-Anosov. 
In <cit.> it is shown that the set of Q^r-Anosov diffeomorphisms 
of a closed surface is an open set (in the C^r topology) of r-expansive diffeomorphisms, 
where r-expansive means N-expansive with N=r.
These examples are Axiom A and show that (N+1)-expansivity does not imply N-expansivity.



  §.§.§ Anomalous cw-expansivity



In <cit.> it was considered another variation of the quasi-Anosov. 
On a surface, as in <ref>, 
it is defined a cw-expansive homeomorphism with what we called an anomalous saddle. 
It is a fixed point whose stable set is connected but not locally connected. 
Let us briefly give a description. 
Let E⊂^2 be the set ({1}∪{1+1/n}_n≥ 2)×[0,1].
It is a countable union of vertical segments.
Consider the auxiliary map T^2→^2 defined as T(p)=p/2. 
Define F^s=(×{0})∪_n≥ 0 T^n(E) and F^u={0}×. 
The anomalous saddle in this case is the origin. 
In <cit.> it is constructed a homeomorphism around (0,0) such that 
its local stable set is F^s and the unstable set is F^u (intersected with a neighborhood of the origin). 
This anomalous saddle is inserted in a Q^r-Anosov diffeomorphism of the previous section.
In the example, the intersection of local stable and unstable sets is totally disconnected, thus implying cw-expansivity.






















 §.§ Invariant continua


Our interest is centered at dynamical systems on surfaces, but some fundamental results on cw-expansivity hold for homeomorphisms on Peano continua, 
i.e., a connected, locally connected and compact metric space.
In this setting, Theorem <ref> plays the role of the Invariant Manifold Theorem <cit.> for hyperbolic diffeomorphisms on smooth manifolds.

The following result is a characterization of cw-expansivity in terms of stable and unstable continua. 
The direct part is known <cit.>*2.
The converse may be new, however it is quite direct from the definitions.



A homeomorphism f X→ X of a compact metric space is cw-expansive if and only if 
 there is ϵ^*>0 such that 
  
   
  * ^σ_ϵ^*⊂^σ and 
   
  * for all ϵ>0 there is δ>0 such that 
   ^σ∩_δ⊂^σ_ϵ
   
   for σ=s and σ=u.



 Direct. Let ϵ^* be such that if A⊂ X is connected and 
 (f^n(A))≤ϵ^* for all n∈ then A is a singleton. 
 Suppose that σ=s and take A∈^s_ϵ^*, that is, 
 (f^n(A))≤ϵ^* for all n≥ 0. 
 If this diameter does not converge to 0 then there are r>0 and n_k→ +∞ such that 
 (f^n_k(A))>r for all k≥ 0. 
 Since (X) is compact we can assume that f^n_k(A)→ C in the Hausdorff metric. 
 We will show that 
 
    (f^n(C))≤ϵ^* for all n∈.

 Since n_k→ +∞, given n∈ there is k_0 such that n_k+n≥ 0 for all k≥ k_0. 
 Then, (f^n_k+n(A))≤ϵ ^* for all k≥ k_0. 
 Since f^n_k+n(A)=f^n(f^n_k(A)) and f^n_k(A)→ C we conclude (<ref>).
 Since (C)≥ r>0 we have that C is not a singleton. 
 This contradicts the cw-expansivity of f and proves that ^s_ϵ^*⊂^σ.
 
 To prove the next part we argue by contradiction. 
 Suppose that there is ϵ>0 and a sequence A_k∈^s such that 
 (A_k)→ 0 and (f^m_k(A_k))>ϵ for all k≥ 0 where m_k≥ 0. 
 Note that m_k→+∞ and that we can assume that ϵ<ϵ^*.
 Since A_k∈^s there is n_k≥ m_k such that (f^n(A_k))≤ϵ for all n≥ n_k.
 We can take a subcontinuum C_k⊂ A_k and another divergent sequence m'_k such that 
 (f^m'_k(C_k))=ϵ and (f^n(C_k))≤ϵ for all n≥ 0. 
 Then, arguing as in the proof of (<ref>), we conclude that 
 a limit continuum of f^m'_k(C_k) contradicts the cw-expansivity. 

 Converse. Let us show that ϵ^* is a cw-expansivity constant. 
 Suppose that (f^n(A))≤ϵ^* for all n∈. 

 Assume, by contradiction, that 
 there is ϵ∈ (0,(A)). 
 For this value of ϵ there is δ>0 such that ^s∩_δ⊂^s_ϵ. 
 Since A∈^u_ϵ^* and ^u_ϵ^*⊂^u 
 there is m≤ 0 such that (f^m(A))<δ. 
 Then f^m(A)∈^s∩_δ. 
 As (A)>ϵ we have that f^m(A)∉^s_ϵ. 
 This is a contradiction that finishes the proof.


The next theorem states the existence of non-trivial stable and unstable continua through each point of the space. 
Moreover, these continua have diameter bounded away from zero. 
Their invariance is in fact given by Proposition <ref>.
The result was first proved independently by Hiraide and Lewowicz in Hi,L for 
the classification of expansive homeomorphisms on compact surfaces. 
It was generalized by Kato in <cit.>*Theorem 1.6 for cw-expansivity. 
A partial result was previously given by Mañé in <cit.> to prove that minimal expansive homeomorphisms can only exist 
on totally disconnected spaces.


For the study of cw-expansive homeomorphisms it is essential to assume some kind of connection of the space. 
To illustrate this point consider that every homeomorphism of a Cantor set is cw-expansive. 
It turns that local connection is a good property to exploit the cw-expansivity. 
There is a minor loss of generality if, in addition, we assume that the space is a Peano continuum. 
Indeed, if X is a locally connected compact metric space then it has a finite number of components. 
Therefore, a homeomorphism f of X will permute this components and 
taking a power of f we will have a finite number of homeomorphisms of Peano continua.


For a set ^*⊂(X), as _δ,^σ and ^σ_ϵ, 
define ^*(x)={A∈^*:x∈ A}.




  If f is a cw-expansive homeomorphism of a Peano continuum X

  then for all ϵ>0 there is δ>0 such that 
  
    ^σ_ϵ(x)∖_δ(x)≠∅

 for all x∈ X and σ=s,u.
















This theorem has the following direct consequences.

[Uniform size of stable continua]
 Note that (<ref>) implies that for all x∈ X there are stable and unstable continua through x 
 of diameter greater than δ. 
 Consequently, these continua meet the boundary of 
 the ball B_δ/2(x). 


[No stable points]

 From the Invariant Continuum Theorem we see that if f is a cw-expansive homeomorphism of a Peano continuum X
 then neither stable nor unstable continua have interior points. 
 This is because if A⊂ X is a stable set with an interior point x then we can take an unstable continuum contained in 
 A that contradicts the cw-expansivity. 
 In particular there are no Lyapunov stable trajectories.


[Surfaces with boundary]

Let us explain why we do not consider surfaces with boundary. 
Suppose that f S→ S is a cw-expansive homeomorphism 
of a compact surface with boundary. 
By Brouwer's Theorem on the Invariance of Domain <cit.> 
we know that ∂ S 
is invariant by f.
Then, the restriction f∂ S→∂ S is cw-expansive.
This gives a contradiction because, 
on one hand there are non-trivial stable continua in ∂ S, 
and on the other hand every non-trivial continuum of ∂ S
has interior points (relative to ∂ S). 
This is the argument to prove that the circle admits no expansive homeomorphisms 
that the author learned from Lewowicz.



 Do cw-expansive homeomorphisms of compact manifolds with non-empty boundary exist? 
For example, does there exist a cw-expansive homeomorphism of the 3-ball?

Does the 3-sphere admit expansive or cw-expansive homeomorphisms?


We say that C⊂ X separates X if X∖ C is not connected.
We will show 
in Theorem <ref>, under a natural assumption on the Peano space X, that no stable set separates. 
Previously we prove a topological lemma.



If X is a continuum and every point has arbitrarily 
small neighborhoods with connected boundary then 
for all ϵ>0 there is δ>0 such that 
if A⊂ X is a closed set that separates X and (A)<δ 
then there is a component V of X∖ A with (V)<ϵ.



  Given ϵ>0 consider a finite open cover 𝒰 of X 
  such that (U)<ϵ for all U∈𝒰 and ∂ U is connected. 
  Take δ>0 such that if (A)<δ then there is U∈𝒰 such that A⊂ U. 
  Suppose that A⊂ X is closed, separates X and (A)<δ. 
Take U_*∈𝒰 such that A⊂ U_*. 
  
Let us show that C=X∖ U_* is connected. 
By contradiction, suppose that C=A∪ B a union of disjoint, closed, non-empty sets. 
Since ∂ U_* is connected, we can assume that ∂ U_*⊂ A. 
Then, B and A∪U_* disconnects X, a contradiction that proves that X∖ U_* is connected.






Let V_* be the component of X∖ A containing X∖ U_*. 
Since A separates X, there is at least another component. 
This component is contained in U_* and has diameter smaller than ϵ.




If f is a cw-expansive homeomorphism of the Peano continuum X and every 
point of X has arbitrarily 
small neighborhoods with connected boundary
then no stable closed set separates X.



Arguing by contradiction assume that A⊂ X is a closed stable set separating X. 
As f is a homeomorphism, every iterate of A separates X.
Since it is stable, taking a positive iterate, we can suppose that it is as small as we want. 
By Lemma <ref> there is a small component U of its complement. 
By Theorem <ref> each point of U has a stable continuum meeting ∂ U. 
Since ∂ U⊂ A we have that ∂ U is stable, therefore, U is a stable set.
Since U has interior points we have a contradiction with Remark <ref>.



If a Peano continuum has no locally separating points then 
every point has arbitrarily small neighborhoods with connected boundary. 
See <cit.> for a proof.




 Theorem <ref> holds if X is a compact manifold with or without boundary. 
 In the one-dimensional case, intervals do not have connected boundary, but since there are not cw-expansive 
 homeomorphisms on one-dimensional manifolds the theorem holds true. 


The following example shows that in Theorem <ref> we need to assume 
that every point has arbitrarily small neighborhoods with connected boundary.


  Consider two copies of an Anosov diffeomorphism of the two-dimensional 
  torus identifying two fixed points. 
  The gluing point has not arbitrarily small neighborhoods with connected boundary. 
  Also, this point forms a stable set and it (locally and globally) separates. 





 §.§ Capacitors



The following technical definition is based on the arguments of the proof of <cit.>*Lemma 2.3.


Given x∈ X, ϵ,r>0, an (ϵ,r,x)-capacitor is a triple (A,G,B) such that: 

 
  * A,B,G⊂ X, A and B are disjoint continua, x∈ G, G is open,
 
  * (∂ G)∩ B_r(x)⊂ A∪ B,
 
  * there is a continuum γ⊂G∩ B_r/2(x) meeting A and B,
 
  * G⊂ B_ϵ(A) or G⊂ B_ϵ (B).




In this case A and B are the plates of the capacitor, 
ϵ is the separation of the plates and r is the radius.
See Figure <ref>.

We say that a capacitor has stable plates if A,B are stable sets for the homeomorphism f X→ X.


In the next result we study unstable continua between two close stable plates.
It generalizes <cit.>*Lemma 2.3
and it will be applied in Theorem <ref> for the study of -expansivity on surfaces. 


Assume that f X→ X is a cw-expansive homeomorphism of a Peano continuum X.
Then, for all r>0 small there is ϵ>0 such that if 
 (A,G,B) is an (ϵ,r,x)-capacitor 

with stable plates then: 
 
  
  * for all y∈ B_r/2(x)∩G there is an unstable continuum from y to A∪ B 
  contained in G∩ B_r(x),
  
  * there is an unstable continuum C⊂G∩ B_r(x) meeting A and B. 
 



Arguing by contradiction assume that there are 
r>0, a sequence of (1/n,r,x_n)-capacitors (A_n,G_n,B_n) with stable plates




and y_n∈ B_r/2(x_n)∩G_n with no unstable continuum from y_n to A_n∪ B_n 
contained in G_n∩ B_r(x_n).
Since X is a Peano continuum, by Theorem <ref> 
for each n there is an unstable continuum C_n⊂ B_r(x_n) 
containing y_n and intersecting the boundary of B_r(x_n). 
Since (∂ G_n)∩ B_r(x_n)⊂ A_n∪ B_n  
we conclude that C_n⊂ G_n. 
Taking subsequences, we can assume that A_n→ A and C_n→ C in the Hausdorff metric. 
By definition of capacitor we have that G_n⊂ B_1/n(A_n) 
which implies that C⊂ A. 
Therefore C is a continuum that is stable and unstable.  
Since (C)>0 we have a contradiction with the 
cw-expansivity of f. 

Given r>0 consider ϵ>0 satisfying the first item. 
Let (A,G,B) be an (ϵ,r,x)-capacitor. 
By definition, there is a continuum γ⊂B_r/2(x) 
and points p^A,p^B
with p^A∈ A∩γ and p^B∈ B∩γ.
Define 

    [ γ^A={y∈γ:∃ an unstable continuum from  y  to A  contained in  G∩ B_r(x)},; γ^B={y∈γ:∃ an unstable continuum from  y  to B  contained in  G∩ B_r(x)}. ]

The sets γ^A and γ^B are non-empty because they contain p^A and p^B respectively. 
They are closed sets and, as we have shown, they cover γ. 
Since γ is connected, 
there is a point z∈γ^A∩γ^B. 




Let f M→ M be an expansive homeomorphism of a compact three-manifold.
In <cit.> it is shown that local stable sets are 
locally connected 
if f is smooth and without wandering points.
By Theorem <ref> we know that a stable set of f 
cannot be homeomorphic to the closure of the set

    C={(x,y,z):-1≤ x≤ 1, 0<z≤ 1, y=sin(1/z)}.













 §.§ Partial expansivity



Consider a homeomorphism f X → X of a compact metric space.
We will use some concepts of topological dimension. We refer the reader to <cit.> for the definitions and basic properties.



Given an integer d≥ -1, we say that f is partially expansive 
with central dimension d and expansivity constant
ϵ>0 
if for every non-trivial compact set C⊂ X with (C)>d 

there is k∈ such that 
(f^k(C))≥ϵ.


 As usual, we say that f 
 is sensitive to initial conditions
 if there is ρ>0 such that for all x∈ M and for all r>0 there are y∈ B_r(x) and 
 n∈ such that (f^n(y),f^n(x))>ρ.



For a homeomorphism f X→ X the following hold:

 
  * f is expansive if and only if it is partially expansive with d=-1,
 
  * f is cw-expansive if and only if it is partially expansive with d=0,

  * if in addition X is a compact manifold of dimension n 
then f is sensitive to initial conditions if and only if 
f is partially expansive with central dimension d=n-1.




Since the arguments are quite direct we only give the details that we consider more relevant.
To prove the first part, note that by definition 
(see <cit.>) the condition (C)>-1 means C≠∅. 
Then, C⊂ X with (C)>-1 is non-trivial if and only if it has at least 
two points. To conclude the stated equivalence one has to note that 
({x,y})=(x,y).

The statement related to cw-expansivity follows because positive dimension is equivalent 
to contain a non-trivial continuum. 

For the last part we recall <cit.>*Corollary 1, p. 46
that if X is a compact n-dimensional manifold 
and C⊂ X then (C)=n if and only if C has non-empty interior.
















Let ϕ× X→ X be a continuous flow. 
We consider the following weak form of expansivity.
We say that a flow ϕ is separating <cit.>
if there is δ>0 (a separating constant) such that if 
(ϕ_t(x),ϕ_t(y))<δ for all t∈ then y=ϕ_s(x) 
for some s∈. 
Examples of separating flows are expansive flows in the sense of 
Bowen-Walters <cit.> and k^*-expansive flows as 
defined by Komuro <cit.>. 



If ϕ is a separating flow then
for all T≠ 0 the homeomorphism 
ϕ_T X→ X is partially expansive with d=1.



Let σ>0 be a separating constant 


and take δ>0 such that 
if C⊂ X is a continuum and (C)<δ then (ϕ_s(C))<σ 
for all s∈[-T,T]. 
Then, if (ϕ_nT(C))<δ for all n∈ we have that 
(ϕ_s(C))<σ for all s∈. 
Therefore, C is an orbit segment and consequently (C)≤ 1. 




















 §.§ Relative expansivity







Given a homeomorphism f X→ X 
it is usual to define the stable set of x∈ X 
as 

    W^s(x)={y∈ X: lim_n→ +∞(f^n(x),f^n(y))= 0}.

and the unstable set 

    W^u(x)={y∈ X: lim_n→ -∞(f^n(x),f^n(y))= 0}.

Consider the equivalence relation x∼_s y, x,y∈ X, if there is a continuum C⊂ X such that 
x,y∈ C and (f^n(C))→ 0 as n→ +∞. 
Similarly we define ∼_u (taking n→-∞). 
The equivalence class of x will be denoted as ^s(x) (and ^u(x)). 
We will give conditions that allow us to prove that W^s=^s and W^u=^u.


Let  be a partition of X.
We say that f is separating mod  if 
 there is δ>0 such that if (f^n(x),f^n(y))<δ for all n≥ 0 then y∈(x)



 If f is separating mod ^s then W^s=^s.



Assume that f is separating mod ^s with δ as in the definition.
Suppose that (f^n(x),f^n(y))→ 0 as n→+∞. 
 Take k such that for all n≥ k it holds that (f^n(x),f^n(y))<δ. 
 Then there is a stable continuum C containing f^k(x),f^k(y). 
 This implies that x,y are in the stable continuum f^-k(C). 
 Then W^s=^s.



 Assuming that f is cw-expansive does the condition ^s=W^s imply 
 that it is separating mod ^s?



 The pseudo-Anosov with 1-prongs on the two-dimensional sphere 
 given in <ref> is not separating mod ^s.



By Proposition <ref>, for all ϵ>0 
 there is a Cantor set C contained in an unstable arc and contained 
 in W^s_ϵ(p) for some p in this unstable arc. 
 Since ^s(p) cuts the unstable arc in a countable set, 
 there is a∈ C that is not in ^s(p). 
 This proves that f is not separating mod ^s.



Does the example in <ref> satisfy W^s=^s? 
The solution could be simple but we were not able to solve it.




 We say that f is expansive mod ^s if for all ϵ>0 there is δ>0 such that 
 if (f^n(x),f^n(y))<δ for all n≥ 0 then x and y are in a common ϵ-stable continuum. 


 

 Note that every homeomorphism expansive mod ^s is separating mod ^s.



 If f X→ X is a cwN-expansive homeomorphism of a Peano continuum 
 and f is expansive mod ^s and f^-1 is expansive mod ^u then f is N-expansive.



Let α>0 be a cwN-expansivity constant for f and take ϵ=α/2. 
Consider δ from Definition <ref> and 
suppose that (f^n(C))<δ for all n∈. 
For x,y∈ C we have that (f^n(x),f^n(y))<δ for all 
n≥ 0. 
Then, there is an ϵ-stable continuum containing x,y. 
We have that there is a 2ϵ-stable continuum containing C. 
Similarly, there is a 2ϵ-unstable continuum containing C. 
Since 2ϵ=α is a cwN-expansivity constant for f we have that 
(C)≤ N. This proves that δ is an N-expansivity constant for f.




 It seems that pseudo-Anosov diffeomorphisms of surfaces without 1-prongs are 
 expansive mod ^s. 
 We know that the pseudo-Anosov with 1-prongs of <ref> is not expansive mod ^s.
 Are the examples in <ref> and <ref> expansive mod ^s? 
 It would be interesting to understand which cw-expansive surface homeomorphisms are
 expansive mod ^s.











































































































§ CONTINUUM THEORY AND DECOMPOSITIONS




In this section we review some results from continuum theory that will be used throughout the article. 
Also, we study decompositions, that will play the role of foliated charts in the next section.



 §.§ Background on Continuum Theory


We recall that a continuum is a compact connected metric space. 
General references for continuum theory are Nadler2,Kur1,Kur.



  §.§.§ Partitions and monotone restrictions


Let (X,) be a compact metric space and denote by (X) the set of subsets of X.
A partition of X is a function 
Q X→(X) such that:

  
  * x∈ Q(x) for all x∈ X,
  
  * y∈ Q(x) if and only if x∈ Q(y) and 
  
  * x∈ Q(y) and y∈ Q(z) implies x∈ Q(z).

A partition Q is monotone if each Q(x) is connected. 
Given Y⊂ X and x∈ Y denote by _x(Y) the component of Y containing x.
For a partition Q X→(X) 
and Y⊂ X define the monotone restriction Q|^m_Y Y→(Y) as 

    Q|^m_Y(x)=_x(Q(x)∩ Y)

for all x∈ Y.


 If Q is a partition of X and Z⊂ Y⊂ X then 
    (Q|^m_Y)|^m_Z=Q|^m_Z.
 



From the definition (<ref>) we see that we have to show that 

    _x(_x(Q(x)∩ Y)∩ Z)=_x(Q(x)∩ Z)

for all x∈ Z. 
To prove the inclusion ⊃ consider a connected set C⊂ Q(x)∩ Z such that x∈ C. 
Since Z⊂ Y we have that C⊂ Q(x)∩ Y. 
Then, C⊂_x(Q(x)∩ Y) because x∈ C.
Given that C⊂ Z we conclude that C⊂_x(_x(Q(x)∩ Y)∩ Z). 
Then _x(Q(x)∩ Z)⊂_x(_x(Q(x)∩ Y)∩ Z).
The converse inclusion is easier to prove. It follows from the fact that _x(Q(x)∩ Y)⊂ Q(x).


We say that a partition Q is upper semicontinuous if for all x∈ X and every open set U containing Q(x) 
there is a neighborhood V of x such that if y∈ V then 
Q(y)⊂ U. 


 
 Q is upper semicontinuous if and only if given x_n→ x such that Q(x_n)→ C in the Hausdorff metric 
then C⊂ Q(x). 
The upper semicontinuity of Q implies that each Q(x) is a closed subset of X.



We say that a partition Q is continuous at x∈ X 
if for every x_n→ x we have that Q(x_n)→ Q(x) in the Hausdorff metric. 
We say that Q is continuous if it is continuous at every 
x∈ X.
A set G⊂ X is residual if it is a 
countable intersection of open and dense subsets of X.



If Q is an upper semicontinuous partition of X then: 

 
  * <cit.>*p. 70-71 there is a residual subset G⊂ X such that Q is continuous at every x∈ G, 
 
  * <cit.>*Theorem 3.9 if in addition X is a continuum then X/Q with its quotient topology is a continuum.[It is clear that X/Q is compact and connected since it is the quotient of the continuum X. 
 In <cit.>*Theorem 3.9 it is shown that X/Q is metrizable.]
 




  §.§.§ Local connection


A continuum is hereditarily locally connected if every subcontinuum is locally connected. 
A convergence continuum A of a compact metric space X is a 
non-trivial subcontinuum of X for which
there is a sequence of continua A_i⊂ X such that A_i→ A in the Hausdorff metric, 
A_i∩ A=∅ and A_i∩ A_j=∅ for all i≠ j. 

[<cit.>*Theorem 10.4]

A continuum X is hereditarily locally connected if and only if X contains no 
convergence continuum. 



[<cit.>*Theorem 3-17]

Every connected, locally connected, complete metric space is arc-connected. 


[Sierpiński's Theorem <cit.>*p. 218] 
 A continuum C is Peano
if and only if for all ϵ>0 there is a finite cover C_1,…,C_n 
of C by connected sets of diameter less than ϵ. 




  §.§.§ Unicoherence


A continuum C is unicoherent 
if given subcontinua A,B⊂ C such that A∪ B=C then A∩ B 
is connected. 
A continuum is hereditarily unicoherent if every subcontinuum is 
unicoherent. 
To show the difference between these concepts and for future reference let us recall the following result.

 

 The union of two subcontinua A,B of the 2-sphere disconnects the sphere if and only if A∩ B is disconnected.


By Janisewski's Theorem we see that the 2-sphere is a unicoherent continuum. 
It is not hereditarily unicoherent because it contains a circle, which is not unicoherent.






  §.§.§ Dendrites


A dendrite is a Peano continuum X containing no simple closed curve.
The points of a dendrite are classified as: 
end point if its complement is connected,
ramification or branch point if its complement has at least three components,
regular point if its complement has two components.
The following results from <cit.>*10 summarizes several properties of dendrites.




Every subcontinuum of a dendrite is a dendrite.





The set of all the ramification points of a dendrite is countable.






If X is a compact metric space then the following statements are equivalent: 

 
  * X is a dendrite,
 
  * X is a hereditarily unicoherent Peano continuum,
 
  * X is connected and any two points of the continuum are 
separated by a third point.







Every dendrite can be embedded in the plane. 
Moreover, the Wazewski's universal dendrite is a dendrite in ^2 which contains a 
topological copy of any dendrite.




















 §.§ Decompositions


The standard theory of foliations is based on a special kind of local partitions in plaques, i.e., foliated charts. 
On a surface S, a C^0 foliated chart is a homeomorphism φ U⊂ S→ (0,1)× (0,1) where U⊂ S 
is an open subset.
If φ=(φ_1,φ_2) then the plaques of U are φ_2^-1(y) for all y∈ (0,1). 
We can define a map Q U→(S) as Q(p)=φ_2^-1(φ(p)).
This map Q is a continuous monotone partition of U.

For a pseudo-Anosov diffeomorphism, the stable and unstable sets form singular foliations. 
At a singular point, an n-prong with n>2, it is not possible to define a local chart as above where the plaques are the local stable sets. 
The partition in local stable sets in a neighborhood of a singularity is monotone and upper semicontinuous. 
The (full) continuity is lost at the singularity. This is the idea we have in mind for the next definition of decomposition, which is a standard concept in continuum theory.

Given a compact metric space X, we will consider a compact subset Y⊂ X. 
We can think that Y is the closure of the open set U in the surface considered above. 
However, the theory is developed in such a way that Y may not be connected and may have empty interior.



A decomposition of Y is a monotone upper semicontinuous partition Q Y→(Y). 
The sets Q(x) are the plaques of the decomposition.


[Extremal examples]

For a compact metric space Y define Q_min(x)={x} and Q_max(x)=_x(Y). 
It holds that Q_min and Q_max are decompositions of Y, a proof can be found in <cit.>*Theorem 24.
For every decomposition Q of Y we have that Q_min(x)⊂ Q(x)⊂ Q_max(x) for all x∈ Y.



For standard foliations, the restriction of a foliated chart to an arbitrary subset may not be a foliated chart. 
This is because the product structure may be lost. 
Next we show that the restriction of a decomposition is a decomposition.



 If Q is a decomposition of the compact set Y
 and Z⊂ Y is compact then Q|^m_Z is a decomposition.



By definition (<ref>) Q|^m_Z is monotone. 
Suppose that x_n→ x and Q|^m_Z(x_n)→ C. 
Taking a subsequence we can also assume that Q(x_n)→ D for some subcontinuum D⊂ X.
Since Q|^m_Z(x_n)⊂ Q(x_n) and Q is upper semicontinuous 
we have that C⊂ D⊂ Q(x) 
and, as C is a continuum contained in Z∩ Q(x), 
we conclude that C⊂ Q|^m_Z(x).
This proves that Q|^m_Z is upper semicontinuous.




 Let P be a continuum and consider a compact metric space Z. 
 Define Y=P× Z and Q Y→(Y) as 
 Q(p,z)=P×{z}. 
If we consider the product topology on Y then
Q is a continuous decomposition of Y.




Given two decompositions Q_i Y_i→(Y_i), i=1,2, 
we say that Q_1 and Q_2 are equivalent if there is a 
homeomorphism h Y_1→ Y_2 such that 
h(Q_1(p))=Q_2(h(p)) for all p∈ Y_1. 
We say that a decomposition is a product structure 
if it is equivalent to a decomposition as in Example <ref>.



 A decomposition Q_1 of a continuum Y is a product structure if and only if 
 there is a decomposition Q_2 of Y 
 such that 
 
    (Q_1(x)∩ Q_2(y))=1

 for all x,y∈ Y.



If Q_1 is a product structure, 
we can assume that Y=P× Z. 
Since Y is a continuum we have that P and Z are continua. 
If Q_1(p,z)=P×{z} we can define Q_2(p,z)={p}× Z. 
Then 

    Q_1(p_1,z_1)∩ Q_2(p_2,z_2)={(p_2,z_1)}

for all (p_1,z_1),(p_2,z_2)∈ P× Z.

To show the converse note that by Proposition <ref>
we know that Y/Q_1 and Y/Q_2 are continua. 
Consider h Y→ Y/Q_1× Y/Q_2 
defined by 
    h(x)=(Q_1(x),Q_2(x)).
 
The condition (<ref>) implies that h is bijective.
Since we consider the quotient topology, 
the projections onto Y/Q_1 and Y/Q_2 are continuous
and h is continuous.
Since Y is compact we have that h^-1 is continuous.
Given that h(Q_1(x))={Q_1(x)}× Y/Q_2 the proof ends.



We say that a compact subset K⊂ Y is a representative set of the decomposition Q
if Q(x)∩ K≠∅ for all x∈ Y.
If K is a representative set we say that Q is a decomposition of (Y,K).


We will consider decompositions of (U,∂ U), with U an open set, 
because for a cw-expansive homeomorphism 
we know that the diameter of the stable and the unstable sets are bounded away from zero. 
Therefore, if U is small, each stable plaque in U meets the boundary of U.




 If Q is a decomposition of (Y,K),
 U⊂ Y is open and U∩ K=∅ then 
 Q|^m_U is a decomposition of (U,∂ U).



From Proposition <ref> we know that Q'=Q|^m_U is a decomposition.
Given x∈ U the fact that Q'(x)∩∂ U≠∅ follows by <cit.>*Theorem 2-16 
or <cit.>*6.1.25.


In the following subsections we study some particular types of decompositions. 
Applications will be given in <ref> for the study of decompositions of two-dimensional discs.



  §.§.§ Dendritic decompositions


Let Q Y→(Y) be a decomposition of the continuum Y. 



 We say that Q is dendritic if Q(x) is a dendrite for all x∈ Y. 
 We say that Q is codendritic if Y/Q is a dendrite with the quotient topology.






Dendritic decompositions are a generalization of one-dimensional foliated charts. 
Codendritic decompositions generalizes codimension one foliated charts.
In <ref> we will study codendritic 
and dendritic decompositions of a two-dimensional disc. Some general properties can be derived in the generality of a continuum Y.



 If Q Y→(Y) is a codendritic decomposition 
 whose plaques have empty interior 
 then there is a residual subset G⊂ Y such that 
 Y∖ Q(x) has 1 or 2 components for all x∈ G.



From Theorem <ref>
we have that the set of ramification points 
of the quotient dendrite Y/Q is at most countable.
Let Q_1,Q_2,… be the ramification points of Y/Q. 
We have that Y∖ Q_i is open in Y. 
It is dense in Y because no Q_i has interior points.

Define G=Y∖∪ Q_i. 
Since for every point x of the residual set G 
we know that Q(x) is not a ramification point, 
we conclude that Y∖ Q(x) has 1 or 2 components.


In the hypothesis of Proposition <ref> we cannot conclude that Y∖ Q(x) has 2 components for all x in a residual subset of Y 
(as could seem natural). See Example <ref>.

With respect to the next proposition, if we think that Q^1 represents stable continua and Q^2 unstable continua, then the condition 
Q^1(z)∩ Q^2(z)={z} is related with cw1-expansivity (recall that 
for cw1-expansivity we require that local stable and unstable continua intersects 
in at most one point).



 Let Q^1, Q^2 be decompositions of the continuum Y such that 
 Q^1(z)∩ Q^2(z)={z} for all z∈ Y. 
 If Q^1 is codendritic then Q^2 is dendritic.



Given x∈ Y consider f Q^2(x)→ Y/Q^1 defined as f(y)=Q^1(y). 
With the quotient topology on Y/Q^1 we have that f is continuous because it is the restriction of the decomposition map 
Q^1. The condition Q^1(z)∩ Q^2(z)={z} for all z∈ Y, gives us that f is injective. 
Since Q^2(x) is compact, f is a homeomorphism onto its image. 
The image of f is a subcontinuum of the quotient dendrite Y/Q^1. 
Then, the result follows by Theorem <ref> (every subcontinuum of a dendrite is a dendrite).














 The converse of Proposition <ref> is not true. Take two one-dimensional foliations on a three-dimensional ball. 
 Both foliations are dendritic but no one is codendritic.










  §.§.§ Smooth and unicoherent decompositions



Our motivation for including the following kind of decompositions is to include 
the example <ref> (the anomalous saddle) in the theory. 
The example presents local stable sets that are not locally connected but hereditarily unicoherent.


A decomposition is 
 hereditarily unicoherent 
 if each plaque is hereditarily unicoherent. 


 If P is an hereditarily unicoherent continuum then
 given x,y∈ P there is a unique minimal continuum containing x and y. 
 If Q is a hereditarily unicoherent decomposition and y∈ Q(x) this minimal continuum will be denoted by Q(x,y). 

 We remark that in continuum theory the term smooth has a particular meaning that is not related with any class of differentiability 
 (or at least the author cannot see any connection). To avoid confusions we will call it -smooth.
 


A decomposition Q of a continuum Y is -smooth
if it is hereditarily unicoherent 
and if x_n→ x, y_n∈ Q(x_n) and y_n→ y then 
Q(x_n,y_n)→ Q(x,y) in the Hausdorff metric.  




 Every -smooth decomposition is dendritic. 



By Theorem <ref>
it is sufficient to prove that 
each plaque is locally connected.
If P is a non-locally connected plaque then there is 
x∈ P without arbitrarily small and connected neighborhoods. 
Let ϵ>0 be such that _x (B_ϵ(x)∩ P) is not 
a neighborhood of x in P. 
Then we can take x_n∈ P such that 
x_n→ x and _x_n(B_ϵ(x)∩ P) is disjoint from 
_x(B_ϵ(x)∩ P). 
Then, (Q(x,x_n)) is bounded away from zero. 
But x_n→ x and Q(x,x)=x. 
This contradicts that Q is -smooth and finishes the proof.




 If Q is a -smooth decomposition of a continuum Y and Z⊂ Y is compact then 
 Q|^m_Z is -smooth.



Since each plaque of Q is hereditarily unicoherent we have that each plaque 
of Q|^m_Z is hereditarily unicoherent. 
Note that if Q(x,y)⊂ Z then Q(x,y)=Q|^m_Z(x,y).
Suppose that x_n→ x, y_n∈ Q|^m_Z(x_n) and y_n→ y. 
We know that Q(x_n,y_n)→ Q(x,y). 
Since Q(x_n,y_n)=Q|^m_Z(x_n,y_n) and Q(x,y)=Q|^m_Z(x,y) we have that Q|^m_Z is -smooth.




§ FOLIATIONS FROM CONTINUUM THEORY


In this section we present a detailed exposition of
our foliations from the viewpoint of continuum theory. 
Since we were not able to find this concept in the literature 
we develop the theory from the most basic concepts. 

In the standard theory of foliations[For precision, fix the meaning of this expression to: the theory developed in <cit.>.]
there are at least two main viewpoints for the definition of foliation:


  * A partition of the manifold into leaves (satisfying certain conditions). 
This approach seems to be preferred in dynamical systems as it appears in <cit.>*5.13 and <cit.>*6.7.

  * A maximal atlas of foliated charts. 
This definition is usual in 
texts on foliations as <cit.>.


Leaves-atlas equivalence. (<cit.>*Theorem 1.2.18)
Both definitions of foliation coincide.

Roughly speaking, the proof is as follows.
Given an atlas, we construct chains of plaques and then, the leaves. 
For the converse implication, 
one needs to recover the plaques from the leaves. 
A plaque is defined as the component of the intersection of a leaf with a local chart. 

In this section we introduce some levels of generalizations of foliations. 
The following table summarizes the definitions and their implications.


As we will see in <ref>, the -foliations do not satisfy the 
leaves-atlas equivalence. 
This depends on the local connection of plaques. 
Then, we introduce the Peano foliations as -foliations with locally connected plaques. 
In Theorem <ref>
we show that Peano foliations satisfy the leaves-atlas equivalence, see Equation (<ref>).
For the study of cw-expansive homeomorphisms we introduce cw-foliations. 



 §.§ -foliations


Let (X,) be a compact metric space and denote by τ the topology of X.
Given two closed sets Y_1,Y_2⊂ X and we say that two decompositions 
Q_i of Y_i, i=1,2, are compatible if 
    Q_1|^m_Y_1∩ Y_2=Q_2|^m_Y_1∩ Y_2.

Recall that U denotes the closure of U.

 An atlas is a collection of compatible decompositions 
 
    ={Q_UU→ (U)}_U∈

 where  is an open cover of X. 
 In this case we say that  is an atlas over .


The atlases can be ordered by inclusion 
and by Zorn's Lemma we have that every atlas is contained in a maximal atlas.


 A -foliation is a maximal atlas. 


A basis  of the topology of X is complete if V∈ and U⊂ V is open then U∈.



Every -foliation is defined over a complete basis.




Let  be a -foliation over .
Let U be an open set contained in V∈. 
Define Q_U=Q_V|^m_U.
To show that U∈ we will show that Q_U is compatible with the decompositions of 
. If W∈ then 

    Q_U|^m_U∩W=
     (Q_V|^m_U)|^m_U∩W=
     Q_V|^m_U∩W=
     Q_W|^m_U∩W

where the second equality follows 
by Proposition <ref>
and the last one is the compatibility of the decompositions of V and W.


Notice that the decompositions Q_U are defined on the closures of the open sets of . 
We will consider Q_U=Q_U|^m_U for U∈.


An open plaque 

is a set of the form
Q_U(x)

for x∈ U∈.




If  is an atlas over the complete basis 
 then the set of open plaques 
{Q_U(x):x∈ U∈}
is a basis of a topology τ_ of X. 
If _1⊂_2 are two of such atlases then τ__1=τ__2⊂τ.



Since  is a cover of X, the open plaques cover X. 
Given x∈ Q_U(y)∩ Q_V(z), U,V∈, 
notice that Q_U(y)∩ Q_V(z)=Q_U(x)∩ Q_V(x) and 
Q_U∩ V(x)⊂ Q_U(x)∩ Q_V(x). 
Since  is complete U∩ V∈ and Q_U∩ V(x) is an open plaque.
This proves that the open plaques form a basis of a topology.
Since U=∪_x∈ U Q_U(x) for all U∈, it holds that τ_⊂τ.

Suppose that _1⊂_2 are atlases over _1⊂_2, respectively.
It is clear that τ__1⊂τ__2.
Given x∈ U∈_2 take V∈_1 such that x∈ V⊂ U. 
Then, Q_V(x)⊂ Q_U(x) and τ__2⊂τ__1.



The topology τ_ will be called the plaque topology. 
Since it does not depend on the atlas  (over a complete basis) representing the -foliation  it will also 
be denoted as τ_.










  §.§.§ Chains of plaques


Let  be an atlas over a complete basis  of the compact metric space (X,). 
A chain of -plaques is a sequence P_1,…,P_n 
of open plaques of  such that P_i∩ P_i+1≠∅ for all i=1,… n-1.
Given x∈ P_1 and y∈ P_n we say that the chain goes from x to y.

For a point x∈ X the -leaf of 
x is the set (x) defined by: y∈(x) if there is a chain of open plaques 
from x to y.






Every component of (X,τ_) is contained in a -leaf.



It follows because the -leaves form a partition of X in τ_-open sets.


We will define a metric on X defining the topology τ_. 
Define _ X× X→ as 

    _(x,y)=inf∑_i=1^n(P_i),

where the infimum is taken over all the chains of open plaques P_1,…,P_n, n≥ 1, with x∈ P_1 and y∈ P_n. 
If x and y are in different leaves we set
_(x,y)=+∞.
It is standard to prove that _ is a metric on X. 


 
If U∈, P_1,…,P_n is a chain of open plaques contained in U and 
 x∈ P_1 then each P_i⊂ Q_U(x).



 Suppose that P_i= Q_U_i(x_i) with U_i∈. 
 Define V_i=U∩ U_i.
 Since P_i⊂ U, by Proposition <ref> we have that
 P_i=Q_V_i(x_i) and  
 P_1⊂ Q_U(x). 
 As P_1∩ P_2≠∅ we conclude that P_2⊂ Q_U(x). 
 By induction, each P_i⊂ Q_U(x).





 The topology defined by _ is τ_.



Given x∈ U∈, consider δ>0 such that 
B_δ(x)⊂ U.
Take y∈ X such that _(x,y)<δ. 
We will show that y∈ Q_U(x). 
Since _(x,y)<δ there is a chain of plaques P_i, i=1,…,n,
such that x∈ P_1, y∈ P_n and ∑_i=1^n(P_i)<δ.
Then (∪_i=1^nP_i)<δ and ∪_i=1^nP_i⊂ U. 
By Lemma <ref> we conclude that 
∪_i=1^nP_i⊂ Q_U(x). 
Consequently y∈ Q_U(x).
To prove the converse consider a ball B={y∈ X:_(x,y)<r}, r>0 and x∈ X. 
Taking U∈ with x∈ U and (U)<r 
we see that Q_U(x)⊂ B and B∈τ_.




Every leaf and X itself are complete (as metric spaces) with respect to _. 



 Let x_n be a Cauchy sequence with respect to _A. 
 The definition of the metric allows us to assume that the sequence is contained in a leaf. 
 Also, its limit (as we will prove that it exists) must be in this leaf. 
 Thus, the proof is reduced to show that X is complete. 
 
 Since ≤_, 
 x_n is a Cauchy sequence with respect to . 
 Then, there is x∈ X such that (x_n,x)→ 0. 
 Take ϵ>0 such that U=B_ϵ(x)∈.
 Take l≥ 1 such that _(x_n,x_l)<ϵ/2 and (x_n,x)<ϵ/2 for all n≥ l. 
 Consider a sequence of open plaques P_n such that x_l,x_n∈ P_n and (P_n)<ϵ/2. 
 Take U_n∈ such that U_n⊂ U and Q_U_n(x_l)=P_n. 
 We know that Q_U_n(x_l)⊂ Q_U(x_l). 
 Since x_n∈ Q_U_n(x_l) and x_n→ x (in ) we conclude that x∈ Q_U(x_l). 
 Therefore, x_n∈ Q_U(x) for all n≥ l. 
 This proves that _(x_n,x)→ 0.





 §.§ Peano foliations





Let X be a compact metric space with topology τ.


An atlas  is a Peano atlas if its open plaques are locally connected in τ. 
A Peano foliation is a maximal Peano atlas.




By maximal Peano atlas we mean maximal among all Peano atlases, see Example <ref> 
for a maximal Peano atlas that is not maximal among all the atlases. 
Every Peano atlas is contained in a Peano foliation and 
every Peano foliations is defined over a complete basis.
Given a topology τ_* on X and Y⊂ X denote by 
τ_*|_Y the relative topology on Y induced by τ_*.



If  is a Peano foliation over  then: 



  * for every open plaque P it holds that τ|_P=τ_|_P,

  * every open plaque is connected in τ_,
  
  * (X,τ_) is locally connected, 
  
  * every leaf is arc-connected in both topologies τ_ and τ,
  
  * every leaf is a component of (X,τ_),
  
  * for every leaf L, U∈ and x∈ L∩ U it holds that 
  
    _x(L∩ U)=Q_U(x).





Let P be an open plaque. By Proposition <ref> we know that τ⊂τ_ 
and then τ|_P⊂τ_|_P.
To show that τ_|_P⊂τ|_P it is sufficient to prove that if P' is another open plaque then 
P∩ P'∈τ|_P.
Take x∈ P∩ P' and suppose that P=Q_U(x) and P'=Q_V(x) with U,V∈. 
Since  is a complete basis of τ there is W∈ such that 
x∈ W⊂ U∩ V and P∩ W is connected (P is locally connected at x). 
Then P∩ W=_x(P∩ W)=_x(Q_U(x)∩ W)=Q_W(x).
Since W⊂ U∩ V we can apply Proposition <ref> to conclude that Q_W(x)⊂ P∩ P'. 
Given that Q_W(x)=P∩ W∈τ|_P we conclude that τ_|_P⊂τ|_P. Then τ|_P=τ_|_P. 





By definition, every open plaque is connected in τ and then 
we have that P is connected in τ_. 
Given that open plaques form a basis for τ_, we conclude that (X,τ_) is locally connected.

To prove that the leaves are arc-connected we first show that 
they are τ_-connected.
Take x,y in a leaf L. 
We know that there is a chain of plaques P_1,…,P_n from x to y. 
Since each plaque is 

τ_-connected, 

∪_i=1,…,nP_i is τ_-connected. 
Fixing x and varying y∈ L we conclude that L is τ_-connected.
Then, we know that each leaf L is connected and locally connected in τ_. 
In addition, by Proposition <ref>, we know that (L,_) is a complete metric space. 
Then, we can apply 
Theorem <ref>

to conclude the arc-connection of L.
This implies that every leaf is contained in a component of (X,τ_). 
The converse inclusion follows by Proposition <ref>.

To prove (<ref>) take a leaf L, U∈, x∈ L∩ U
and define C=_x(L∩ U).
By the definition of leaf, it is clear that Q_U(x)⊂ C.
We will show the converse inclusion. 
For this purpose, since C is a union of open plaques, we can write 

    C=∪_j∈ JP_j
 
with P_j=Q_U_j(x_j) and some index set J.
By Proposition <ref> we can suppose that U_j⊂ U for all j∈ J.
Define 
    J_1={j∈ J: P_j∩ Q_U(x)≠∅}
 and J_2=J∖ J_1.
Since Q_U_j=Q_U|^m_U_j, if j∈ J_1 then P_j⊂ Q_U(x)⊂ C. 

Define V_1=∪_j∈ J_1 P_j and V_2=∪_j∈ J_2 P_j. 
Define V'_1=V_1∩ Q_U(x) and V'_2=V_2∩ Q_U(x).
Then Q_U(x)⊂ V'_1∪ V'_2. 
Since Q_U(x) is locally connected with respect to τ, 
we conclude that 
V'_1,V'_2∈τ|_Q_U(x). 
Since V'_1∩ V'_2=∅, V'_1≠∅ and Q_U(x) is connected in τ we conclude 
that V'_2=∅.
Then Q_U(x)⊂∪_j∈ J_1Q_U_j(x_j). 
Since Q_U_j(x_j)⊂ Q_U(x) for all j∈ J_1
we conclude that Q_U(x)= ∪_j∈ J_1Q_U_j(x_j). That is, Q_U(x)=C=_x(L∩ U).


We will give two examples of -foliations with particular properties 
to show the necessity of the hypothesis of Theorem <ref>.  
First we show that a leaf may not be connected in the plaque topology (for a non-Peano foliation).



Consider the plane continuum 
X=X_1∪ X_2 where X_1={0}×[-1,1] and X_2={(x,sin(1/x)):x∈ (0,1]}. 
On X consider the complete basis =τ (the relative open subsets of the plane with its usual topology).
Let  be the -foliation defined as Q_U(x)=_x(U) for every U∈τ.
We have only one leaf L=X. 
We see that this leaf is not connected in τ_ since it has two components X_1 and X_2.
Note that X itself is an open plaque that is not locally connected. Then  is not a Peano foliation. 
It is an exercise for the interested reader to check which conclusions of Theorem <ref> are satisfied in this example.
In Example <ref> we give a decomposition of a square with a leaf like the set X.


In the following variation of the previous example we show that (X,τ_A) may not be locally connected.



For n a non-negative integer define

    X_n={(x,(3+sin(1/x))/4^n)∈^2: x∈ (0,1]}

and X=({0}× [0,1])∪ ([0,1]×{0})∪⋃_n≥ 0 X_n. 
See Figure <ref>. 
As in the previous example consider the -foliation 
defined by Q_U(x)=_x(U) for every U∈τ.
We have that (0,0) has not connected neighborhoods in the plaque topology.
 


We think that the following result is interesting because it allows us to characterize Peano foliations in terms 
of the plaque topology.


For an atlas  over a complete basis

the following statements are equivalent:

 
  *  is a Peano atlas,
 
  * τ|_P=τ_|_P for every open plaque P.




The direct part follows by Theorem <ref>.
To prove the converse, let P be an open plaque. 
We have to prove that P is locally connected in τ. 
Given x∈ P we can take a small neighborhood U∈ of x. 
By definition, the plaque Q_U(x) is connected in τ. 
Also, Q_U(x) is a neighborhood of x in the plaque topology. 
Since, by hypothesis, τ|_P=τ_|_P, 
we conclude that Q_U(x) is a (connected) neighborhood of x in τ|P. 
Then, P is locally connected with respect to τ.


We say that a leaf is plaque-compact if it is compact in the plaque topology.



Let  be a Peano foliation.
An -leaf is plaque-compact if and only if it is a Peano continuum in the relative topology of τ.




Assume that L is a plaque-compact leaf. 
Since the plaque topology is finer than the relative topology, Proposition <ref>, we have that 
L is compact in the relative topology.

Given ϵ>0 we define the complete basis 

    _ϵ={U∈τ:(U)<ϵ}.

We assume that ϵ is so small that _ϵ⊂, where  is the complete basis of .
Consider the following cover of L 
    _L={Q_U(x):x∈ L∩ U, U∈_ϵ}.


Since L is plaque-compact, there is a finite subcover. 
The plaques of this subcover have diameter smaller than ϵ. 
Applying Sierpiński's Theorem <ref> 
we conclude the local connection of L in the relative topology.

Converse. 
Assume that the leaf L is a Peano continuum in τ.
Take a cover _L={P_i}_i∈ I of L by open plaques. 
Given x∈ L take U_x∈ such that x∈ P_i=Q_U_x(x). 
Since L is locally connected and  is a complete basis 
there is V_x∈ such that x∈ V_x⊂ U_x
and V_x∩ L is connected. 
Then, by Theorem <ref>,
Q_V_x(x)=V_x∩ L.
Since {V_x∩ L:x∈ L} is a cover of L by relative open sets of L and 
L is compact we can take x_1,…,x_n such that V_x_1,…,V_x_n cover L. 
Since V_x⊂ U_x, we have that 
{Q_U_x_1(x_1),…,Q_U_x_n(x_n)} is a finite subcover of _L. 
This proves that L is compact in the plaque topology.



In Example <ref> we see that a compact leaf (in the relative topology) 
may not be plaque-compact (it fails to be locally connected). 



A locally connected leaf may not be compact. 
Take a plane flow (with X a compact invariant annulus) with limit cycles. 
An orbit converging to a cycle is locally connected but it is not compact. 


Note that the examples given above (Examples <ref> and <ref>) are not Peano foliations but they satisfy (<ref>) in Theorem <ref>. 
The next subsection is devoted to describe an example of a -foliation not satisfying (<ref>).



  §.§.§ Devil's backgammon


In Theorem <ref> we proved that for Peano foliations, if 
L is a leaf then for every U∈ and x∈ L∩ U it holds that 
_x(L∩ U) is the plaque of x in U. 
That is, knowing the leaves and the basis  we can recover the plaques. 
This is the leaves-atlas equivalence mentioned in the beginning of this section. 
The next example shows that a maximal atlas (with non-locally connected plaques) can contain strictly more information than 
the leaves and the basis, i.e., does not satisfy (<ref>).

We start constructing a continuum X⊂ [0,1]× [0,1]. 
Let K⊂ [0,1] be the ternary Cantor set. 
A point x∈ K can be expressed as x=∑_i2/3^n_i for some increasing sequence n_i of integers. 
Consider the function g K→[0,1] 
such that if x=∑_i2/3^n_i∈ K then g(x)=∑_i1/2^n_i.
For each y∈[0,1] there are 1 or 2 preimages by g. 
For x∈ K let I_x be the closed segment line from (x,0) to (g(x),1). 
Define 

    X=([0,1]×{0})∪⋃_x∈ K I_x.

We have that X is an arc-connected continuum. Note that it is not locally connected.
It can also be obtained by deleting open triangles from the square, as shown in Figure <ref>.
 

 Consider the open sets 
 
    [ U={(x,y)∈ X:y<2/3},; V={(x,y)∈ X:y>1/3}. ]
 
 Define Q_U(x,y)=U for all (x,y)∈ U. 
 Given p∈ V suppose that p∈ I_a. 
 Define 
 
    J_p={[                  I_a  if  g^-1(g(a))={a},;             I_a∪ I_b if  g^-1(g(a))={a,b} ].

and Q_V(p)=_p (J_p∩ V). 
The decompositions Q_U,Q_V define a -foliation .
It is clear that it is not a Peano foliation since it has non-locally connected open plaques.
Note that there is only one leaf, namely L=X. 

In this example we see that
for p∈ V we have that _p(L∩ V)=L∩ V and it is not Q_V(p).
As we said, this means that the plaques cannot be recovered by the leaves and the basis.



















 §.§ Cw-foliations


It must be recalled that we have in mind the study of the distribution of stable and unstable 
continua of cw-expansive homeomorphisms on surfaces and Peano continua. 
This guides us in selecting the properties that we require in the following definition. 

Recall that a subset of X is meagre if it can be expressed as 
the union of countably many nowhere dense subsets of X. 
A set is nowhere dense if its closure has empty interior.

















An atlas  is a cw-atlas if its -leaves:

 
  * are a countable union of open plaques,
 
  * are meagre sets and 
 
  * have diameter bounded away from zero.

A cw-foliation 
is a -foliation with a cw-atlas.


More observations about this definition must be given.


Note that requiring that a leaf is a countable union of plaques does not imply that the leaf has a countable basis. 
The plaques in this countable union may not be small.



A foliation of a compact manifold M in the standard sense 
is a cw-foliation if and only if the dimension of the leaves is positive and 
less than (M). On a surface it means one-dimensional.



In the standard theory of foliations the leaves are immersed submanifolds 
and have a countable basis for its leaf topology. Also, the intersection of a leaf 
with a local chart is a countable number of plaques. 
For Peano cw-foliations these properties can be recovered.
A Peano cw-foliation is a Peano foliation with a cw-atlas.



If  is a Peano cw-foliation


then: 

 
  * τ_|_L has a countable basis for every leaf L,
 
  * for every leaf L and every U∈τ the set L∩ U has a countable number 
of components with respect to both topologies τ and τ_.




Since X is a compact metric space it has a countable basis. 
By Theorem <ref> we know that τ|_P=τ_|_P, for every open plaque P.
We conclude that each plaque has a countable basis.
Consequently, as every leaf is a countable union of plaques, we have that each leaf has a countable basis. 

Consider a leaf L and U∈τ.
We know that L∈τ_. 
From Proposition <ref> we have that U∈τ_. 
Then U∩ L∈τ_.
We know that τ_|_L has a countable basis.
Therefore, τ_|_U∩ L has a countable basis. 
By Theorem <ref>, knowing that U∩ L∈τ_, 
we conclude that U∩ L is locally connected with respect to τ_.
Then[Every locally connected topological space with a countable basis has 
 at most a countable number of components. 
 This can be proved as follows. 
 Since the space is locally connected, its components are open 
 and given that the space has a countable basis each component contains an open set of the countable basis. 
 As the components are disjoint, there is at most a countable number of components.] 
 U∩ L has a countable number of components 
with respect to τ_.
Since τ⊂τ_ the same holds for τ.



For future reference (Proposition <ref>) we give one more result.
Recall from (<ref>) that _δ is the complete basis containing 
all the open sets of diameter smaller than δ.



 If  is a cw-atlas over the complete basis  of X then 
 there is δ>0 such that for all x∈ U∈_δ 
 it holds that: 
 
  
  * Q_U(x)∩∂ U≠∅ and 
  
  * x is not a τ-interior point of Q_U(x).
 



From the definition of cw-atlas we know that 
there is δ>0 such that every leaf has diameter greater than δ. 
Take x∈ U∈_δ. 
Since the leaf of x has diameter greater than δ there is y∉U in the leaf of x. 
Consider a chain of plaques P_1,…,P_n from x to y. 
Let P_m be the first of these plaques that is not contained in U. 
By Lemma <ref> the plaques P_1,…,P_m-1 are contained in Q_U(x).
Take z∈ P_m-1∩ P_m. 
Since P_m is connected we have that _z(P_m∩ U) cuts ∂ U.
Given that _z(P_m∩ U)⊂ Q_U(x) we have that Q_U(x)∩∂ U≠∅.

Since each leaf is a meagre set, no plaque has interior points. 

The next example shows that the converse of Proposition <ref> is not true.



 Let K⊂ [0,1] be a Cantor set. 
 Define X=K×[0,1]×[0,1]∪ [0,1]×[0,1]×{0}. 
 Define the projection π X⊂^3→ given by π(x,y,z)=y and the 
 decomposition  as (x,y,z)=π^-1(y). 
 For δ∈(0,1/2) and the Euclidean metric on X, define Q_U(p)=|^m_U(p). 
 It defines an atlas over _δ. 
 No leaf has interior points and every leaf has diameter greater than δ. 
 But this atlas does not define a cw-foliation because the leaves are not a countable union plaques.





 §.§ Continuous atlas





We say that an atlas  over a complete basis  is continuous if the set 

    {U∈: Q_U is continuous}

is a basis of the topology of X. The continuity of Q_U means the continuity of the 
map Q_UU→(Q_U) with respect to the Hausdorff metric.








The stable foliation of a pseudo-Anosov diffeomorphism 
with 1-prongs (as in <ref>)
defines a continuous atlas. 
In Theorem <ref> we will show that continuous Peano cw-foliations of compact surfaces (i.e., continuous dendritations),
are in fact foliations (in the standard sense) with a finite number of 1-prongs. 


[Hiraide's generalized foliations]
Let (X,) be a Peano continuum. 
A partition  of X is a generalized foliation Hi89, AH if for all 
x∈ X there are non-trivial arc-connected subsets D,K⊂ X 
with D∩ K={x}, a connected open neighborhood N of x in X 
and a homeomorphism ϕ_x D× K→ N

such that: 

 
  * ϕ_x(y,x)=y for all y∈ D, ϕ_x(x,z)=z for all z∈ K,
 
  * for every leaf L there is an at most countable set B⊂ K such that N∩ L=ϕ_x(D× B). 

Generalized foliations are a main concept for the study of expansive homeomorphisms with canonical coordinates.
Notice that the stable foliation of the pseudo-Anosov diffeomorphism of <ref>
is not a generalized foliation.
We can show that every generalized foliation defines a continuous atlas as follows.
Given x∈ X consider a local coordinate ϕ D× K→ N around x. 
Take D' and K' compact neighborhoods of x in D and K respectively.
Let U be an open set containing x such that U= ϕ(D'× K'). 
Since on D× K we consider the product topology, 
we have that |^m_U is continuous. 




§ EXPANSIVITY AND CW-FOLIATIONS



The results that we have obtained will be now applied in the study of the 
stable and unstable cw-foliations determined by a cw-expansive homeomorphism. 
Also, we will give sufficient conditions for cw1-expansivity to imply expansivity.



 §.§ Stable and unstable cw-foliations



Let f X→ X be a homeomorphism of a compact metric space. 
Recall the definitions of stable and unstable continua given in (<ref>).


A decomposition of a closed subset of X is stable if every plaque is stable.


An atlas  is stable if each decomposition is stable. 
A ^s-foliation is a maximal stable atlas.
The unstable versions of these definitions are analogous. 


As before, maximal stable atlas means maximal among all the stable atlases, see Example <ref>.
For a closed subset Y⊂ X define 
Q^s_Y Y→(Y) 
by y∈ Q^s(x) if there is a stable continuum 
C⊂ Y such that x,y∈ C. 
For every homeomorphism f and closed subset Y⊂ X 
we have that Q^s_Y is a monotone partition of Y. 
Similarly we define Q^u_Y considering unstable continua.
Define:

    [ ^s={U∈τ: Q^s_U is a stable decomposition},;                           ^s={Q^s_U:U∈^s}.;                                            ]

Note that ^s and ^s depend on f. 
The sets ^u and ^u are defined similarly considering the inverse of f.

We will show that if 
f is cw-expansive then
^σ is a cover of X 
and that in this case 
^σ is a stable atlas of a cw-foliation.
By definition it is clear that ^σ is a maximal stable (unstable) atlas. 
We define the stable and unstable cw-foliations of f as ^s, ^u respectively.

An atlas  over  is invariant if:

 
  * f^n(U)∈ and 

  * Q_f^n(U)(f^n(x))=f^n(Q_U(x)) for all x∈U

for all n∈ and all U∈. 
The purpose of this section is to prove the following result.



 If f is a cw-expansive homeomorphism of a Peano continuum then 
 ^s,^u are invariant cw-foliations without plaque-compact leaves.


The proof is developed in some lemmas that do not assume that f is cw-expansive.



If Q^σ_Y is a stable (unstable) decomposition, σ=s,u, and Z⊂ Y is closed then 
    Q^σ_Z=Q^σ_Y|^m_Z.










 






Take x∈ Z and y∈ Q^s_Z(x). 
Then there is a stable continuum C such that x,y∈ C⊂ Z. 
Since Z⊂ Y, we have that C⊂ Q^s_Y(x). 
Since x∈ C and C is connected we have that C⊂ Q^s_Y|^m_Z(x). 
Consequently y∈ Q^s_Y|^m_Z(x) and Q^s_Z(x)⊂ Q^s_Y|^m_Z(x).

Let us show the converse inclusion. 
Take x∈ Z and y∈ Q^s_Y|^m_Z(x). 
Recall that Q^s_Y|^m_Z(x)=_x(Q^s_Y(x)∩ Z).
Since Q^s_Y is a stable decomposition we have that 
_x(Q^s_Y(x)∩ Z) is a stable continuum.
Given that _x(Q^s_Y(x)∩ Z)⊂ Z we have that 
_x(Q^s_Y(x)∩ Z)⊂ Q^s_Z(x) and y∈ Q^s_Z(x).
Then Q^s_Y|^m_Z(x)⊂ Q^s_Z(x).




If ^σ covers X, σ=s,u, then: 


 
  * ^σ is a stable atlas over the complete basis ^σ, 
 
  * every stable (unstable) continuum is contained in an open ^σ-plaque,
 
  * Q^σ_X(x) is the ^σ-leaf of x for all x∈ X, 
 
  * every ^σ-leaf is a countable union of open ^σ-plaques.





Assume that ^s covers X.
By Lemma <ref> we know that ^s is a complete basis. 
  Consequently, ^s is a stable atlas.
Let C⊂ X be a stable continuum.
Since ^s covers X
we can take n≥ 0 and U∈ such that f^n(C)⊂ U∈^s. 
Then C⊂ Q^s_f^-n(U)(x) for all x∈ C. 
Since ^s is f-invariant, Q^s_f^-n(U)(x) is a stable open plaque containing C. 
It is clear that the leaf of x is contained in Q^s_X(x). The converse inclusion follows by the previous item.

Let us show that each leaf is a countable union of open plaques. 
Take δ>0 such that every open subset of diameter smaller than δ belongs to ^s. 
Then B_δ/2(x)∈^s for all p∈ X 
and we can consider the open plaque Q^s_B_δ/2(x)(x).
For x∈ X and n≥ 0 define 

    P_n=f^-n(Q^s_B_δ/2(f^n(x))(f^n(x))).

We have that P_n is a stable open plaque. 
We have proved that the leaf of x is Q^s_X(x). 
Then P_n⊂ Q^s_X(x) for all n≥ 0. 
Given y∈ Q^s_X(x) let us show that there is k≥ 0 such that y∈ P_k. 
If y∈ Q^s_X(x) then there is a stable continuum C containing x and y. 
Since C is stable, there is k≥ 0 such that (f^k(C))<δ/2. 
Given that f^k(x)∈ f^k(C) and that f^k(C) is a stable continuum 
we have that f^k(C)⊂ Q^s_B_δ/2(f^k(x))(f^k(x)).
Then y∈ P_k. This proves that Q^s_X(x)=∪_n≥ 0P_n, a countable union of plaques.



 
 If f is a homeomorphism of a compact metric space X and there are ϵ,δ>0 such that 
 ^σ∩_δ⊂^σ_ϵ and 
 ^σ_2ϵ⊂^σ 
 for σ=s,u then ^σ covers X. 



We give the proof for σ=s.
Let Y⊂ X be a compact subset with (Y)≤δ.
We will show that Q^s_Y is a stable decomposition of Y.
Take x∈ Y. 
By definition we have that Q^s_Y(x) is connected for all x∈ Y. 
We have that Q^s_Y(x) is the union of all the stable 
continua C contained in Y and containing x. 
Since ^s∩_δ⊂^s_ϵ and 
(Y)≤δ we have that each C∈^s_ϵ.
This implies that 

    (f^k(Q^s_Y(x))≤2ϵ for all k≥ 0.

Since ^s_2ϵ⊂^s 
we have that Q^s_Y(x) is a stable continuum. 
Take x_n→ x in Y with Q^s_Y(x_n)→ C.
From (<ref>) we know that (f^k(C))≤2ϵ for all k≥ 0. 
Therefore, we conclude that C is stable. 
Then, C⊂ Q^s_Y(x) and the proof ends.













By Proposition <ref> 
we know that f satisfies the hypothesis of Lemma <ref>. 
Then ^σ covers X. 
Applying Lemma <ref>
we conclude that ^σ is a complete basis, that ^σ is an atlas 
and that every ^σ-leaf is a countable union of open ^σ-plaques. 
By Theorem <ref> we know that ^σ_ϵ(x)∖_δ(x)≠∅.
Then, every leaf has diameter greater than δ.
 The cw-expansivity and the condition ^σ_ϵ(x)∖_δ(x)≠∅ 
 imply that each stable plaque has empty interior. 
 Therefore, stable and unstable leaves are meagre sets.
This proves that ^σ are cw-foliations.

Suppose that L is a plaque-compact leaf of ^s. 
Let {P_i}_i∈ I be a cover of L 
by stable plaques. 
Since L is plaque-compact we can assume that I is finite. 
Then (f^n(L))→ 0 as n→+∞. 
This gives us stable leaves of arbitrarily small diameter, a contradiction to Theorem <ref>.



























 The quasi-Anosov diffeomorphism <ref>
 is expansive and Theorem <ref> can be applied. 
 In this example we see that the plaques of ^s and ^u may not have constant dimension. 
 There are stable plaques of dimension 1 and 2.


[One-dimensional expanding attractor]

 Let g T^2→ T^2 be a derived from Anosov diffeomorphism of the two-dimensional torus
 and denote by X⊂ T^2 the non-trivial basic set (an expanding attractor). 
 Let f X→ X be the restriction of g. 
 We have that X is locally a product of a Cantor set and an arc and every leaf is dense in X. 
 The dynamics of f on X expands the length of each arc contained in X 
 and every proper subcontinuum is an unstable arc. 
 That is, given an open set U⊂ X, such that U≠ X, 
 the unstable plaque of x∈ U is, simply, its component in U (an arc). 
 Then, the unstable cw-foliation is a Peano foliation.
 It is remarkable that in this case, 
 the unstable atlas is compatible with the trivial decomposition Q X→(X) given by 
 Q(x)=X for all x∈ X. However, X, as a plaque of this decomposition Q, is neither unstable nor locally connected. 
 Then, a maximal atlas extending a Peano unstable atlas may not be neither unstable nor Peano.




For a cw-expansive homeomorphism of a Peano continuum, is it true that every continuum contained on a stable leaf is stable?  

 

 Can a stable leaf of a cw-expansive homeomorphism of a Peano continuum 
be compact in the relative topology?
By Theorem <ref> we only know that it cannot be 
plaque-compact.






 §.§ Generating pairs


If ^1 and ^2 are complete bases then ^1∩^2 is a complete basis. 
If ^i={Q^i_U:U∈^i} is an atlas over the complete basis ^i, i=1,2, 
then both atlases are defined in a common complete basis, namely =^1∩^2. 

The following concept is the key to prove in <ref>
that cw1-expansivity implies expansivity on a compact surface. 




Given two atlases ^1 and ^2 of the continuum X, we say that 
they generate (the topology of X) if for all B_ϵ(x)∈ there is 
δ>0 such that if y∈ B_δ(x) then 
Q^i_B_ϵ(x)(x)∩ Q^j_B_ϵ(x)(y)≠∅ 
for {i,j}={1,2}.


 
A pair of transverse foliations of a smooth manifold (with the standard meaning of these concepts) 
form a generating pair of cw-foliations. 
Another example is formed by 
the stable and the unstable singular foliations of a pseudo-Anosov map of a compact surface, 
even with singularities and 1-prongs.[We were tempted to say transversal instead of generating pair in Definition <ref>, 
but at 1-prongs the foliations look closer to a tangency than to a transversal point, at least from author's viewpoint.] 




For r>0 and x∈ X define 

    su_r(x)={y∈ X:∃ A^s∈^s_r(y),A^u∈^u_r(x) with  A^u∩ A^s≠∅}




Also define y∈ us_r(x) if x∈ su_r(y).
If f X→ X is cw-expansive then the cw-foliations ^s and ^u generate if and only if
su_ϵ(x)∩ us_ϵ(x) is a neighborhood of x, for all x∈ X and for all ϵ>0.  
The proof follows by the definitions.



Notice that for pseudo-Anosov singular foliations 
the size of the ball covered by su_r(x) is not uniform.
For x close to a singularity, the maximal ball contained in su_r(x) is small. 



[Quadratic tangencies]
 Let ^1 be the foliation of ^2 whose leaves are horizontal lines y=c for c∈. 
 Define ^2 by y=x^2+c for c∈. 
 It is easy to prove that these foliations do not generate at tangency points of the foliations. 
These kind of tangencies are present in the wandering set of the 
 Q^r-Anosov diffeomorphisms of <ref>.



Consider the vertical cw-foliation ^1 of the plane and a cw-foliation ^2 as in Figure <ref>. 
In this case, for a point x as in the figure, it holds that 
^2|^m_B_ϵ(x)(x)∩^1|^m_B_ϵ(x)(y)≠∅ 
for all y close to x. 
But, exchanging the foliations, we can find points y arbitrarily close to x such that
^1|^m_B_ϵ(x)(x)∩^2|^m_B_ϵ(x)(y)=∅.
This kind of non-generating points appear: 1) at the non-wandering set of the examples in <ref>
and 2) at wandering points in the example of <ref>. 



 




  If su_r(x) is a neighborhood of x, for all x∈ X and for all r>0 then X is locally connected. 



  Given x∈ X and a neighborhood U of x take ρ>0 such that B_ρ(x)⊂ U. 
  Take r∈ (0,ρ/2). 
  We know that su_r(x) is a neighborhood of x. 
  For each y∈ su_r(x) there are C^s∈^s_r and C^u∈^u_r 
  such that x∈ C^u, y∈ C^s and C^u∩ C^s≠∅. 
  Then, (C^s∪ C^u)≤ 2r and (x,y)<ρ. 
  Then su_r(x)⊂ B_ρ(x). 
  Since su_r(x) is a union of connected sets C^s∪ C^u, with a common point x, 
  we conclude that su_r(x) is connected. This proves the local connection of X at x. 
  Since x is arbitrary, X is locally connected.






 §.§ Cw1-expansivity versus expansivity


We now investigate the following question. 

Does cw1-expansivity imply the expansivity of a homeomorphism f on a Peano continuum X? 


The answer is affirmative in the examples known by the author and in 
Proposition <ref> we gave some information on this problem.
In Theorem <ref> we will prove it for X a compact surface.
On a Cantor set X the identity is (trivially) cw1-expansive but not expansive.
Even assuming the connection of X, the local connection of X is needed in the question because of the following example. 



Let f_1 T^2→ T^2 be an Anosov diffeomorphism with a fixed point p. 
Denote by X_2 the closure of {1/n:n∈^+} in the usual topology of . 
Define Y=T^2× X_2 and g Y→ Y as g(x,y)=(f_1(x),y). 
Now, collapsing the fixed points {(p,0),(p,1),(p,1/2),…} to a single point we obtain a 
continuum X, see Figure <ref>. 
Also, we have a homeomorphism f X→ X induced by g. 
We have that f is not expansive because the distance (f^n(x,1/k),f^n(x,1/l)) is small for 
all n∈ whenever k and l are sufficiently large.
By construction, f is cw1-expansive. 
Notice that X is a non-locally connected continuum. 




Under certain hypothesis, see Theorem <ref>,
we can prove that cw1-expansivity implies expansivity. 
The following lemma is a part of its proof,
it is separated in order to simplify the notation. 
We assume that f is not expansive because we will argue by contradiction. 



If f is not expansive 

and us_r(x) is a neighborhood of x for all x∈ X and for all r>0,
then for all ϵ>0 there are 
x,y∈ X such that 
x≠ y, (f^n(x),f^n(y))<ϵ 
for all n∈ and x,y∈ A with 
A∈^s_ϵ∪^u_ϵ.



For ϵ>0 given, 
consider δ_1>0 and a finite set Z⊂ X such that 
B_δ_1(z)⊂ us_ϵ/3(z) for each z∈ Z and 
X=∪_z∈ ZB_δ_1(z).
Take δ_2∈(0,ϵ/3) such that if v,w∈ X and (v,w)<δ_2 then v,w∈ B_δ_1(z) 
for some z∈ Z.
Since f is not expansive there are v,w∈ X such that (f^n(v),f^n(w))<δ_2 for all n∈ 
and v≠ w. 
Take z∈ Z such that v,w∈ B_δ_1(z). 
Then v,w∈ su_ϵ/3(z).
By the definitions, we can take 
A_v^s, A_w^s∈^s_ϵ/3 and 
A_v^u,A^u_w∈^u_ϵ/3
such that z∈ A_v^s∩ A_w^s, A_v^s∩ A_v^u≠∅, A_w^s∩ A_w^u≠∅, 
v∈ A_v^u and w∈ A_w^u. 






Take v_1∈ A^s_v∩ A^u_v and w_1∈ A^s_w∩ A^u_w. 
Since A^s_v,A^s_w∈^s_ϵ/3 with a common point, namely z, 
their union is a 2ϵ/3-stable continuum. 
Since v_1,w_1∈ A^s_v∪ A^s_w we have that 

    (f^n(v_1),f^n(w_1))<2ϵ/3 for all n≥ 0.

Also (f^n(v_1),f^n(w_1)) 
is smaller or equal than 
    (f^n(v_1),f^n(v))+(f^n(v),f^n(w))+(f^n(w),f^n(w_1)).

For n≤ 0 we have that 

    [      (f^n(v_1),f^n(v))≤ϵ/3,; (f^n(v),f^n(w))<δ_2<ϵ/3 and;      (f^n(w),f^n(w_1))≤ϵ/3. ]

Therefore (f^n(v_1),f^n(w_1))<ϵ for all n≤ 0. 
Recalling (<ref>) we conclude that 

    (f^n(v_1),f^n(w_1))<ϵ

for all n∈.

Recall that A^s_v∪ A^s_w is a 2ϵ/3-stable continuum containing v_1 and w_1. 
If v_1≠ w_1 then we take x=v_1, y=w_1 and the ϵ-stable continuum A=A^s_v∪ A^s_w.
If v_1=w_1 then we take x=v, y=w and the 2ϵ/3-unstable continuum A=A^u_v∪ A^u_w.


Notice that by Remark <ref> and Proposition <ref> the hypothesis of the next result imply that X is locally connected.



If f X→ X is a cw1-expansive homeomorphism of a continuum X
and ^s,^u is a generating pair 
then f is expansive. 



Take γ>0 such that 

    if A^s∈^s_γ and A^u∈^u_γ then (A^s∩ A^u)≤ 1.

Consider δ∈(0,γ/3) such that

    if (A)≤ 2δ then (f^± 1(A))<γ/3

for every A⊂ X.
Consider ϵ∈ (0,δ) such that 

    if (a,b)<ϵ then there is z∈ X such that a,b∈ su_δ(z)∩ us_δ(z)

for any a,b∈ X.
Assume by contradiction that f is not expansive.
  From Lemma <ref> there are x,y∈ X and 
  A^s_0∈^s_ϵ such that x≠ y, (f^n(x),f^n(y))<ϵ for all n∈
  and x,y∈ A^s_0. 
  If the lemma gives an ϵ-unstable continuum then change f by f^-1 in what follows. 

In this paragraph we will show that there is n≥ 0 such that f^-n(x) and f^-n(y) are not in a common 
2δ-stable continuum. 
Arguing by contradiction, 
assume that for each n≥ 0 there is A^s_n∈^s_2δ containing f^-n(x) and f^-n(y). 
Since f is cw-expansive and δ is small, we have that (f^n(A^s_n))→ 0 as n→∞. 
Since x,y∈ f^n(A^s_n) we conclude that (x,y)=0, contradicting that x≠ y.

Let n_0 be the first positive number satisfying the condition proved in the previous paragraph. 
Then f^-n_0+1(x) and f^-n_0+1(y) are in a common 2δ-stable continuum A^s_n_0-1. 
From (<ref>) we have that (f^-1(A^s_n_0-1))<γ/3. 
Define a=f^-n_0(x), b=f^-n_0(y) and A^s_*=f^-1(A^s_n_0-1). 
We have that A^s_* is a γ/3-stable continuum containing a and b. 
Also, a and b are not in a common 2δ-stable continuum.

Since (a,b)<ϵ  
we can apply condition (<ref>) to conclude that there is z∈ X such that a,b∈ su_δ(z). 
Then, there are 
A^s_a,A^s_b∈^s_δ and 
A^u_a,A^u_b∈^u_δ such that 
z∈ A^u_a∩ A^u_b, a∈ A^s_a, b∈ A^s_b, 
A^s_a∩ A^u_a≠∅ and
A^s_b∩ A^u_b≠∅.
We have that A^s_a∩ A^s_b=∅ because a and b are not in a common 2δ-stable continuum. 
Therefore, we can take p∈ A^s_a∩ A^u_a and
q∈ A^s_b∩ A^u_b such that p≠ q. 
Since δ<γ/3 we have that p and q are in a γ-stable continuum, namely, A^s_a∪ A^s_*∪ A^s_b. 
Also, p and q are in the 2δ-unstable continuum A^u_a∪ A^u_b. 
This contradicts (<ref>) and proves that f is expansive.


 
The quasi-Anosov diffeomorphism of <ref> is expansive but
it does not satisfy 

the hypothesis 
of Theorem <ref>.





In this case, the set su_δ(x)∩ us_δ(x) is not a neighborhood of a wandering point x. 
This means, if it is true that cw1-expansivity implies expansivity on a Peano continuum, 
then new arguments will be needed for its proof.





§ DENDRITATIONS OF SURFACES



In this section we will study cw-foliations of compact surfaces. 
Applications to cw-expansive surface homeomorphisms will be given.



 §.§ Cw-decompositions of discs



Let D be a metric space homeomorphic to the Euclidean disc 

    {(x,y)∈^2:x^2+y^2≤ 1}.

As usual, ∂ D is the boundary of the disc. 


 A decomposition of D is a cw-decomposition 
 if it is codendritic and hereditarily unicoherent. 
 That is, the quotient space is a dendrite and each plaque is hereditarily unicoherent.


We say that Q is n-dimensional if Q(x)=n for all x∈ D∖∂ D. 
As in <ref>,  denotes the topological dimension. 



If Q is a decomposition of D then the following statements are equivalent:

 
  * Q is a cw-decomposition,
 
  * Q(x)∩∂ D≠∅ and Q is 1-dimensional for all x∈ D.




(1→ 2). 
Arguing by contradiction, suppose that Q(x)∩∂ D=∅ for some x∈ D.
Since Q is codendritic there is y∈ D close to x such that the plaque Q(y) separates D and 
Q(y)∩∂ D=∅. 
By Janisewski's Theorem <ref> Q(y) is not unicoherent, contradicting that Q is a cw-decomposition.
To prove that Q is 1-dimensional, note that each plaque Q(x) has dimension 0, 1 or 2. 
If some plaque has dimension 2 then it has interior points and then it is not hereditarily unicoherent. 
For an interior point x of the disc it holds that (Q(x))≠ 0, because it is a continuum meeting ∂ D.

(2→ 1). Let us first show that Q is hereditarily unicoherent. Arguing by contradiction suppose that 
Q(x) is not hereditarily unicoherent. 
Then there are two continua A,B such that Q(x)=A∪ B and A∩ B is disconnected. 
By Janisewski's Theorem Q(x) separates D. Let U be a component of D∖ Q(x) 
disjoint from ∂ D.
This implies that U⊂ Q(x), contradicting that Q(x) is one-dimensional. 

Now we prove that Q is codendritic.
First we show that if x,y∈ D  
and Q(x)∩ Q(y)=∅ then there is z∈ D such that Q(z) separates Q(x) from Q(y).   
Let I,J be two arcs contained in ∂ D such that Q(x)∩∂ D⊂ I 
and Q(y)∩∂ D⊂ J. 
Suppose that I,J are minimal with this property.
Since Q(x) and Q(y) are disjoint we have that I∩ J=∅. 
Denote by α and β the open arcs in ∂ D∖ (I∪ J).



Take an arc Γ⊂ D from a∈α to b∈β separating Q(x) and Q(y) 
as in Figure <ref>.


Define the sets

    [ Γ_α={u∈Γ: Q(u)∩α≠∅},; Γ_β={u∈Γ: Q(u)∩β≠∅}. ]

We have that a∈Γ_α and b∈Γ_β, which implies that they are non-empty. 
Since Γ is disjoint from Q(x) 
and Q(y), the upper semicontinuity of Q implies that Γ_α and Γ_β 
are closed, recall Remark <ref>. 
As every plaque intersects ∂ D we have that Γ=Γ_α∪Γ_β. 
Given that Γ is connected, we find z∈Γ_α∩Γ_β. 
Therefore, Q(z) separates Q(x) from Q(y).

By Proposition <ref> we know that D/Q is a continuum
and from 
Theorem <ref> 
we conclude that Q is codendritic.










 If Q is a cw-decomposition and x_1,x_2,x_3∈ D are such that 
 Q(x_i) does not separate Q(x_j) from Q(x_k) for {i,j,k}={1,2,3} 
 then there is x∈ D such that Q(x_1),Q(x_2),Q(x_3) are in different components of D∖ Q(x). 



Knowing that D/Q is a dendrite the result follows by general properties of such spaces.



 On a dendrite there can be four points that cannot be separated simultaneously with a single point. 
 Consider for example a dendrite with the shape of the letter H and try to separate with one single point the four ends. 
 It is easy to construct a decomposition Q such that D/Q has this topology.
 See Figure <ref>.
 



 It seems that given any tree T (i.e., a dendrite with a finite number of ramification points) 
 there is a decomposition Q of D such that D/Q is homeomorphic to T. 
 It also seems that this is still true for an arbitrary dendrite T, but a proof is not so clear for the author.



In the study of complex dynamics, several definitions of laminations can be found. 
Following <cit.>, a lamination is a closed equivalence relation on ∂ D 
such that the convex hull of different equivalence classes are disjoint. 
The convex hull is considered with respect to (a differentiable structure and) a Riemannian metric on D that makes it isometric with an Euclidean disc. 
There is a strong link between cw-decompositions  and laminations:
if Q is a cw-decomposition of D then the restriction of Q to the boundary of D is a lamination.






 If  is a cw-foliation over the complete basis  of a closed surface S then
 there is δ>0 such that Q_ U is a cw-decomposition for all U∈ with (U)<δ. 




It follows by 









Propositions <ref> and <ref>.


































 §.§ Dendritations and generic leaves


The following is the main definition of the paper.
It combines the properties of Peano foliations, 
cw-foliations and the plane topology. 


A dendritation is a Peano cw-foliation of a compact surface.



By Remark <ref> we know that compact surfaces with non-empty boundary admits no cw-expansive homeomorphisms. 
Then, we will assume that the surface has empty boundary.
Recall from (<ref>) the definition of the complete basis _δ.



 If ={Q_U:U∈} is an atlas of a dendritation, δ>0 is sufficiently small, U∈_δ and 
 D= U is a disc then 
 each plaque Q_D(x) is a dendrite and 
 the quotient D /Q_D is a dendrite.



By definition, we know that  is a cw-atlas. 
Take δ from Proposition <ref> such that Q_D is a cw-decomposition. 
By the definition of cw-decomposition we know that D/Q_D is a dendrite 
and that each plaque is a hereditarily unicoherent continuum. 
By hypothesis, the plaques are locally connected.
Applying Theorem <ref> we conclude the
plaques are dendrites. 



 Knowing that each plaque is a dendrite 
it is natural to look for more topological information given that they are contained in the plane. 
Unfortunately, we must deal with the worst kind of dendrites because every dendrite can be embedded in the plane. 
Recall the Wazewski's universal dendrite, Theorem <ref>.
However, generic dendrites in a dendritation are nice, as we will now explain.


We will study the properties of the plaque of a point in a residual subset of the disc. 




A property is generic if it holds in a residual subset.
For this purpose we recall that
in <cit.> Moore proved  that every family of mutually disjoint triods of the plane is countable. 
An alternative proof can be found in <cit.>. 
Recall that a triod is a union of three arcs, 
homeomorphic to ([-1,1]×{0})∪ ({0}× [0,1]) in the plane (three segments with a common end point).













If Q is a cw-decomposition of the disk D then there is a residual set G⊂ D such that 
(Q(x)∩∂ D)≤ 2
for all x∈ G.



First we show that there is a residual set G_1⊂ D such that Q(x)∩∂ D is totally disconnected 
for all x∈ G_1.
Let A⊂∂ D be a countable dense subset. 
If Q(x)∩∂ D contains a non-trivial arc then 
A∩ Q(x)≠∅. 
Therefore, at most a countable family of plaques intersects ∂ D in a non totally disconnected set. 
The complement of the union of such plaques gives our residual set G_1.

By Proposition <ref> we know that Q is codendritic. 
Then, by Proposition <ref> there is a residual set G_2⊂ D such that 
D∖ Q(x) has 1 or 2 components for all x∈ G_2.




By Janisewski's Theorem this implies that Q(x)∩∂ D has 1 or 2 components. 
For x∈ G=G_1∩ G_2 we have that Q(x)∩∂ D has 1 or 2 points.


In the examples known to the author of decompositions defined from stable sets of cw-expansive homeomorphisms 
it is true that 
(Q(x)∩∂ D)=2 on a residual set. 
In general, for arbitrary decompositions, this is not true as the next example shows.



We will construct a cw-decomposition Q of a disc D 
such that for every residual set G⊂ D there is x∈ G whose dendrite Q(x) does not separate D. 
 Consider a pseudo-Anosov diffeomorphism on a hyperbolic surface. 
 Denote by U={(x,y)∈^2:x^2+y^2<1} the universal cover of the surface and denote 
 by F̃^s the lifting to U of the stable singular foliation of the pseudo-Anosov diffeomorphism. 
 On the closed disc D'=U (the closure in the usual topology of ^2) 
 consider the dendritic decomposition where
 Q'(p) is the closure of F̃^s(p). 
 Consider the annulus A={(x,y)∈^2:1≤ x^2+ y^2≤ 2} and the disc D=U∪ A. 
 We will extend Q' to D. 
 For p∈∂ U define A_p={λ p:1≤λ≤ 2} a line segment from p to ∂ D contained in the line through p and the origin. 
 For p∈ D' define 
 
    Q(p)=Q'(p)∪⋃_q∈ Q'(p)∩∂ D' A_q.

 It is easy to prove that Q defines a dendritic decomposition on D.

 Let us show that for a generic point p∈ A we have that Q(p) is a non-separating arc. 
 The proof follows the ideas in <cit.>*Lemma 3.1. 
 Let D_n be the closed disc of radius 1-1/n centered at the origin. 
 Define T_n=Q^-1(Q(D_n)). 
 Notice that 
    ⋃^∞ T_n={p∈ D:Q(p) separates  D}.

 Also A∖ T_n is open and dense in A. 
 Define G=A∖(∪^∞ T_n).
 Therefore, G is residual in A and for all p∈ A the plaque Q(p) is an arc of the form A_q (containing p), 
 and this arc does not separate D.


A decomposition is arc-connected if each plaque is arc-connected.



 If Q is an arc-connected cw-decomposition on D then there is a residual set G⊂ D 
 such that Q(x) is an arc for all x∈ G.



If a plaque is not an arc then it has a ramification point and contains a triod. 
By Moore's Theorem <cit.> at most a countable number of disjoint triods can be embedded in the plane.
Then, the set of plaques with ramification points is countable. 
Denote by Q_1, Q_2,… such plaques.
Since each Q_n is closed and has empty interior, its complement in the disc D, U_n=D∖ Q_n is an open and dense subset of D. 
Then, G=∩_n≥ 1 U_n is a residual set of points x such that Q(x) is an arc.


If  is a dendritation of a surface we say that y∈(x) is a ramification point if 
(x)∖{y} has at least 3 components in the plaque topology.



If  is a dendritation of a compact surface S then there is a residual set G⊂ S such that (x) has no ramification for all x∈ G.
For all x∈ G the leaf (x) with the plaque topology is a one-dimensional manifold: , [0,+∞), [0,1] or the circle.




By Proposition <ref> and Theorems <ref>, <ref>
we know that: τ_ is a metric topology, the leaves are arc-connected in τ_ and 
each leaf has a countable basis with respect to τ_.



Arguing as in the proof of Proposition <ref> we have that the number of 
leaves with ramification points is at most countable. 
Since each leaf is a meagre set (by the definition of cw-foliation), the set of points in leaves without ramifications is a residual set G.
Then, the result follows by the classification of one-dimensional manifolds, see for example <cit.>.



On a residual set, as in Theorem <ref>, the one-dimensional manifolds can occur. 
For example, a torus can be foliated by circles. 
Also, a minimal flow on the torus has all the leaves homeomorphic to . 
Using Example <ref> we can define a cw-foliation of the sphere (identifying two discs by the boundary) 
with a residual set of 
leaves homeomorphic to one-dimensional manifolds with boundary.



It would be interesting to extend Theorem <ref>
without assuming the local connection of the plaques.


In the next result we will use the stable and unstable 
cw-foliations ^s,^u that were constructed in <ref> for an arbitrary 
cw-expansive homeomorphism of a
Peano continuum (as our compact surface S).



If f S→ S is cw-expansive homeomorphism and ^s,^u are dendritations then 
 there is a residual set G⊂ S such that for all x∈ G 
 it holds that ^s(x) and ^u(x) are homeomorphic to  or [0,+∞).





Take from Theorem <ref> two residual sets G^s and G^u corresponding to ^s and ^u respectively 
and define G=G^s∩ G^u. 
For x∈ G we know that the stable and the unstable leaf of x is a one-dimensional manifold. 
In Theorem <ref> we proved that no leaf ^σ(x) is plaque-compact.
By Theorem 
<ref>, this implies that no leaf ^σ(x) is a Peano continuum in the relative topology of τ (the topology of the surface).
Since [0,1] and the circle are Peano continua 
we know from Theorem <ref> that ^s(x) and ^u(x) is homeomorphic to  or [0,+∞) for each x∈ G.



 In the examples known to the author, for every cw-expansive homeomorphism of a compact surface there is
 a residual set of points whose stable and unstable leaves are homeomorphic to  
 and just a countable number of leaves are homeomorphic to [0,+∞).




 §.§ Graph like continua and decompositions


In this section we develop a technique to construct examples of decompositions.

Let I, J⊂ be compact intervals. 
Consider a continuum with empty interior C⊂ I× J 
disjoint from I×∂ J and 
such that for each x ∈ I the set C∩ [{x}× J] is connected and non-empty. 
In this case we say that C is a graph like continuum. 
For such continuum C define 

    [ U^+={(x,y)∈ I× J:  if  (x,y')∈ C then  y>y'},; U^-={(x,y)∈ I× J:  if  (x,y')∈ C then  y<y'}. ]

We have that I× J is the disjoint union of C, U^+ and U^-.



We will define a decomposition of I× J associated to a graph like continuum C. 
Define v I×→^2 by v(p)=(0,(p,C)),
where (p,C)=min{(p,q):q∈ C}. 
Since v is Lipschitz, it defines a continuous flow ϕ on I× with equilibrium points at every point of C. 
For p∈ I× J define Q_C(p)=C if p∈ C and 

    Q_C(p)={q∈ I× J:∃ t∈ such that ϕ_t(p),ϕ_t(q)∈ I×∂ J}

if p∉ C.
Recall from <ref> that 
the continuity of a decomposition is with respect to the Hausdorff metric.


 If C⊂ I× J is a graph like continuum then 
 Q_C is a cw-decomposition and: 
 
  
  * Q_C(p) is the graph of a continuous map I→ J for all p∉ C,
  
  * Q_C is continuous at every p∉ C, 
  
  * Q_C is continuous (at every point in C) if and only if C∩∂ U^+=C∩∂ U^-,
  
  * [I× J]/Q_C is an arc.
 



 Fix p∈ U^+. 
 Define L^± as the components of I×∂ J contained in ∂ U^± respectively.
 Take T(p)∈ such that ϕ_T(p)(p)∈ L^+. 
 Since L^± are transverse to ϕ we have that T U^+→ is continuous. 
 From the definition of Q_C we have that Q_C(p)=ϕ_-T(p)(L^+), given that p∈ U^+. 
 Therefore, Q_C(p) is the graph of a continuous map for all p∈ U^+. 
 In a similar way this conclusion holds for all p∈ U^-. 
 This implies that each Q_C(p) is a continuum meeting ∂ (I× J). 
 Also, no Q_C(p) has interior points. 
 
 In order to conclude that Q_C is a decomposition it only remains to prove 
 the upper semicontinuity. 
 The continuity of the flow ϕ implies that the functions whose graphs are Q_C(p), for p∉ C, 
 varies with continuity. This proves the continuity of Q_C at the points p∉ C. 
 Now consider p∈ C and p_n→ p and assume without loss of generality that p_n∈ U^+. 
 We have that T(p_n)→-∞. Take q_n∈ Q_C(p_n) with q_n→ q. 
 By the continuity of T we have that q∈ C. This proves the upper semicontinuity of Q_C at p∈ C. 
 Therefore Q_C is a decomposition. 
 
 With the previous notation, we have that Q_C(p_n)→ C∩∂ U^+. 
 Therefore, Q_C is continuous at points in C if and only if C∩∂ U^+=C∩∂ U^-.
 In order to conclude that the quotient is an arc note that except L^±, every plaque separates the rectangle I× J. 
 This also implies that Q_C is codendritic and a cw-decomposition.
 



In the square [-1,1]× [-1,1] consider the graph like continuum

    C=([-1,1]×{0})∪ ({0}× [0,1/2]).

The decomposition Q_C induced by C is shown in Figure <ref>. 
We see that it can be continuous at some x while not being continuous at some y∈ Q_C(x).





In the square [-2,2]× [-2,2] consider the graph like continuum

    C=({0}×[-1,1])∪{(x,sin(1/x)):0<|x|≤ 2}.

We have that Q_C is a continuous decomposition. 
It is illustrated in Figure <ref>.




 Consider K⊂ [0,1] a Cantor set and define the graph like continuum
 
    C=(K×[1/3,2/3])∪([0,1]×{1/3}).
 
Consider the decomposition Q_C in the square [0,1]×[0,1].  
See Figure <ref>.
If we think of Q_C as a cw-foliation of this square and we consider 
a complete basis of small open sets then we see that C is a leaf 
that is not a countable union of plaques. 





 §.§ The foliated box







The purpose of the present section is to give a characterization of the standard foliated two-dimensional box (a product structure of a rectangle).



Let Q D→(D) be a cw-decomposition. 
If P is an arc-connected plaque that separates D and Q is continuous at every x∈ P 
then P is an arc and P∩∂ D has exactly two points.



By Janisewski's Theorem <ref>[In this case the Jordan's closed curve Theorem could also be applied.], we have that P∩∂ D is disconnected because P is a continuum separating D. 
Take a,b∈∂ D in different components of P∩∂ D. 
Since P is arc-connected there is an arc γ⊂ P from a to b. 
We can assume that γ∩∂ D={a,b}.

We will show that γ= P. 
Arguing by contradiction assume that there is z∈ P∖γ.
Denote by D_1,D_2 the components of D∖γ and suppose that z∈ D_1.
Take y∈γ and x_n∈ D_2 with x_n→ y. 
Since Q is continuous at every point of P, it is continuous at y. 
But z is not in the limit of Q(y_n). 
This contradiction proves that P=γ.


 
We wish to remark the necessity of the hypothesis in Proposition <ref>.
The arc connection is needed by Example <ref>, where C is separating and continuous but not arc-connected. 
In Example <ref> we see that Q must be continuous at every point of C in order to conclude that it is an arc, even being arc-connected and separating.
We need to assume that C is separating in order to conclude that it is an arc. An example is illustrated in Figure <ref>.





If Q D→(D) is a continuous arc-connected cw-decomposition 
then D/Q is an arc. 



By definition of cw-decomposition we know that D/Q is a dendrite.
In order to prove that it is an arc it is sufficient to prove that 
if Q(x) separates D/Q then it separates in two components and this is a direct consequence of Proposition <ref>.


 Proposition <ref> is not true if we do not assume the arc-connection of each Q(x). 
 An example can be constructed based on Wada's lakes. 
 See Figure <ref>.
 A similar decomposition was previously considered in <cit.>*Remark 4.11. 
 It is a continuous decomposition Q of a disc D such that Q/D is a triod.



For the following proof we recall that a size function is a continuous function μ(D)→ 
such that: μ(C)≥ 0 for all C∈(D) with equality if and only if C is a singleton; 
if C⊂ C' and C≠ C' then μ(C)<μ(C'). 
Size functions exist <cit.> and in fact were introduced by Whitney in <cit.> 
for the study of regular families of curves (very similar to our case).
The following result characterizes standard foliated boxes 
(or product structure as defined in <ref>).
Its proof uses techniques from <cit.>*Theorem 17A.





If Q D→(D) is a -smooth, continuous
cw-decomposition 
with two non-trivial plaques contained in ∂ D
then it is a product structure.



In Proposition <ref> we proved that 
 every -smooth decomposition is dendritic (in particular, arc-connected). 
Then, we can apply Proposition <ref> to conclude that the quotient D/Q is an arc. 
Therefore, it has two non-separating plaques.
Let E_1, E_2⊂∂ D be the non-separating plaques of Q. 
Take two arcs A_1,A_2⊂∂ D intersecting E_1 and E_2 in one point to each one, as shown in Figure <ref>.

Since D/Q is an arc, Q(z) separates E_1 from E_2 for all z∈ D.
Then Q(z)∩ A_i≠∅ for all z∈ D and i=1,2.
By Proposition <ref> we have that Q(z)∩ A_i is a singleton.
Define π D→ A_1 by {π(z)}=A_1∩ Q(z). 
Proposition <ref> also gives us that 
each Q(z) is an arc. 
Define Γ D→(D) such that Γ(z)⊂ Q(z) is the arc from z to π(z). 

Let μ(D)→ be a size function 
and define h D→A_1× [0,1] by 

    h(z)=(π(z),μ(Γ(z))/μ(Q(z))).

Since Q has not trivial plaques we have that μ(Q(z))≠ 0 for all z∈ D. 
Let us show that h is continuous. 
As Q(z)∩ A_1 is a singleton for all z∈ D we have that π is continuous.


Given that Q is -smooth, we conclude that Γ is continuous.
Since μ is continuous we conclude that h is continuous. 

To prove that h is injective suppose that h(z)=h(z'). 
Then π(z)=π(z') and μ(Γ(z))/μ(Q(z))=μ(Γ(z'))/μ(Q(z')). 
Therefore z,z' are in the same plaque and μ(Q(z))=μ(Q(z')), 
which implies that μ(Γ(z))=μ(Γ(z')). 
Since every plaque is an arc we conclude that z=z' and the injectivity of h. 
Similar arguments prove the surjectivity of h
and we conclude that h is a homeomorphism.
Denote by Q̃ the decomposition of the rectangle A_1× [0,1] in horizontal lines {a}× [0,1] for a∈ A.
Since it holds that h(Q(z))=Q̃(h(z)) 
the proof ends.


It is natural to ask if a 
continuous arc-connected cw-decomposition Q D→(D)
must be a product structure. In the next example we show that this is not always the case.

 

The cw-decomposition illustrated in Figure <ref> 
is not a product structure because it is not -smooth.























 §.§ Smooth dendritations

Let  be a cw-foliation on a closed surface S with atlas ={Q_ U:U∈}. 







We say that cw-foliation  is -smooth if 
each Q_ U is a -smooth decomposition.


By Proposition <ref> we know that monotone restrictions of a -smooth cw-decomposition are -smooth. 
Then, a -smooth cw-foliations is defined ever a complete basis.




Every -smooth cw-foliation of a closed surface S is 
a standard foliation with a finite set of prong-singularities and no 1-prongs.



Consider a cw-atlas ={Q_ U:U∈} of the 
-smooth cw-foliation . 
By Proposition <ref> we have that  is a dendritation.
Take U∈ such that D=U is a disc and define Q=Q_D.

In this paragraph we will show that 
if z is an end point of the dendrite Q(z) then z∈∂ D.

Arguing by contradiction suppose that 
there is z∈ D∖∂ D that is an end point of its plaque. 
Take z_n∈ D∖ Q(z) such that z_n→ z.
Since Q is codendritic, for each n≥ 1 there is a plaque P_n separating Q(z) from z_n.
Consider an arc A_n⊂ P_n separating Q(z) from z_n such that 
A_n∩∂ D={x_n,y_n}.
Consider a Riemannian metric on D and for each n≥ 1 a geodesic arc γ_n from z to z_n. 
Since A_n separates z from z_n, we can take z'_n∈γ_n∩ A_n. 
Since γ_n is a geodesic arc we have that z'_n→ z. 
Taking subsequences we can assume that x_n→ x and y_n→ y with x,y∈ Q(z)∩∂ D. 
Given that Q is -smooth we have that A_n→ A where A is an arc from x to y.
Since z'_n∈ A_n and z_n→ z we have that z∈ A. 
The upper semicontinuity of Q gives us that A⊂ Q(z).
Since z∉∂ D we have that z is not an end point of Q(z). 
This contradiction proves that every end point of each plaque is in the boundary of D.

Now we will show that
 for all compact set K⊂(D) 
 the set of points x∈ K such that x is a ramification point of Q(x) is finite.

The result of the previous paragraph implies that for all x∈ D
every component of Q(x)∖{x} meets ∂ D. 
Now suppose that there is a sequence y_n→ y such that 
y_n∈ K and each y_n is a ramification point of Q(y_n). 
Denote by A_n=Q(y_n)∩∂ D. 
We have that each A_n has at least three points. 
We can take a_n,b_n∈ A_n such that (a_n,b_n)→ 0 
and y_n∈ Q(a_n,b_n)[For two points a,b in a common plaque, Q(a,b) is the arc in the plaque from a to b.]. 
This contradicts that Q is -smooth 
and proves that the set of ramification points can only accumulate on ∂ D.
Then, there is a finite number of ramification points in the whole surface.

Take x∈ S and a small disc D around x. 
Assume that in D∖{x} there are no ramification points. 
We know that Q_D(x) separates D in at least two components. 
Some of these components may be far from x and only a finite number of them contain x in its boundary, see Figure <ref>. 
Denote by D_1,…,D_n the closures of such components. 


Since the only possible ramification point in D is x, we know that in each D_i every plaque is an arc.
As the end points of the plaques are in the boundary, each plaque is an arc starting and ending at the boundary. 

Given y in the interior of D_i, the arcs Q_D_i(x) and Q_D_i(y) determine two arcs A, B contained in the boundary of D_i 
as shown in Figure <ref>.


Now we will prove that there is a neighborhood V of 
x in D_i such that every z∈ V 
separates Q_D_i(z) in two arcs γ_A(z) and γ_B(z) such that 
γ_A(z) cuts A and is disjoint from B and 
γ_B(z) cuts B and is disjoint from A.
Arguing by contradiction assume first that there is z_n→ x such that Q_D_i(z_n)∩ B=∅. 
We can take two points a_n,b_n∈ A∩ Q_D_i(z_n) such that (a_n,b_n)→ 0. 
This contradicts that Q_D_i is -smooth. 
Therefore, if z is close to x then Q_D_i(z) cuts A and B.
A similar argument proves that if z is sufficiently close to x then 
one component of Q_D_i(z)∖{z} cuts A and not B while the other component 
cuts B and not A.



Define K as the union of all the arcs starting and ending in A and contained in a plaque of Q_D_i. 
Considering that the arcs defining K may be trivial, we have that A⊂ K. 
In this way K is a continuum.
For u,v∈ D_i in a common plaque define (u,v) as the arc from u to v inside the plaque. 
For u∈ V define π(u) as the first point of γ_A(u) in A. 
Define Γ V→(D_i) as 

    Γ(u)=(u,π(u))∪ K.


In this paragraph we show that Γ is continuous. 
Take u∈ V and u_n→ u. 
We can assume that Γ(u_n) converges and π(u_n)→ v∈γ_A(u)∩ A. 
Since Q is -smooth we have that (u_n,π(u_n))→(u,v). 
It is clear that (u,π(u))⊂(u,v). Consequently, Γ(u)⊂limΓ(u_n). 
Since (π(u),v)⊂ K and 

    (u,v)=(u,π(u))∪(π(u),v)
 we have that 
limΓ(u_n)⊂Γ(u). This proves the continuity of Γ.

Consider a size function μ(D_i)→ 
and define φ V→ as φ(u)=μ(Γ(u)). 
The function φ is continuous and increases along a plaque from A to B. 
The level sets of φ and the plaques give coordinates and define a product structure 
in a neighborhood of x in D_i (contained in V). 

Arguing in the same way in the sectors D_1,…,D_n an n-prong structure is defined around x. 
As we proved, n≠ 1 and the set of n-prongs with n≥ 3 is finite. This finishes the proof.







 §.§ Continuous dendritations












Let ={Q_ U:U∈} be an atlas of a dendritation  over the complete basis  of a surface S. 

 

 We say that a dendritation  is continuous 
 if for all x∈ U∈ there is a closed disc D⊂ U 
 such that x is in the interior of D and Q_D is continuous.


This definition of continuity is stronger
than the one given in <ref>

because we require the continuity on discs instead of arbitrary neighborhoods.

Denote by Q_1 the decomposition of D=[-1,1]× [-1,1] in horizontal plaques (a foliated box). 
Consider the equivalence relation on D generated by (x,1)∼(-x,1) for all x∈ [-1,1]. 
Denote by Q_2 the induced decomposition on the quotient space D̃=D/∼. 
The point {(0,1)} in D̃ is called 1-prong of Q_2.



Every continuous dendritation is a foliation possibly with a finite number of 1-prongs.



By Proposition <ref> there are no ramification points, that is, 
every plaque is an arc. 
Take x∈ S and a disc D around x with Q_D continuous. 
From Proposition <ref> we know that D/Q is an arc. 
Then there are exactly two non-separating plaques that will be called E_1, E_2.
From Proposition <ref>, every separating plaque cuts the boundary of D 
at exactly two points.
Reducing D with two separating plaques (if needed) we can assume that E_1 and E_2 are non-trivial (are not singletons).

In this paragraph we will show that if E_1,E_2⊂∂ D then Q is a product structure.
For this purpose we will show that Q is -smooth and apply Theorem <ref>.
Arguing by contradiction assume that Q(x_n,y_n) does not converge to Q(x,y) 
assuming that x_n→ x, y_n→ y and y_n∈ Q(x_n). 

Taking a subsequence we can assume that Q(x_n,y_n)→ C for a continuum C⊂ D. 
Since Q(x_n,y_n)⊂ Q(x_n) and Q is continuous we have that 
C⊂ Q(x). 
Therefore, C is an arc because Q(x) is an arc. 
Moreover, x,y∈ C.
Since C≠ Q(x,y), we have that y (or x) is an interior point of the arc C.
Let z∈ C be the end point of C that is not in Q(x,y). 
Consider z_n∈ Q(x_n,y_n), an interior point, 
such that z_n→ z. 
Since the dendritation is continuous
there is a disc D', a neighborhood of z, 
such that Q'=Q|^m_D' is continuous. 
We can assume that D' is so small that y,y_n∉ D' for all n≥ 1. 
Therefore, Q'(z_n) is a sequence of arcs separating D'. 
We can find n_1,n_2,n_3 such that
z and Q'(z_n_i) are in the same component of D'∖ Q'(z_n_j) if i≠ j, i,j∈{1,2,3}. 
Then Q' has at least three non-separating plaques.
This contradicts Proposition <ref> in the disc D'
and proves that Q is -smooth.
Since E_1,E_2 are non-trivial we can apply 
Theorem <ref> to conclude that Q is a product structure.

If x∈ E_1 (i.e., E_1 is not contained in the boundary of D) 
we can cut D along E_1 and reduce the proof to the previous case (E_1,E_2⊂∂ D). 
In this case we obtain that Q is a 1-prong decomposition around x. 
This also proves that the number of 1-prongs is finite.


In the following result the word foliation means a one-dimensional C^0 foliation in the standard sense (without singular points).



 A dendritation is a foliation if and only if it is continuous and -smooth.



It follows by Theorems <ref> and <ref>.


































































































































































































 §.§ Cw_ F-expansivity and dendritations











If two arcs α,β in a surface meet at x we say that they are topologically transverse at x
if the components of α∖β are in 
different components of D∖β, where D is a disc separated by β and containing α.
See Figure <ref>.



The following result gives some ideas of how the stable and the unstable cw-foliations of a -expansive 
homeomorphism are distributed in the surface.



 If f is a -expansive homeomorphism of a compact surface S then: 
 
  
  * ^s and ^u are f-invariant dendritations and no leaf is a Peano continuum, 

  
  * there is a residual set G⊂ S such that ^s(x) and ^u(x) are non-compact 
  one-dimensional manifolds for all x∈ G, 
  
  * there is a subset H⊂ G that is dense in S such that ^s(x) and ^u(x) are topologically transverse at x for all x∈ H.
 











By Theorem <ref> we know that ^s and ^u are f-invariant cw-foliations 
without plaque-compact leaves. 
We start proving the local connection of stable and unstable plaques to conclude that they are dendritations.
Arguing by contradiction, let P⊂ S be a non-locally connected stable continuum. 
By Theorem <ref>
we can consider P_n a sequence of subcontinua of P such that 
P_i∩ P_j=∅ if i≠ j and (P_n,P_*)→ 0 for some non-trivial continuum P_*⊂ P. 
Take x∈ P_* and a small disc D around x such that every P_n separates D 
for every n≥ n_0, for some n_0. 
We can assume that D=B_r(x) for some r>0.
Denote by Q^s and Q^u the stable and the unstable cw-decompositions of D, respectively. 
Consider x_n∈ P_n∩ D such that x_n→ x and each Q^s(x_n) separates D. 
Taking a subsequence if needed we can assume that Q^s(x_n) converges to a stable continuum C⊂ Q^s(x). 
Moreover, we can assume that the sequence of plaques Q^s(x_n) is monotonous (i.e., if i<j<k then Q^s(x_j) separates 
Q^s(x_i) from Q^s(x_k)).
For the value of r>0 fixed above (D=B_r(x)) 
take ϵ>0 given by Theorem <ref>.
Denote by G_n the component of D between C and Q^s(x_n). 
Since Q^s(x_n)→ C in the Hausdorff metric, there is n_1>0 such that 
Q^s(x_n)⊂ B_ϵ(C) for all n≥ n_1. 
Consequently, G_n⊂ B_ϵ(C) for all n≥ n_1. 
If ϵ is small, we can take y_n∈∂ B_r/2(x)∩ Q^s(x_n) 
and an arc γ_n⊂ B_r/2(x) from x_n to y_n.
Taking a subarc of γ_n we can assume that γ_n⊂G_n∩ B_r/2(x) 
meeting C and Q^s(x_n).
This proves that (C,G_n,Q^s(x_n)) is an (r,ϵ,x)-capacitor for all n≥ n_1. 
Applying Theorem <ref> we can take an unstable plaque Q^u(z) cutting C and Q^s(x_n). 
This unstable plaque cuts Q^s(x_m) for all m≥ n. 
Since the plaques Q^s(x_n) are contained in the stable plaque P (considered at the beginning of the proof) 
we have a contradiction with the -expansivity of f.












This proves that ^s and ^u are dendritations. 

Since -expansivity implies cw-expansivity we can now apply Theorem <ref> 
to obtain the residual set G.

By Theorem <ref> no stable leaf is plaque compact and 
by Theorem <ref>, a leaf is plaque compact if and only if it is a Peano continuum. 
Therefore, no stable leaf is a Peano continuum.

Let D⊂ S be a small disc and take an interior point x∈ D. 
We can take a sequence x_n∈ G such that x_n→ x. 
Consider a stable arc γ^s_n from x_n to y_n∈∂ D. 
We can assume that y_n→ y and γ_n→ C where C is a dendrite containing x and y. 
Taking a subsequence we can suppose that y_n is monotone in ∂ D. 
Consider four consecutive arcs γ^s_n+1,…,γ^s_n+4. 
As before, for n large, we can apply Theorem <ref> to the capacitor determined by 
γ^s_n+1 and γ^s_n+4 to find a point p∈ G between γ^s_n+2 and γ^s_n+3 
whose unstable arc cuts γ^s_n+1 or γ^s_n+4. 
Suppose that it cuts γ^s_n+1. 
Then, it cuts γ^s_n+2. 
By the -expansivity of f,
the unstable arc of p intersects γ^s_n+2 in finitely many points.
By the construction, at least one of this cuts is topologically transverse. 




 §.§ Cw1-expansivity



In this section we show that every cw1-expansive homeomorphism of a compact surface is expansive.




Let Q^1,Q^2 be two cw-decompositions of D such that 
Q^1(x)∩ Q^2(x)={x} for all x∈ D, 
∂ D is the union of two arcs α and β with extreme points p,q 
and α⊂ Q^1(a) for some a∈∂ D. Then
for all x∈α, x∉{p,q}, 
it holds that Q^2(x)∩β≠∅.



Suppose that for some x∈α, x≠ p,q, we have that 
Q^2(x)∩β=∅. 
We know that D/Q^2 is a dendrite.
Since Q^2(x)∩ Q^1(x)={x} we have that Q^2(x) is an end of D/Q^2.
In the quotient dendrite we can find a point say Q^2(y) that separates D/Q^2 
and is arbitrarily close to Q^2(x). 
Then Q^2(y) cuts Q^1(a) in at least two points, contradicting our hypothesis.



Given two cw-decompositions Q^1,Q^2 of a disc D we say that ∂ D is a (Q^1,Q^2)-rectangle  
if there are x_1,y_1,x_2,y_2∈∂ D
such that ∂ D is contained in the ordered union Q^1(x_1)∪ Q^2(y_1)∪ Q^1(x_2)∪ Q^2(y_2).




 Let Q^1,Q^2 be two cw-decompositions of a disc D such that Q^1(x)∩ Q^2(x)={x} for all x∈ D and 
 ∂ D is a (Q^1,Q^2)-rectangle. Then there is 
 a homeomorphism h D→ [0,1]×[0,1] sending the plaques of Q^1 and Q^2 onto horizontal and vertical 
 segments, respectively.



We know that D/Q^1 and D/Q^2 dendrites. 
Let us show that they are arcs. 
For this purpose we will show that they have no ramification point. 
Suppose by contradiction that Q^1(x) (similarly for Q^2) is a ramification of the dendrite D/Q^1 for some x∈ D.
Then Q^1(x) separates D into at least three components. 
Consequently, Q^1(x)∩∂ D has at least three components.
Since ∂ D is a (Q^1,Q^2)-rectangle, we have that Q^1(x) cuts in at least two points 
to a plaque of Q^2 in the boundary of D. This contradiction proves that D/Q^1 is an arc. 

Let h D→ [D/Q^1]× [D/Q^2] as h(x)=(Q^1(x),Q^2(x)).
We have that h is continuous and injective. 
In order to conclude that it is a homeomorphism it is sufficient to prove that 
it is surjective. 
The surjectivity in this case means that Q^1(x) cuts Q^2(y) for all x,y∈ D, 
and this follows by Lemma <ref>.




If Q^1,Q^2 is a pair of dendritic cw-decompositions of D and 
Q^1(x)∩ Q^2(x)={x} for all x∈ D then (Q^1,Q^2) is a generating pair.



Fix a point x∈ D. 
Taking a small subdisc around x we can assume that the intersection of each component of 
Q^i(x)∖{x} with ∂ D is a single point, i=1,2.
In this way D∖ Q^i(x) has a finite number of components.
By Lemma <ref> we know that each component of D∖ Q^1(x) is separated by a component of Q^2(x)∖{x}. 
Therefore, the components of Q^1(x)∖{x} and Q^2(x)∖{x} are alternated in the disc. 
Denote by D' a disc contained in D bounded by an arc α^1⊂ Q^1(x), an arc α^2⊂ Q^2(x)∖{x} and 
an arc γ⊂∂ D. 
Assume that α^1 and α^2 are consecutive. 

By Lemma <ref> we have that for each interior point 
y∈α^1 there is an arc β^2_y (of Q^2) from y to γ. 
Also, for each interior point 
z∈α^2 there is a stable arc β^u_z from z to γ. 
Since α^s and α^u are consecutive, if z and y are close to x 
we have that β^s_y cuts β^u_z. 
In this way we obtain a (Q^1,Q^2)-rectangle and we can apply Proposition <ref> to prove that both decompositions 
generate a neighborhood of x in D'.




Repeating this argument in each sector we conclude 
that Q^1 and Q^2 generate a (full) neighborhood of x. 




Every cw1-expansive homeomorphism of a compact surface is expansive.



If f is cw1-expansive then by Proposition <ref> we have that ^s and ^u generate.
Then, the result follows by Theorem <ref>. 


As usual, Ω(f) denotes the non-wandering set of f.


For a homeomorphism f S→ S of a compact surface the following statements are equivalent: 
 
  
  * f is cw1-expansive,
  
  * f is expansive,
  
  * f is 2-expansive and Ω(f)=S,
  
  * f is conjugate to a pseudo-Anosov diffeomorphism without 1-prongs.
 



We have that 2→ 1 on arbitrary metric spaces. 
The equivalence of 2 and 4 was shown in Hi,L. 
The equivalence of 2 and 3 follows by <cit.>.
We have that 1 → 2 by Theorem <ref>. 





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Departamento de Matemática y Estadística del Litoral, Universidad de la República,

Gral. Rivera 1350, Salto-Uruguay

E-mail: artigue@unorte.edu.uy
