Generic Dynamics on Compact Metric Spaces
Alfonso Artigue
math.DS








Generic Dynamics on Compact Metric Spaces
    Alfonso Artigue[Email: artigue@unorte.edu.uy. Adress: Departamento de Matemática y Estadística del Litoral, Universidad de la República, Gral. Rivera 1350, Salto, Uruguay.]
    December 30, 2023
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We prove that generically and
modulo a topological conjugacy there is only one 
dynamical system.







§ INTRODUCTION


In this article we will study generic dynamical systems. 
A property in a complete metric space is generic 
if the set of the points satisfying this property is residual.
Let us explain the space of dynamical systems that we will consider.

To define a dynamical system we need a phase space. 
For this purpose we will consider compact metric spaces. A key point of the paper is that no phase space is fixed. 
Instead, 
given a complete metric space (,) we consider all the compact subsets of . 
Considering the results in this paper, we suggest to think that  is Euclidean ^n or Urysohn universal space 
 (that is defined in <ref>).
We remark that  contains an isometric copy of each compact metric space.
The idea is that, when performing a perturbation of a dynamical system, 
we are allowed to perturb its domain too.

When a compact subset M⊂ is fixed, a dynamical system is a homeomorphism of this set,
a continuous surjective map or a set-valued map. 
A function f M→ M will be identified with its graph {(x,f(x)):x∈ M}. 
If M is compact and f is continuous then 
its graph is a compact subset of M× M.
Thus, an arbitrary compact subset of M× M can be regarded as 
a generalized continuous function. In this way we arrive to set valued maps. 
If f⊂ M× M is compact, then we define f(x)={y∈ M:(x,y)∈ f}. 
Therefore, in this paper, a dynamical system is a compact subset of M× M, for some compact subset M⊂. 
There is a technical detail, we assume that f⊂ M× M and its inverse are onto. 
The inverse of f is f^-1={(y,x)∈ M× M:(x,y)∈ f} 
and f is onto if for all y∈ M there is x∈ M such that (x,y)∈ f. 
The set of dynamical systems, that depends on , will be denoted as ().

In order to define and study generic properties on (), we need a topology for this set. 
Given that we defined dynamical systems as 
certain compact subsets of ×, 
it is natural to consider the Hausdorff metric. It will be denoted as , and in this way 
we obtain the metric space of dynamical systems 

    ((),),
 
which is the object of study of the present paper.

To state our main result suppose that (,) is Polish and perfect (as for example ^n and ). 
We remark that every surjective map, in particular every homeomorphism, 
of a compact metric space has an isometric conjugate in ((),) (where  is the Urysohn universal space).
In Theorem <ref> we will show that generically and
modulo a topological conjugacy there is only one dynamical system in (). 
In other words, we show that there is a single homeomorphism in () whose conjugacy class is a dense G_δ subset of ().
This dynamical system is a homeomorphism of a Cantor space, known as the Special Homeomorphism <cit.>.

From the following viewpoint, this result is quite natural. 
On one hand: 



 a generic compact metric space is a Cantor space. 


This is true if, for instance, we consider the space of compact subsets of a complete and perfect metric space (see <cit.>*the paragraph below Lemma 1). 
It is also true with respect to the Gromov-Hausdorff metric <cit.>*Corollary 5.
On the other hand: 


 on a fixed Cantor space, a generic homeomorphism is conjugate to the Special Homeomorphism.


This result was first proved by Kechris and Rosendal <cit.>. 
In <cit.>, Akin, Glasner and Weiss gave a concrete construction.
Thus, on a generic compact metric space (= a Cantor space) a generic homeomorphism is conjugate to the Special Homeomorphism. 
What we prove in Theorem <ref> is: 


 a generic (homeomorphism of a compact metric space) is conjugate to the Special Homeomorphism.


The key of our approach is the space ((),) which allows us to measure the distance between 
dynamical systems defined on different spaces.

A sketch of the proof of Theorem <ref> is as follows. First, we show in Proposition <ref> 
that a generic f∈() is a homeomorphism of a Cantor space. 
From this result it follows the density of the conjugacy class of the Special Homeomorphism in (). 
To prove the genericity we consider a map 

    φ(K,)× H(K)→_CH()
 
defined as φ(h,f)=h f h^-1 (a homeomorphism of h(K)). 
There, K is a fixed Cantor space, (K,) is the space of continuous injective maps from K to , H(K) is the space of homeomorphisms of K and 
_CH()⊂() is the set of homeomorphisms of Cantor spaces. 
If  denotes the set of homeomorphisms f∈ H(K) conjugate to the Special Homeomorphism, 
we will show that φ((K,)×) is a dense G_δ subset of _CH(), which easily finishes the proof.





Some dynamical properties of the Special Homeomorphism are known, for instance,
it has vanishing topological entropy <cit.>, the pseudo-orbit tracing property <cit.> 
and that it is not conjugate to a subshift <cit.>. 
The reader may see <cit.> for a survey and more results on dynamical systems on Cantor spaces.
In Corollary <ref> we give 
an application of Theorem <ref> to the dynamics of homeomorphisms of compact manifolds.


In <cit.>, Hochman considered another extension of the genericity of the Special Homeomorphism. 
In this paper it is proved, among other results, the genericity of the Special Homeomorphism 
in the space of subshifts of the Hilbert cube.  
This space is topologically universal for dynamical systems, 
i.e., it contains a topological conjugate of every dynamical system 
of a compact metric space (see <cit.>*2.5). 
We say that a homeomorphism 
φ X→ Y is a topological conjugacy (or simply, conjugacy) between 
two maps f X→ X and g Y→ Y if g φ=φ  f. 
If, in addition, φ is an isometry we say that it is an isometric conjugacy.
The isometric universality of 
the space () 
is the key for our next result. 

The second result that we obtained, Theorem <ref>, 
is the projection of Theorem <ref> 
onto the space of dynamical systems modulo isometric conjugacies.
If () denotes the set of compact subsets of  then 
it is natural to consider the quotient space 

    _()=()/∼

where ∼ is the equivalence relation of isometry (between compact subsets of ).
For = we obtain the so called Gromov-Hausdorff space.
In a recent paper, Arbieto and Morales <cit.> extended this idea to dynamical systems. 
If in () we consider the equivalence relation ∼ of isometric conjugacies (see <ref>) we obtain 
the space of dynamical systems 

    _()=()/∼.

In <cit.> the authors studied the notion of topological stability, defining the topology of _() 
 via a quasi-metric (see Remark <ref>). 
In this paper we give a natural metric for this space. 
It is defined in <ref>
and the compatibility with Arbieto-Morales metric is proved in Theorem <ref>. 
In Theorem <ref>
we will show that if (,) is Polish, perfect and ultrahomogeneous 
then a generic f̃∈_() is conjugate to the Special Homeomorphism. 
A space is ultrahomogeneous if every isometry between compact subsets can be extended to a global isometry, 
see <ref>.

It could not be clear why we do not fix =. 
On one hand, since  is universal we are considering all the 
compact metric spaces. But, on the other hand, it is more general not to fix . 
Let us say more. During the preparation of this work we found that 
the key property (for our purposes) of  is the ultrahomogeneity, not the universality. 
Besides, allowing =^n (ultrahomogeneous too) we are including a quite standard space. 
The Gromov-Hausdorff distance can be defined in ^n (allowing only isometries of ^n). 
This space is studied, for example, by Mémoli in <cit.> where also some applications are given.

As we said, in the spaces of dynamical systems we will consider set-valued maps. 
It could be natural to consider only continuous maps or homeomorphisms, 
but this would imply that the metric  would not be complete (see Remark <ref>). 
In fact, we will show that the metric completion of the space of homeomorphisms is 
the space of set-valued dynamics. That is, 
((),) is complete (Proposition <ref>)
and homeomorphisms are dense in () (Proposition <ref>). 
These results assume that (,) is complete and perfect.


I thank Mauricio Achigar, Ignacio Monteverde and José Vieitez for useful conversations 
during the preparation of this work.
Also, I thank the referee for calling my attention to <cit.>*Theorem 1.2, which allowed me to improve the proof 
of Theorem <ref>. 



§ DYNAMICAL SYSTEMS


In this section we give some preliminaries on metric spaces and 
the basic properties of (). 
In <ref> we prove Theorem <ref>.











 §.§ Cantor spaces and the Hausdorff distance



We say that a topological space is perfect if it has no isolated points.
A metric space is a Cantor space if it is compact, perfect and totally disconnected. 


 Any two Cantor spaces are homeomorphic. 
Every open subset of a complete and perfect space contains a Cantor space.




For an arbitrary metric space (N,) denote by (N) the set of compact subsets of N.
The Hausdorff distance between X,Y∈(N) 
is defined as

    _H(X,Y)=inf{ϵ>0:X⊆ B_ϵ(Y) and Y⊆ B_ϵ(X)},

where B_ϵ(x)={y∈ N:(x,y)<ϵ} and B_ϵ(X)=∪_x∈ X B_ϵ(x).






If (N,) is complete (compact) then ((N),_H) is complete (compact).


Fix a Cantor space K⊂ and let H(K) be the space of homeomorphisms of K. 
Suppose that h A→ B is a function with A,B⊂.
In this case we define 

    h=sup_a∈ A(a,h(a)).

Let H(A,B) be the set of homeomorphisms h A→ B.
For a family  of subsets of  let ()=sup{(U):U∈}.
The next result gives a particular way to calculate the Hausdorff distance between Cantor spaces.


 If A,B⊂ are Cantor spaces then 
 
    _H(A,B)=inf_h∈ H(A,B)h.
 



The inequality ≤ is clear (and holds for arbitrary A and B homeomorphic).
Define ρ=_H(A,B) and for δ>0 let ϵ=3δ+ρ.








Let _A be a clopen partition of A with (_A)<δ. 
Take an injective function f_1_A→ B such that for each U∈_A there is x_U∈ U with (x_U,f_1(U))<ρ+δ. 
For each U∈_A consider a clopen set U'⊂ B such that 
f_1(U)∈ U', ^1_B={U':U∈_A} are pairwise disjoint and (^1_B)<δ.

Suppose that ^1_B does not cover B. 
Let ^2_B be a clopen partition of B∖∪^1_B with (^2_B)<δ. 
Take an injective function f_2^2_B→ A such that 
for each V∈^2_B there is y_V∈ V with 
(y_V,f_2(V))<ρ+δ.
Let '_B=^1_B∪^2_B and 
define f_3'_B→_A as 
f_3(U')=U for all U'∈^1_B and
if V∈^2_B then f_3(V)∈_A contains y_V.
Consider a clopen partition '_A of A, refining _A, 
admitting f_4'_B→'_A bijective with f_4(V)⊂ f_3(V) for all V∈'_B. 
Let h B→ A be a homeomorphism such that h(V)=f_4(V) for all V∈'_B.
Since ('_A),('_B)<δ, and for each V∈'_B there are x∈ V and y∈ h(V) such that 
(x,y)<ρ+δ we conclude that h<ρ+3δ=ϵ. 

















 §.§ Polish spaces


A topological space is said to be Polish if it is separable and admits a compatible complete metric.
A set R⊂ X is G_δ if it is a countable intersection of open subsets of X.


Let φ X→ Y be a continuous open surjection with X and Y Polish spaces. 
If R is a dense G_δ subset of X then 

    φ_#(R)={y∈ Y:φ^-1(y)∩ R is dense in φ^-1(y)}

is a dense G_δ subset of Y.



 

From this result we deduce the following consequence which is the key for Theorems <ref> and <ref>.
 


Let φ X→ Y be a continuous open surjection with X and Y Polish spaces. 
If R is a dense G_δ subset of X and R=φ^-1(φ(R)) then φ(R) is a 
dense G_δ subset of Y.



 Since R=φ^-1(φ(R)), we have that y∈φ(R) if and only if φ^-1(y)⊂ R 
 if and only if φ^-1(y)∩ R≠∅. 
  This proves that φ(R)=φ_#(R). 
  Thus, the result follows by Theorem <ref>.


The next classical result allows us to conclude that a G_δ subset of a Polish space is Polish.



In a complete space, a subset
is G_δ if and only if it is completely metrizable.





 §.§ Spaces of dynamical systems


Let (,) be a metric space.
On ^2=× we consider the max-metric _2^2×^2→ given by 

    _2((a,b),(c,d))=max{(a,c),(b,d)}.
 
Consider the projections π_i^2→ given by π_i(a_1,a_2)=a_i for i=1,2. 
Define the set 

    ()={f∈(^2):π_1(f)=π_2(f)}.

A dynamical system is any f∈() (i.e., a surjective relation). 
Denote by  the Hausdorff distance of (^2) associated to _2.
For f∈() define M(f)=π_i(f), and we say that f is a dynamical system on M(f). 
Note that since the projections are continuous and f is compact, we have that M(f) is compact.
In () there are three classes: 

 
  * homeomorphisms of M⊂,
 
  * surjective continuous maps of M⊂, 
 
  * set valued maps of M⊂, where for each x∈ M, f(x)={y∈ M:(x,y)∈ f} may not be a singleton.

In any case, if f∈() we define f^-1={(y,x)∈^2:(x,y)∈ f}∈().



The distance between f,g∈(), (f,g), by definitions 
(<ref>) and (<ref>) is the infimum ϵ>0 
satisfying the following conditions:

 
  * for all (x,y)∈ f there is (x',y')∈ g such that (x,x')<ϵ and (y,y')<ϵ,
 
  * for all (x,y)∈ g there is (x',y')∈ f such that (x,x')<ϵ and (y,y')<ϵ.




 Since (x,y)↦ (y,x) is an isometry of ^2, we have that
(f,g)=(f^-1,g^-1) for all f,g∈().


If M,N∈() and f,g M→ N are continuous functions,
the C^0-distance is defined as

    _C^0(f,g)=sup_x∈ M(f(x),g(x)).

Let C(M,N) be the set of all the continuous functions f M→ N.






The metric  is compatible with the C^0-topology of C(M,N).





















It is well known that the space C(M,M) with _C^0 is complete. 
By Proposition <ref>,  is compatible with the topology induced by _C^0. 
Let us show that C(M,M) is not complete with respect to . 
Consider M as the interval [0,1] (or an isometric copy in ) and define 
f_n(x)=x^n for all n∈ and all x∈ [0,1]. 
Define f⊂ [0,1]× [0,1] as f=([0,1]×{0})∪({1}× [0,1]).
It is easy to see that (f_n,f)→ 0. 
Thus, f_n is a Cauchy sequence with respect to . 
Its limit exists in (), is the set-valued function f, but f is not a function. 
This example explains why we consider non-injective maps and set-valued functions, even if our interest is on 
homeomorphisms, the set-valued functions completes the space of homeomorphisms (with respect to , which is 
the metric that allows us to measure the distance between functions on different spaces).




For all f,g∈() it holds that _H(M(f),M(g))≤(f,g).
Given X∈() denote by i_X the identity map of X. 
 It holds that _H(X,Y)=(i_X,i_Y) for all X,Y∈().


Since Cantor homeomorphisms play a key role we introduce the following notation

    _CH()= {f∈():f is a homeomorphism of a Cantor space}.

The next result extends Proposition <ref>, and will be used to estimate the distance between Cantor homeomorphisms. 



 If g,j∈_CH() then (g,j)<δ if and only if there are homeomorphisms
 h_1,h_2 M(g)→ M(j) such that h_1,h_2<δ and h_2  g=j  h_1.



Assume that (g,j)<δ. 
By Proposition <ref>, there is a homeomorphism h g→ j with h<δ. 
Let π_i M(j)× M(j)→ M(j) be the canonical projections, for i=1,2, and 
define 
h_1(x)=π_1(h(x,g(x)) and h_2(g(x))=π_2(h(x,g(x))). 
That is, h(x,g(x))=(h_1(x),h_2(g(x)))∈ j and j(h_1(x))=h_2(g(x)).
Finally note that l_i≤h for i=1,2.


We give two elementary results that will be used in <ref>.



Finite relations are dense in ().



Given f∈() and ϵ>0, as M(f) is compact, we can take a finite set Q⊂ M(f) with _H(Q,M(f))<ϵ. 
Define g⊂ Q^2 by 
(a,b)∈ g if and only if 
there is (x,y)∈ f with _2((x,y),(a,b))<ϵ (i.e., (x,a)<ϵ and (y,b)<ϵ). 
By the definition of g we have that (f,g)<ϵ and M(g)=Q is finite.




If f∈() and f(x)={y∈ M(f):(x,y)∈ f} is a singleton for all x∈ M(f) then f is a continuous function. 



It is clear that f is a function. 
Given a convergent sequence x_n∈ M(f) if f(x_n) has an accumulation point y≠ f(x) then, 
f is compact, (x,f(x)),(x,y)∈ f. Which contradicts our hypothesis.





 §.§ On a complete and perfect space


We start assuming that (,) is complete.
This implies that ^2 is complete with the max-metric (<ref>).



If (,) is complete then ((),) is complete. 



Since ^2 is complete we have that ((^2),) is complete (Proposition <ref>). 
It remains to show that () is closed in (^2).
Suppose that f_n is a sequence in () converging to f∈(^2). 
Since the projections π_i are continuous, π_i(f_n) converges to π_i(f). 
Therefore, π_1(f)=π_2(f) and f∈().
















The next result 
is known in the space of homeomorphisms of a fixed Cantor space Sh89,Kimura.



If (,) is complete and perfect then 
the set of subshifts of finite type on Cantor spaces is

dense in ().



Given f∈(), by Proposition <ref>, we know that there is a finite relation 
g∈() that is -close to f.
Suppose that M(g)={q_1,…,q_k}.
Let A be the k× k matrix defined by 
A_ij=1 if (q_i,q_j)∈ g and A_ij=0 otherwise. 
This matrix induces a subshift of finite type in the set of symbols Q={q_1,…,q_k}. 
Let Σ={a∈ Q^: (a_i,a_i+1)∈ g for all i∈}. 
Since Σ may not be perfect, consider Σ_*=Σ×{0,1}^.
For each q∈ Q define the Cantor space Σ_q={(a,b)∈Σ_*:a_0=q}. 

Suppose ϵ>0 is given. By Proposition <ref>, for each q∈ Q there is a homeomorphism onto its image 
φ_qΣ_*→ 
such that φ(Σ_q)⊂ B_ϵ(q). 
Define N=φ(Σ_*) and 
the homeomorphism h N→ N as h=φσφ^-1, 
where σΣ_*→Σ_* is the shift map.
It is clear that M(h)=N and D(h,g)<ϵ. 
Since g is conjugate to the subshift of finite type σ on Σ_* the proof ends.



Since subshifts of finite type are homeomorphisms, by Proposition <ref> we have that 
if (,) is complete and perfect then homeomorphisms are dense in (). 
Thus, by Proposition <ref>, we have that the space of set valued dynamical systems is the completion of 
the space of homeomorphisms (with respect to ).






If (,) is complete and perfect then 
_CH() is a dense G_δ subset of ().



We start showing that the set 

    {f∈():f is a homeomorphism}

is a dense G_δ subset of ().
For ϵ>0 given define 

    A_ϵ={f∈():(f(x))<ϵ, ∀ x∈ M(f)}.
 
By Proposition <ref> we know that homeomorphisms are dense in (). 
In particular, for each ϵ>0, A_ϵ is dense in (). 
Let us show that ()∖ A_ϵ is closed. 
Suppose that f_n is a convergent sequence in () such that 
for some x_n∈ M(f_n) we have (f_n(x_n))≥ϵ for all n∈. 
Let f be the limit of f_n.
The limit (in the Hausdorff metric) of f_n(x_n) is contained in f(x), and we conclude that (f(x))≥ϵ. 
This proves that A_ϵ is open. 
By Proposition <ref> we know that f∈() is a homeomorphism if and only if f,f^-1∈ A_ϵ for all ϵ>0. 
For n∈, let U_n be the open and dense set of dynamical systems f such that f,f^-1∈ A_1/n. 
The set ∩_n∈U_n is the set of the homeomorphisms f∈().

To conclude the proof we will show that the set

    {f∈():M(f) is a Cantor space}

is a dense G_δ subset of ().
Given M∈() define 

    M=sup_x∈ M(_x(M)),

where _x(M) denotes the connected component of M containing x.
For ϵ>0 let 
    A_ϵ={f∈():M(f)<ϵ}.

It is easy to see that A_ϵ is open in (), for each ϵ>0. 
From Proposition <ref>
we have that A_ϵ is dense in ().
Define 

    A_0={f∈():M(f)=0}.

That is, f∈ A_0 if and only if M(f) is totally disconnected.
Since A_0=∩_n≥ 1 A_1/n we have that A_0 is a dense G_δ set. 

By Proposition <ref>
we know that homeomorphisms on perfect sets 
are dense in 
(). 
For ϵ>0 define

    F_ϵ={f∈(): ∃ x∈ M(f) s.t.  B_ϵ(x)={x}},

where B_ϵ(x) is the open ball in M(f). Also consider

    F_0={f∈():M(f) is perfect}.

It is easy to see that F_ϵ is closed in (). 
Consequently, ()∖ F_ϵ is open and dense in (). 
Since 

    F_0=∩_n≥ 1(()∖ F_1/n)

we have that F_0 is a dense G_δ set. 

Therefore F_0∩ A_0 is a dense G_δ set. 
Note that f∈ F_0∩ A_0 if and only if M(f) is a Cantor space.




 §.§ The Special Homeomorphism



In this section we assume that (,) is Polish and perfect. 


 Let K be a Cantor set and (,) Polish and perfect. 
 Define 
 
    (K,)={h∈ C(K,):f is injective}.

Consider _n a sequence of clopen partitions of K such that (_n)→ 0 and let 

    U_n={h∈ C(K,):h(A)∩ h(B)=∅ if A,B∈_n, A≠ B}.

It is easy to see that each U_n is open (and dense), and that 
∩_n∈U_n=(K,). 
Consequently, (K,) is a G_δ subset of C(K,), and by Theorem <ref> it is a Polish space.




 
If (,) is Polish and perfect then 

    ^SH()= {f∈(): f is conjugate to the Special Homeomorphism}

is a dense G_δ subset of ().



Let K⊂ be a fixed Cantor space.
Consider the map 

    φ(K,)× H(K)→_CH()
 
defined as φ(h,f)=h f h^-1. 
Let 
    ={f∈ H(K):f is conjugate to the Special Homeomorphism}.

Define 

    R=(K,)×.

By KeRo,AGW, we have that R is a dense G_δ subset of (K,)× H(K). 
Notice that φ(R)=^SH() and φ^-1(φ(R))=R.
Therefore, in order to finish the proof we need to check the remaining hypothesis of Corollary <ref>.

It is clear that φ is continuous and surjective. 
We will show that φ is open. 
For this purpose, fix h∈(K,) and f∈ H(K). 
Let U⊂(K,)× H(K) be an open set containing (h,f) and take ϵ>0 such that 
if _C^0(f,f')<ϵ and _C^0(h,h')<ϵ then (h',f')∈ U. 
Let g h(K)→ h(K) be the homeomorphism g=φ(h,f).
As h^-1 is uniformly continuous, there is δ>0 such that 

    if x,y∈ h(K), (x,y)<2δ then (h^-1(x),h^-1(y))<ϵ.

Suppose that j∈() is a homeomorphism of a Cantor space 
such that (g,j)<δ. 
By Proposition <ref>, consider h_1,h_2 with
l_i<δ. 
Let h'=h_1  h∈(K,) and f'=h^-1  h_1^-1  j  h_1  h∈ H(K).
Since h_1<δ we have that _C^0(h,h')<δ. 
Also

    _C^0(g,h_1^-1jh_1)=_C^0(g,h_1^-1h_2g)<2δ

and 

    _C^0(gh,h_1^-1jh_1h)<2δ.

This and (<ref>) implies that 

    _C^0(h^-1gh,h^-1h_1^-1jh_1h)<ϵ

and _C^0(f,f')<ϵ.	
Thus, (h',f')∈ U. Since 
φ(h',f')=j we conclude that φ is open and the proof ends.




The next result is an application of the densitity of the conjugacy class 
of the Special Homeomorphism. This kind of result was suggested to the author by Mauricio Achigar.



 If f M→ M is a homeomorphism of a compact manifold (M,) then 
 for all ϵ>0 there are an ϵ-dense Cantor set K⊂ M and a 
 homeomorphism g K→ K conjugate to the Special Homeomorphism 
 such that _C^0(g,f|_K)<ϵ. 




Given ϵ>0, take δ∈(0,ϵ/2) such that if 
x,x'∈ M and (x,x')<δ then (f(x),f(x'))<ϵ/2. 
Applying Theorem <ref> with =M, we have that there is g∈(M) conjugate to the Special Homeomorphism with 
 (f,g)<δ. 
 Let K=M(g). 
 Then, Remark <ref>, 
 _H(K,M)<(f,g)<δ<ϵ, where M=M(f), i.e., K is ϵ-dense in M. 
 
Since (f,g)<δ, for all x∈ K there is x'∈ M such that 
(x,x')<δ and (g(x),f(x'))<δ. 
By the uniform continuity of f, we have that (f(x),f(x'))<ϵ/2. 
Thus, 

    (g(x),f(x))≤(g(x),f(x'))+(f(x'),f(x))<δ+ϵ/2<ϵ

and _C^0(f,g)<ϵ.




§ DYNAMICS MODULO ISOMETRIC CONJUGACIES


In this section we study the equivalence relation of isometric conjugacy 
between dynamical systems in (). 
In <ref> we apply Theorem <ref> to conclude that the Special Homeomorphism 
is also generic in this setting.
In <ref> we show that our approach is topologically equivalent to <cit.>.



 §.§ Metric spaces



Given two metric spaces (X,^X) and (Y,^Y) a surjective function 
φ X→ Y is an isometry 
if ^Y(φ(x),φ(x'))=^X(x,x') for all 
x,x'∈ X. 
In this case we write X∼ Y and we say that Y is an isometric copy of X. 
We say that a metric space (X,) is homogeneous if for all x,y∈ X there is an isometry φ M→ M such that y=φ(x).

A metric space is universal if it contains an isometric copy of every 
separable
metric space. 
An example of a universal space is =l^∞, the Banach space 
of bounded real sequences a→ with the sup-norm
a_∞=sup_n∈|a_n|.
Indeed, given a separable metric space (M,ρ) with a dense sequence {c_n}_n∈ 
consider φ M→ as φ(x)=a where a_n=ρ(x,c_n)-ρ(c_n,c_1). 
The set φ(M)⊂ is an isometric copy of M.
The function φ is known as a Fréchet embedding.
Since the translations of  are isometries, we have that  is homogeneous.
The space  is complete but not separable. 
Another classical universal space is 
the Banach space of continuous functions from [0,1] to  with the sup-norm, see <cit.>.

















A metric space (,) is ultrahomogeneous if any isometry φ A→ B between two
compact subsets of , extends to an isometry φ'→.
We remark that Euclidean ^n is not universal but it is Polish, perfect and ultrahomogeneous (see <cit.>*Corollary 1).
A metric space is Urysohn universal if it is 
ultrahomogeneous, universal and Polish. 
The next result is due to Urysohn.


Up to isometry,
there is a unique Urysohn universal space. 



The ultrahomogeneous condition is usually stated for A,B finite sets, but from Huhu,Hol 
the formulation given above is equivalent. See also <cit.>*Exercise (b'), p. 82.


The Urysohn universal space will be denoted as (,). 
It is clear that  is perfect.
The interested reader may see <cit.>*p. 32 for more on ultrahomogeneous spaces.

For a metric space (,), the equivalence relation of isometry ∼ on the space ((),_H) gives rise 
to a quotient space 

    _()=()/∼.

For X∈() denote by X̃={Y∈():X∼ Y} its equivalence class.
Given X,Y∈() define 

    _H̃(X̃,Ỹ)=inf{_H(A,B):A∼ X, B∼ Y, A,B∈()}.

This is the so called Gromov-Hausdorff distance. 
As we learned in <cit.>, it was first defined by Edwards <cit.>.
The next result summarizes some standard properties. 


If (,) is complete and ultrahomogeneous then
_H̃ is a complete metric in _() 
compatible with the quotient topology.


It is interesting to remark how these metrics are related for =^n,. 
Denote by ^^n and ^ the Euclidean metric and the metric of Urysohn universal space, respectively.
In <cit.>*Theorem 2 it is shown that 

    ^_H̃(X,Y)≤^^n_H̃(X,Y)≤ c_n√(^_H̃(X,Y))

for all X,Y∈(^n). The constants c_n depends only on the dimension of ^n.
Naturally, to compute ^_H̃(X,Y) we are taking isometric copies in . 




 §.§ Isometric conjugacies


Let (,) be a metric space. 
We define an equivalence relation on () as 
f∼ g if there is an isometry φ M(f)→ M(g) such that 
φ^-1  g  φ=f. 
In this case we say that φ is an isometric conjugacy.





 Suppose that f∼ g with an isometry φ M(f)→ M(g). 
 Let ψ M(f)× M(f)→ M(g)× M(g) be the isometry ψ(x,x')=(φ(x),φ(x')). 
 We will show that ψ(f)=g. 
 We know that (x,y)∈ f iff (x,y)∈φ^-1  g  φ, 
 that is (φ(x),φ(y))∈ g, i.e. ψ(x,y)∈ g.  
 This implies that f and g are isometric.




 Two maps f,g∈ may have isometric graphs but they may not be conjugate. 
 In Figure <ref> we see an example. 
 Note that f and f^-1 are isometric sets for all f∈().
 




 §.§ The quotient space


Consider the quotient of () by isometric conjugacies 

    _()=()/∼
 with the quotient topology and 
denote by π()→_() the canonical projection. 
Recall from <ref>
that  is the Hausdorff distance in (^2).
Define 

    (f̃,g̃)=inf(f',g')

where inf is taken over all f'∼ f and g'∼ g.
It is clear that 

    (f,g)≥(f̃,g̃)

for all f,g∈().



 If (,) is ultrahomogeneous then: 
 
  
  *  is a metric compatible with the quotient topology,
  
  * it holds that π(B_r(f))=B_r(f̃) for all r>0, f∈() and π is an open map,
  
  * If in addition (,) is complete then (_(),) is complete.
 



First we show that π(B_r(f))=B_r(f̃). 
By (<ref>) we have that π(B_r(f))⊂ B_r(f̃). 
To prove the other inclusion suppose that (f̃, g̃)<r.
By definition, there are f'∼ f and g'∼ g such that (f',g')<r. 
Thus, there is an isometry φ M(f')→ M(f) conjugating f and f'. 
As  is ultrahomogeneous, there is an isometry ψ→ with ψ|_M(f')=φ. 
Let g”∈() be defined as 
 g”={(ψ(x),ψ(y))∈^2:(x,y)∈ g'}. 
Since ψ is an isometry, we have that (f,g”)=(f',g')<r and g”∼ g.
Therefore, B_r(f̃)⊂π(B_r(f)).

To prove that  is a metric, note that  is non-negative and symetric. 
The triangular inequality follows from the ultrahomogeneity of . 
If (f̃,g̃)=0 then by  
Ascoli Theorem <cit.>*p. 234 we have that f∼ g.

By (<ref>) we have that π is continuous with respect to , which implies that 
every open set U⊂_() is open in the quotient topology. 
Conversely, suppose that U is open in the quotient topology. 
Then, π^-1(U) is open in (). 
For each f∈π^-1(U) there is r>0 such that B_r(f)⊂π^-1(U). 
Thus, π(B_r(f))=B_r(f̃)⊂ U. This proves that U is open with respect to . 

To prove that _() is complete, let f̃_n∈_() be a Cauchy sequence. 
Take a subsequence f̃_n_k such that 
∑_k=1^∞(f̃_n_k,f̃_n_k+1)<∞. 
Since  is ultrahomogeneous there is g_k∈() such that 
g_k∼ f_n_k and 
(g_k,g_k+1)<2(f̃_n_k,f̃_n_k+1) for all k∈. 
In this way, g_k is a Cauchy sequence. 
Since  is complete, by Proposition <ref> 
we have that () is complete and g_k is convergent with limit g∈(). 
Since π is continuous, g̃_k=f̃_n_k converges to 
g̃. 
As f̃_n is a Cauchy sequence with a convergent subsequence, we conclude that 
it is convergent and the proof ends.




If (,) is Polish, perfect and ultrahomogeneous
then 

    ^SH_()={f̃∈_(): f  is conjugate to the Special Homeomorphism}

is a dense G_δ subset of _().



The projection π()→_() is continuous, surjective and open (Proposition <ref>).
By Theorem <ref> we have 
that 
^SH() is a dense G_δ subset of ().
Note that π(^SH())=^SH_() and π^-1(π(^SH()))=^SH().
Therefore, the result follows by Corollary <ref>.




 §.§ Arbieto-Morales metric


Given two metric spaces (X,^X) and (Y,^Y) a function 
φ X→ Y is an ϵ-isometry if 

    |^Y(φ(x),φ(x'))-^X(x,x')|<ϵ

for all x,x'∈ X and ^Y_H(φ(X),Y)<ϵ.
Let _ϵ(X,Y) be the set of ϵ-isometries from X to Y.
The C^0-Gromov-Hausdorff distance <cit.> 
between the maps f X→ X and g Y→ Y is defined as 

    [                                       (f,g)= inf{ϵ>0: 
     ∃φ∈_ϵ(X,Y)
      and ψ∈_ϵ(Y,X)
      s.t.;                                                 _C^0(g  φ,φ  f)<ϵ and _C^0(ψ  g,f  ψ)<ϵ}. ]

Recall _C^0 from (<ref>). 
Note that  is defined for maps and not for arbitrary relations. 

[<cit.>*Theorem 1]

 The C^0-Gromov-Hausdorff distance satisfies the following properties: 
 
  
  * (f,g)=0 if and only if f∼ g, 
  
  * (f,g)=(g,f) and 
  
  * (f,g)≤ 2((f,h)+(h,g)),
 
 for all continuous maps f,g,h of compact metric spaces. 
 In <cit.> a function as  satisfying these conditions is called pseudo quasi-distance with coefficient 2 (in the 
 triangular inequality). 
 Notice that  is invariant by isometric conjugacies, i.e., if f∼ f' and g∼ g' then 
 (f,g)=(f',g').




 If φ X→ Y is an ϵ-isometry 
 then there is a metric ^Z in the disjoint union Z=X∪ Y such that 
 ^Z(x,x')=_X(x,x') for all x,x'∈ X, 
 ^Z(y,y')=_Y(y,y') for all y,y'∈ Y, 
 ^Z_H(X,Y)≤ 2ϵ and 
 (x,φ(x))=ϵ for all x∈ X.



Since in <cit.> the authors consider dynamical systems on all the compact metric spaces, 
the next proof is only given for =.



The restriction of  to continuous maps in _() is 
topologically equivalent to .



First we will show that 

    (f,g)≤ 2(f̃,g̃)

Suppose that (f̃,g̃)<δ. 
Taking isometric conjugacies we can assume that (f,g)<δ. 
For each x∈ X=M(f) there is y∈ Y=M(g) such that 
(x,y)<δ and (f(x),g(y))<δ. 
This defines a function φ X→ Y as φ (x)=y 
satisfying (x,φ (x))<δ and (f(x),g(φ (x)))<δ for all 
x∈ X. 

We will show that φ∈_2δ(X,Y).
Since (f,g)<δ, given y∈ Y there is x∈ X such that 
(x,y)<δ and (f(x),g(y))<δ. 
Then 
    (y,φ (x))≤(y,x)+(x,φ (x))<δ+δ=2δ.

This proves that _H(φ (X),Y)<2δ. 
Given x,x'∈ X note that 

    |(φ (x),φ (x'))-(x,x')|≤(φ (x),x)+(x',φ (x'))<2δ.

Thus, φ∈_2δ(X,Y).
Given x∈ X we have that

    (g(φ (x)),φ (f(x)))≤(g(φ (x)),f(x))+(f(x),φ (f(x)))<2δ.

This proves that _C^0(g  φ ,φ   f)<2δ. 
In a similar way we can define the required ψ∈_2δ(Y,X). 
This proves (<ref>). 


Given a map g∈() and ϵ>0 take 
δ∈ (0,ϵ/3) such that if u,v∈ M(g) and (u,v)<δ then (g(u),g(v))<ϵ/3.
Suppose that (f,g)<δ.
By definition, there are
φ∈_δ(X,Y)
and
ψ∈_δ(Y,X)
such that _C^0(g  φ,φ  f)<δ and 
 _C^0(ψ  g,f  ψ)<δ.
By Lemma <ref>, we can (in addition) assume that 
(x,φ(x))<δ
for all x∈ X.
Then

    (f(x),g(φ(x)))≤(f(x),φ(f(x)))+(φ(f(x)),g(φ(x)))
     <δ+δ=2δ.

Since ^Y_H(φ(M(f),M(g))<δ, 
for all y∈ Y there is x∈ X such that (y,φ(x))<δ. 
Then

    (x,y)≤(x,φ(x))+(φ(x),y)<δ+δ=2δ.

Also

    (f(x),g(y)) ≤   (f(x),φ(f(x)))+(φ(f(x)),g(φ(x)))
       +(g(φ(x)),g(y))
    ≤   δ+δ+ϵ/3<ϵ.

As explained in Remark <ref>, this proves that (f,g)<ϵ.
















































































































































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