Groups of smooth diffeomorphisms of Cantor sets embedded in a line
Dominique Malicet Emmanuel Militon






Groups of smooth diffeomorphisms of Cantor sets embedded in a line
    Dominique Malicet, Emmanuel Militon
    December 30, 2023
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Let K be a Cantor set embedded in the real line ℝ. Following Funar and Neretin, we define the diffeomorphism group of K as the group of homeomorphisms of K which locally look like a diffeomorphism between two intervals of ℝ. Higman-Thompson's groups V_n appear as subgroups of such groups. In this article, we prove some properties of this group. First, we study the Burnside problem in this group and we prove that any finitely generated subgroup consisting of finite order elements is finite. This property was already proved by Rover in the case of the groups V_n. We also prove that any finitely generated subgroup H without free subsemigroup on two generators is virtually abelian. The corresponding result for the groups V_n was unknown to our knowledge. As a consequence, those groups do not contain nilpotent groups which are not virtually abelian.  





§ INTRODUCTION


We call Cantor set any compact totally disconnected set K such that any point of K is an accumulation point.

When studying the dynamics of an action of a group G on a closed surface, it is convenient to look at minimal subsets of the action. Recall that a G-invariant closed nonempty subset K of our surface is a minimal subset for the action of G on our surface if every orbit of points in K is dense in K. This is equivalent to saying that K is minimal for the inclusion relation among G-invariant closed nonempty subsets. Zorn's lemma ensures that such subsets always exist. A typical case which can occur is the case where this minimal subset turns out to be a Cantor set K.

In this article, we will restrict to the case where our Cantor set is embedded in a line, i.e. embedded in a one-dimensional submanifold diffeomorphic to ℝ.

We will give two equivalent definitions of the group we are interested in. 

 
Let r be an integer greater than or equal to 1 or + ∞. Let K be a Cantor set contained in a line L which is C^r-embedded in a manifold M with dim(M) ≥ 2. The group of C^r-diffeomorphisms of K, denoted 𝔡𝔦𝔣𝔣^r(K), is the group of restrictions to K of C^r-diffeomorphisms f of M such that f(K)=K.



The isomorphism class of this group is independent of the embedding of the line L in M and of the manifold M, as long as dim(M) ≥ 2. This is a consequence of the second definition below, which is independent of L and M, and of the equivalence between the two definitions. However, if we look at the same group in the case where M is a circle, we only obtain a strict subgroup of the latter group as elements of the group have to preserve a cyclic order.


 If G is a group acting on a manifold M by C^r-diffeomorphisms with such a Cantor set K as a minimal set, then there exists a nontrivial morphism G →𝔡𝔦𝔣𝔣^r(K). If we understand well the group 𝔡𝔦𝔣𝔣^r(K), we can obtain information on which group can act on M with such a minimal invariant set.


We now give a second definition of our group: the group 𝔡𝔦𝔣𝔣^r(K) is the group of homeomorphisms of K which locally coincide with a C^r-diffeomorphism of an open interval of ℝ (see precise definition below). We will prove the equivalence between the two definitions in Section 2 (see Proposition <ref> for a precise statement).

 
Let K be a Cantor set contained in ℝ.  The group 𝔡𝔦𝔣𝔣^r(K) is the group of homeomorphisms f of K such that, for any point x in K, there exists an open interval I of ℝ containing x and a C^r-diffeomorphism f̃: I →f̃(I) such that f̃_|I ∩ K =f_|I ∩ K.


 we can adapt this second definition to other kinds of regularity. For instance, in this article, we will denote by 𝔡𝔦𝔣𝔣^1+Lip(K) the group of homeomorphisms of K which locally coincide with a C^1+Lip-diffeomorphism between two intervals of ℝ, that is a C^1-diffeomorphism f̃ such that log(f̃') is Lipschitz continuous.


The generalizations of those two definitions to the case r=0 are not equivalent, but we will use this second definition to define the group 𝔡𝔦𝔣𝔣^0(K) : it is the group of homeomorphisms of K which coincide locally with a homeomorphism between two intervals of ℝ.

In the article <cit.>, Funar and Neretin have computed these groups in many cases and provided examples of Cantor subsets for which these groups are trivial. They have in particular computed this group in the case where K is the standard ternary Cantor subset, which we call K_2.  Let us recall first the construction of K_2. Start with the segment [0,1]. Cut this interval into three equal pieces [0,1/3], [1/3,2/3] and [2/3, 1]. Now, throw out the middle segment: we obtain a new compact set [0,1/3] ∪  [2/3, 1]. Now remove the middle third of each of these intervals : we obtain the compact subset [0,1/9] ∪ [2/9,1/3] ∪ [2/3,7/9] ∪ [8/9,1]. Then repeat the procedure for each of the obtained intervals. We obtain a decreasing sequence of compact subsets : the intersection of this sequence is the subset K_2 (see Figure <ref>). More generally, if we remove n-1 evenly spaced intervals at each step instead of one, we obtain a Cantor set which we denote by K_n.



For convenience, we will call interval of K_2 the intersection of K_2 with one of the intervals appearing in this construction (this corresponds to cylinders in symbolic dynamics). The set of intervals of K_2 is a basis of the topology of K_2. This basis consists of clopen subsets, i.e. subsets which are closed and open. The set of intervals of K_2 is in bijective correspondence with the set of vertices of a rooted binary tree (see Figure <ref>).

We now give a procedure to construct elements of 𝔡𝔦𝔣𝔣^∞(K_2) (see Figure <ref> for an example of a diffeomorphism of K_2).

Step 1: Choose two finite partitions of K_2 by intervals of K_2 which have the same cardinality. We denote these partitions by { I_i,  1 ≤ i ≤ m } and { J_j,  1 ≤ j ≤ m }.


Step 2: Take a permutation σ∈𝒮_m. This enables to define an element f of the group 𝔡𝔦𝔣𝔣^∞(K_2) in the following way. The restriction of f to I_i is the unique orientation preserving affine map which sends the interval I_i onto the interval J_j. Such a map sends I_i∩ K_2 to J_j∩ K_2.

Step 3: Chose a subset A of the second partition { J_j,  1 ≤ j ≤ m } and flip each of the intervals in A, i.e. compose the diffeomorphism obtained in Step 2 with the diffeomorphism whose restriction to J_j is the identity if J_j∉ A and is the symmetry with respect to the midpoint of J_j if J_j∈ A.




Of course, we can define a similar algorithm to construct elements of 𝔡𝔦𝔣𝔣^r(K_n).


[Funar-Neretin]
For any r ≥ 1 or for r=∞, the group 𝔡𝔦𝔣𝔣^r(K_n) is the group consisting of elements constructed following the above procedure.

 
Strictly speaking, Funar and Neretin proved only the case n=2 but their proof also applies in the case of the Cantor sets K_n for n>2. Notice that this group does not depend on the regularity r ≥ 1 : this seems to be a consequence of the “regular” shape of this Cantor set.

Inside 𝔡𝔦𝔣𝔣^r(K_n), there is a natural subgroup : the subgroup consisting of elements which are constructed using only the first two steps of the above procedure. This subgroup is the well-known Higman-Thompson group V_n (see the introduction of the article <cit.> for more background on Higman-Thompson groups).

In this article, we prove some general results about the groups diff^r(K). These theorems suggest that those groups share common features with rank one simple Lie groups. In what follows, we fix a Cantor set K embedded in ℝ.



 §.§ Burnside property


 A group is periodic if any of its elements has finite order.


In 1902, Burnside asked whether there exist finitely generated periodic groups which are infinite (see <cit.>). Nine years later, Schur managed to prove that any finitely generated periodic group which is a subgroup of GL_n(ℂ) has to be finite. Much later, in the 60's, Golod and Shafarevich proved in <cit.> that there exist infinite finitely generated periodic groups. Many more examples were constructed later.


 
Any periodic finitely generated subgroup of 𝔡𝔦𝔣𝔣^1+Lip(K) is finite.


We are not able to lower the regularity to C^1 in this theorem for the moment. However, observe that the same theorem is false in the case of the group of homeomorphisms of a Cantor set as any finitely generated group is a subgroup of this group. To see this, observe that any infinite countable group G acts continuously and faithfully on { 0,1 }^G, which is a Cantor set for the product topology and recall that any two Cantor sets are homeomorphic.

We prove Theorem <ref> in Section 3 of this article.

As a consequence, as the Higman-Thompson groups V_n are subgroups of groups of C^∞-diffeomorphisms of Cantor sets, we have a new proof of the following theorem by Rover (see <cit.>).

[Rover]
Let n ≥ 2. Any finitely generated periodic subgroup of V_n is finite.


The proof of Theorem <ref> relies on an adaptation of standard 1-dimensional tools from dynamical systems, namely Sacksteder's theorem and the Thurston stability theorem.



 §.§ Subgroups without free subsemigroups on two generators


In this section, we will consider finitely generated subgroups of the group 𝔡𝔦𝔣𝔣^1+Lip(K) without free subsemigroups on two generators. This includes all finitely generated groups of subexponential growth. In particular, this includes
nilpotent finitely generated subgroups, which we define below. 

Fix a group G. If H and H' are subgroups of G, we define [H,H'] as the subgroup of G generated by elements of the form [h,h']=hh'h^-1h'^-1, where h ∈ H and h' ∈ H'.

We define a sequence (G_n)_n ≥ 1 of subgroups of G by the following relations:

    G_1=G


    ∀ n ≥ 1, G_n+1=[G_n,G].


A group G is said to be nilpotent if there exists n ≥ 1 such that G_n={ 1 }. If the group G is nilpotent and nontrivial, its degree is the smallest integer n such that G_n is nontrivial and G_n+1 is trivial. A typical example of finitely generated nilpotent group is the Heisenberg group ℋ with integer coefficient. This group ℋ is the group of upper unitriangular 3 × 3 matrices with integer coefficients.

The following theorem states that subgroups of 𝔡𝔦𝔣𝔣^1+Lip(K) without free subsemigroups on two generators are close to being abelian.

 
Let G be a finitely generated subgroup of 𝔡𝔦𝔣𝔣^1+Lip(K) without free subsemigroups on two generators. Then the group G is virtually abelian.


Recall that, by definition, a group is virtually abelian if it contains a finite-index subgroup which is abelian. We cannot hope for a better conclusion in Theorem <ref>. Indeed, take K=K_2 the standard ternary Cantor set. Let S be the group of diffeomorphisms of K_2 which permute the intervals [0,1/9], [2/9,1/3], [2/3,7/9], [8/9,1]  and whose restriction to each of these intervals is orientation-preserving and affine. The group S is isomorphic to the finite group 𝒮_4, the symmetric group on 4 elements. Take also an infinite order element f of 𝔡𝔦𝔣𝔣^∞(K_2) which is supported in [0,1/9]∩ K_2, meaning that it pointwise fixes the points outside [0,1/9]∩ K_2. Then the subgroup of 𝔡𝔦𝔣𝔣^∞(K_2) generated by f and S is virtually abelian : it contains the group ℤ^4 as a finite index subgroup. But it is not abelian. Notice that, with this kind of construction, we can obtain any virtually abelian group as a subgroup of a group of diffeomorphisms of a Cantor set. 

 Observe that finitely generated groups of subexponential growth contain no free subsemigroup on two generators. Hence subgroups of 𝔡𝔦𝔣𝔣^1+Lip(K) of subexponential growth are virtually abelian. 

 Notice that Theorem <ref> implies Theorem <ref> as periodic groups contain no free subsemigroup on two generators and finitely generated abelian periodic groups are known to be finite. However, we use Theorem <ref> (and even a stronger version of it which is Proposition <ref>) to prove Theorem <ref>. That is why we will first prove Theorem <ref> in this article.


 In Theorem <ref>, the regularity cannot be lowered to a C^1 regularity. Indeed, Farb and Franks proved in <cit.> that any torsion-free finitely generated nilpotent group is a subgroup of the group of homeomorphisms of the compact interval [0,1]. For each nilpotent subgroup that they construct, it is possible to find a Cantor subset which is invariant. Hence any torsion-free finitely generated nilpotent group is a subgroup of the group of C^1 diffeomorphisms of some Cantor set.


If we only assumed that our group contained no free subgroups, we could not prove that our group is virtually abelian. Indeed, the Thompson group F is contained in V_2 and is hence a subgroup of the group of diffeomorphisms of the standard ternary Cantor set. But this group is finitely generated, contains no free subgroup on two generators and is not abelian. For more information about the group F and references for proofs of these results, see Section 1.5 of <cit.>. The best we can hope for finitely generated subgroups without a free subgroup on two generators is that they have a finite orbit. We will explore this question in a forthcoming article.

We prove Theorem <ref> in Section 4 of this article.

Observe that the corresponding statement for Higman-Thompson's groups V_n was unknown, as far as we know. We state it as a corollary.


Let n ≥ 2. Any subgroup of the group V_n without free subsemigroup on two generators is virtually abelian.


As the derived subgroup of a finitely generated nilpotent group is finitely generated and has no free subsemigroups on two generators, we obtain the following statement.

 
Any finitely generated nilpotent subgroup of 𝔡𝔦𝔣𝔣^1+Lip(K) is virtually abelian.


In particular, there is no Heisenberg group with integer coefficients as subgroup of 𝔡𝔦𝔣𝔣^r(K) for r ≥ 2. 

Higman-Thompson groups V_n were already known to satisfy this corollary by a result by Bleak, Bowman, Gordon Lynch, Graham, Hughes, Matucci and Sapir <cit.> about distorted cyclic subgroups.

As a consequence of this corollary, we obtain the following statement, which is related to the Zimmer conjecture.


Let r ≥ 2 and Γ be a finite index subgroup of SL_n(ℤ) (or any almost simple group which contains a nonabelian nilpotent group whose derived subgroup is infinite). Any morphism Γ→𝔡𝔦𝔣𝔣^r(K) has a finite image. 



As the group Γ contains a nonabelian nilpotent group whose derived subgroup is infinite and by Corollary <ref>, any morphism Γ→𝔡𝔦𝔣𝔣^r(K) has an infinite kernel. But the group Γ is almost simple, which means that any normal subgroup of Γ is either finite or a finite index subgroup of Γ. This implies that the kernel of a morphism Γ→𝔡𝔦𝔣𝔣^r(K) is a finite index subgroup of Γ: the image of this morphism is finite.


Navas proved in <cit.> that subgroups of C^1+Lip-diffeomorphisms of the half-line without free subsemigroups on two generators are abelian. To prove Theorem <ref>, we try to adapt his techniques. However, this adaptation is not easy :  any diffeomorphism of the half-line preserves the natural order on the half-line whereas, in our case, the diffeomorphisms do not a priori preserve any order on our Cantor set. Moreover, Navas is able to lower the regularity to C^1+bv whereas we have to stick to the C^1+Lip regularity.

Acknowledgement: The second author wants to thank Isabelle Liousse for a conversation which was the origin of this article.



§ EQUIVALENCE BETWEEN THE TWO DEFINITIONS




Let r ≥ 1 be an integer or r=∞. In this section, we prove the equivalence between Definitions <ref> and <ref> of 𝔡𝔦𝔣𝔣^r(K). 

Let M be a differential manifold with dim(M) ≥ 2. Let L be a real line which is C^r-embedded in M. We identify the line L with the real line ℝ. Denote by 𝔡𝔦𝔣𝔣^r(K)_1 the group of C^r-diffeomorphisms of K according to Definition <ref>, that is the group of restrictions to K of C^r-diffeomorphisms of M which preserve K. Denote by 𝔡𝔦𝔣𝔣^r(K)_2 the group of C^r-diffeomorphisms of K according to Definition <ref>, that is the group of homeomorphism of K ⊂ L=ℝ which locally coincide with C^r-diffeomorphisms between two intervals of ℝ.

We prove the following statement.

 
The map

    [ 𝔡𝔦𝔣𝔣^r(K)_1           ↦ 𝔡𝔦𝔣𝔣^r(K)_2 ]

which to a homeomorphism f of K in 𝔡𝔦𝔣𝔣^r(K)_1, associates f, is well-defined, as it takes values in 𝔡𝔦𝔣𝔣^r(K)_2, and is onto.


This amounts to showing that, if g is a C^r-diffeomorphism of M which preserves K, then the restriction g_|K belongs to 𝔡𝔦𝔣𝔣^r(K)_2 and that, if we denote by f a homeomorphism of K in 𝔡𝔦𝔣𝔣^r(K)_2, then there exists a C^r-diffeomorphism g of M such that g_|K=f.


Let g be a C^r-diffeomorphism of M which preserves K. Take a chart φ defined on an open subset U of M onto ℝ^dim(M) such that K ⊂ U and φ(L ∩ U)= ℝ×{ 0 }^dim(M)-1. For instance, you can take as open subset U a tubular neighbourhood of a segment of L which contains K. Finally, denote by Π the “projection on L” φ^-1∘ p_1∘φ, where p_1: ℝ^dim(M)→ℝ is the projection on the first coordinate.

Let x_0∈ K. We will prove that the differential of Π∘ g_|L at the point x_0 does not vanish. Hence, by the inverse function theorem, the map Π∘ g_|L is a C^r-diffeomorphism of an open neighbourhood I of the point x_0 in L onto an open neighbourhood J of the point g(x_0) in L. Moreover, g(I ∩ K) ⊂ K ⊂ U ∩ L so that Π∘ g_|I ∩ K=g_| I ∩ K and the map g_|K satisfies Definition <ref>.

It remains to show that d(Π∘ g_|L)(x_0) ≠ 0. As the compact subset K is a Cantor set, there exists a sequence (y_n)_n ∈ℕ of elements of K ∖{ x_0} which converges to the point x_0. Observe that, for any n, the point g(y_n) belongs to K ⊂ L ∩ U. Hence the partial derivative 

    ∂/∂ x_1( φ∘ g ∘φ^-1)(φ(x_0))

belongs to ℝ×{ 0 } and does not vanish as g is a diffeomorphism. Hence  

    ∂/∂ x_1( p_1∘φ∘ g ∘φ^-1)(φ(x_0)) ≠ 0

and d(Π∘ g_|L)(x_0) ≠ 0.

Now, let f be a homeomorphism of K which satisfies Definition <ref>. By compactness of K, there exists a partition (K_i)_1 ≤ i ≤ l of K which satisfies the following conditions.


  * Each subset K_i is a clopen subset of K.

  * For any index 1 ≤ i ≤ l, there exists an open interval I_i of ℝ such that K_i= I_i∩ K.

  * The intervals I_i are pairwise disjoint.

  * For any 1 ≤ i ≤ l, there exists a diffeomorphism f̃_i defined on I_i onto an open interval J_i such that f̃_i|K_i=f_|K_i.

  * The intervals J_i are pairwise disjoint.

To obtain such a partition, take first any cover (K'_i)_1 ≤ i ≤ l' by clopen subsets such that conditions 1., 2. and 4. are satisfied. Take finite partitions of the clopen sets K'_i∖∪_j<i K'_j and throw away empty sets to obtain a partition such that conditions 1., 2. and 4. are satisfied. We cannot take the sets K'_i∖∪_j <i K'_j directly as those sets might not satisfy the second condition. Finally, shrink the obtained intervals I_i in such a way that the two remaining conditions hold.

Take closed intervals I'_i⊂ I_i in such a way that the compact set K_i is contained in the interior of the interval I'_i and let J'_i= f̃_i(I'_i). Finally, use the isotopy extension property (Theorem 3.1 p. 185 in <cit.>) to extend the map

    [          f̃: ⋃_i=1^l I'_i            → ⋃_i=1^l J'_i;                   x ∈ I_i            ↦      f̃_i(x) ]

to a diffeomorphism g in Diff^r(M). In <cit.>, the isotopy extension property is stated only for one disk but, with an induction, it is not difficult to prove this property for a union of disjoint closed disks (here closed intervals, which are one-dimensional disks).




§ BURNSIDE PROPERTY


In this section, we prove Theorem <ref>. To prove it, we will use Definition <ref> of the group 𝔡𝔦𝔣𝔣^1+Lip(K). Hence we see our Cantor set K as a subset of the real line L=ℝ.

Before starting the proof of Theorem <ref>, we need some definitions.

First, elements of 𝔡𝔦𝔣𝔣^1(K) have a well-defined derivative at each point of K. Indeed, fix a diffeomorphism f in 𝔡𝔦𝔣𝔣^1(K) and a point x_0 of K. Then there exists an open interval I of ℝ which contains the point x_0 and a C^1-diffeomorphism f̃: I →f̃(I) such that f_I ∩ K=f̃_I ∩ K. Then the derivative f̃'(x_0) does not depend on the chosen extension f̃. Indeed,

    f̃'(x_0)= lim_[ x → x_0;   x ∈ K ]f(x)-f(x_0)/x-x_0

and the right-hand side of this equality depends only on f. We call this number the derivative of f at x_0 and we denote it by f'(x_0).

As we can define the notion of derivative for the elements of our group, we also have a notion of hyperbolic fixed point which is recalled in the following definition.


Let f be a diffeomorphism in 𝔡𝔦𝔣𝔣^1(K). Let x_0 be a point of K. We say that the point x_0 is a hyperbolic fixed point for f if the following hold.


  * f(x_0)=x_0.

  * |f'(x_0)| ≠ 1.



Observe that any diffeomorphism f in 𝔡𝔦𝔣𝔣^1(K) with a hyperbolic fixed point x_0 is an infinite order element as the sequence ((f^n)'(x_0))_n ≥ 0, which is a geometric sequence, is infinite.

The proof of Theorem <ref> relies on the following proposition, which is a consequence of a theorem by Sacksteder.

 
Let G be a finitely generated subgroup of 𝔡𝔦𝔣𝔣^1+Lip(K). Then one of the following properties holds.


  * The group G contains an element with a hyperbolic fixed point. 

  * Any minimal invariant subset for the action of G on K is finite.






 §.§ Proof of Proposition <ref>


Before proving the proposition, we have to recall the definition of a pseudogroup of diffeomorphisms of the real line ℝ.

 
A set 𝒢 of C^r-diffeomorphisms  g: dom(g) →ran(g) between two open subsets dom(g) and ran(g) of ℝ is called a pseudogroup of C^r-diffeomorphisms of ℝ if


  * The set 𝒢 is stable under composition, that is, if two elements g and h belong to mathcalG with ran(h) ⊂dom(g), then the composition gh of those elements belongs to 𝒢.

  * The set 𝒢 is stable under inverses, that is, for any element g in 𝒢, its inverse g^-1 belongs to 𝒢.

  * The identity of ℝ belongs to 𝒢.

  * The set 𝒢 is stable under restrictions, that is, if g is an element of 𝒢 and A is an open subset of dom(g) then the diffeomorphism g_|A:A → g(A) belongs to 𝒢.



Let 𝒢 be a pseudogroup of diffeomorphisms of ℝ. A subset A of ℝ is invariant under 𝒢 if, for any point x of A and any element g of 𝒢, we have

    x ∈ran(g) ⇒ g(x) ∈ A.

A compact subset A of ℝ is a minimal invariant set for the action of 𝒢 if the set A is nonempty and invariant under 𝒢 and minimal for the inclusion relation among invariant nonempty compact subsets of ℝ.
A subset 𝒮 of 𝒢 is called a generating set of 𝒢 if any element g of 𝒢 is the restriction of a product of elements of 𝒮. We say that a pseudogroup 𝒢 is finitely-generated if it admits a finite generating set.


Proposition <ref> is a consequence of the following theorem by Sacksteder (see <cit.> for a proof).

 
Let 𝒢 be a pseudogroup of C^1+Lip-diffeomorphisms of ℝ. Suppose that the following conditions hold.


  * The pseudogroup 𝒢 has a Cantor set K' as a minimal set of its action on the line. 

  * There exists a finite set 𝒮 of elements of 𝒢 which generates 𝒢 whose domains have a compact intersection with K'.

Then the pseudogroup 𝒢 contains an element with a hyperbolic fixed point.


To apply this theorem, we need to connect the action of our group G to a pseudogroup. This is achived by the following proposition, which is roughly a consequence of the equivalence between the two definitions of the group 𝔡𝔦𝔣𝔣^r(K).

 
Let G be a finitely generated subgroup of  𝔡𝔦𝔣𝔣^1+Lip(K). Then there exists a pseudogroup 𝒢 of C^1+Lip-diffemorphisms of the line L such that


  * For any diffeomorphism h of the pseudogroup 𝒢 and for any point x of dom(h) ∩ K, there exists an open neighbourhood O of x and an element g of the group G such that h_|O ∩ K=g_|O ∩ K.

  * For any element g in G and any point x in K, there exists an open neighbourhood O of x and an element h of 𝒢 such that O ⊂dom(h) and g_|O ∩ K=h_|O ∩ K.

  * There exists a finite generating set 𝒮 of 𝒢 such that, for any element h in 𝒮, the set dom(h) ∩ K is compact.



In particular, the pseudogroup 𝒢 preserves K and, for any point x of K, the orbit of x under the action of G is also the orbit of x under the action of 𝒢. The last property will enable us to apply Sacksteder's theorem. Before proving the above proposition, we use it to prove Proposition <ref>.


Denote by 𝒢 the pseudogroup associated to G as in Proposition <ref>. Let K_min⊂ K be a minimal invariant set for the action of G on the Cantor set K. Then the set K_min is a minimal invariant set for the action of the pseudogroup 𝒢. Suppose that the set K_min is infinite. Then the set K'_min of accumulation points of K_min is closed and nonempty. As the set K'_min is also G-invariant, by minimality of K_min, we have K'_min=K_min and the set K_min is a Cantor set. By Theorem <ref>, there exist x ∈ K_min⊂ K and an element h ∈𝒢 such that x ∈dom(h) and x is a hyperbolic fixed point for h. Hence, by Proposition <ref>, some element of G has a hyperbolic fixed point.





Let g be a diffeomorphism in 𝔡𝔦𝔣𝔣^1+ Lip(K). By Definition <ref>, there exists a diffeomorphism ĝ from an open neighbourhood O_g of K in L to another neighbourhood of K in L. For each connected component O_i,g of O_g (which is an open interval with compact intersection with K), let γ_i,g= ĝ_|O_i,g.
If S is a symmetric finite generating set of G, let 𝒮 be the finite set consisting of diffeomorphisms of the form γ_i,s, where s ∈ S. 

We claim that, if 𝒢 denotes the pseudogroup generated by 𝒮, then 𝒢 satisfies the desired properties.

Let us prove it by induction on word length. Fix x ∈ K. For any element g in the group G, let us denote by l_S(g) the minimal number of factors required to write g as a product of elements of S. Let us prove the following statement by induction on n: for any element g ∈ G with l_S(g) = n, there exists an element h of 𝒢 such that g=h_|K on a neighbourhood of x in K.

For n=1, this property holds by construction of 𝒮. Suppose this property is true for some n and let us prove it for n+1. Let g ∈ G and suppose l_S(g)=n+1. Write g=sg_1, where l_S(g_1)=n and s belongs to S. By induction hypothesis, there exists h ∈𝒢 such that g_1=h on a neighbourhood of x. Let O_i,s be the connected component of O_s which contains g_1(x). Then g=γ_i,sh_|K on a neighbourhood of x in K. This completes the induction.

For any element h in 𝒢 and any point x in dom(h), we denote by l_𝒮,x(h) the minimal number of factors required to write h as a product of elements of 𝒮 on a neighbourhood of x. To finish the proof of Proposition <ref>, it suffices to prove the following statement by induction on n. For any element h in 𝒢 and for any point x in dom(h) ∩ K, if l_𝒮,x(g) ≤ n, then there exists an element g of G such that g=h_|K on a neighbourhood of x in K. This induction is straightforward to carry out and is left to the reader.




 §.§ End of the proof of Theorem <ref>


Let G be a finitely generated periodic subgroup of 𝔡𝔦𝔣𝔣^1+Lip(K). Let us prove that G is finite. We will use the following lemma.

 
For any point x in K, there exists a G-invariant clopen neighbourhood U of x such that the action of G on U factors through a finite group action.


Before proving this lemma, we use it to prove Theorem <ref>.


By Lemma <ref>, there exists a cover (U_1,U_2, …, U_r) of K by G-invariant clopen sets on which the action of G factors through a finite group action. Changing U_i to 

    U_i- ⋃_1 ≤ j ≤ i-1 U_j
 if necessary, we can suppose that the sets U_j are pairwise disjoint. For any i, we denote by G_i the group of restrictions to U_i of elements of G. The groups G_i are finite and the restriction maps define a morphism 

    G →∏ _i=1^r G_i.

This morphism is one-to-one as the open sets U_i cover K. Hence G is finite.


Now, we prove Lemma <ref>. This lemma will be a consequence of the following lemma which will be proved afterwards.

 
For any minimal invariant subset K_min⊂ K for the action of G on K, the set K_min is finite and there exists a G-invariant clopen neighbourhood U of K_min on which the action factors through a finite group action.



Take the closure G.x of the orbit of x under the group G. Take a minimal set K_min of the action of G on the compact set G.x. Apply Lemma <ref> to find a G-invariant clopen neighbourhood U of K_min on which the action factors through a finite group action. As K_min⊂G.x, there exists g in G such that g(x) ∈ U. As U is G-invariant, the point x belongs to U and the lemma is proved.


We now prove Lemma <ref>.

 As elements of G are finite order elements, no element of G has a hyperbolic fixed point on K. By Proposition <ref>, the compact set K_min has to be a finite set. Let G_2 be the subgroup of G consisting of diffeomorphisms which pointwise fix K_min. It is a finite index subgroup of G and it is finitely generated as a finite index subgroup of a finitely generated group. Notice that the derivative of any diffeomorphism in G_2 at each point of K_min is either 1 or -1 as we observed that the group G contained no elements with a hyperbolic fixed point. Take the subgroup G_1 of G_2 consisting of elements whose derivative at each point of K_min is 1: it is still a finite index (finitely generated) normal subgroup of G.



  Fix a point x in K_min.  There exists a G_1-invariant clopen neighbourhood of x on which G_1 acts trivially.


Suppose this claim holds and let us see how to finish the proof of Lemma <ref>. 

By the claim, there exists a G_1-invariant clopen neighbourhood U' of x on which G_1 acts trivially. Observe that the open set

    U= ⋃_g ∈ G g(U')

is a G-invariant clopen neighbourhood of K_min. As G_1 is a normal subgroup of G, it acts trivially on U. As G_1 is a finite index subgroup of G, the action of G on U factors through a finite group action. It remains to prve the claim.

 Fix a finite generating set S of G_1. By Definition <ref>, there exists an open interval I of ℝ which contains x such that, for any element s of the generating set S, there exists a diffeomorphism s̃ : I →s̃(I) such that s_|I ∩ K=s̃_|I ∩ K. As the derivative of the diffeomorphism s̃ at x is 1>0, the diffeomorphism s̃ is orientation preserving.

We claim that any element of G_1 pointwise fixes I ∩ K. Indeed, otherwise, there would exist a diffeomorphism s in the generating set S and a point y in I ∩ K such that s(y) ≠ y. Replacing s with s^-1 if necessary, we can suppose that s(y) lies in the interval of ℝ delimited by x and y. As s̃ is an orientation-preserving diffeomorphism between intervals of ℝ, it is strictly increasing and the sequence (s^n(y))_n ≥ 0=(s̃^n(y))_n ≥ 0 is infinite. This is a contradiction as elements of G_1 are finite order elements.






 §.§ A generalization of Theorem <ref>


In the rest of the article, we will need the following generalization of Theorem <ref>.

 
Let F be a closed subset of K and G be a subgroup of 𝔡𝔦𝔣𝔣^1+Lip(K) which consists of elements which preserve F. Denote by G(F) the group of restrictions to F of elements of G. Suppose the group G(F) is periodic. Then


  * The group G(F) is finite.

  * Let G_1 be the subgroup of G consisting of elements which pointwise fix F and have a positive derivative at each point of F. Then G_1 is a finite index subgroup of G. 



Observe that the second conclusion of this proposition implies the first one. Indeed, the second conclusion implies that the subgroup H of G consisting of diffeomorphisms which pointwise fix F has a finite index and the group G(F) is isomorphic to the quotient of G by the subgroup H. However, in the following proof, we will first prove the first conclusion and then use this first conclusion to obtain the second conclusion.

As the proof of this proposition is sometimes really similar to the proof of Theorem <ref>, we will skip some details.



First, let us prove by contradiction that any minimal invariant set for the action of G(F) on F is finite. Suppose the action of the group G(F) on F has an infinite minimal invariant subset K_1⊂ F. Then the set K_1 has to be a Cantor set. Hence, by Proposition <ref>, the action of G(F) on K_1 has a hyperbolic fixed point. It is impossible as the group G(F) consists of finite order elements by hypothesis.

We then need the following claim.


The action of the group G(F) on F has only finite orbits.



Suppose there exists a point p of F such that the orbit G.p is infinite. Take a minimal invariant set K_2⊂G.p for the action of G(F) on F. We just saw that the set K_2 has to be finite. Take the finite index subgroup G_2 of G(F) consisting of elements which pointwise fix the finite set K_2 and have a positive derivative at each point of K_2. As in the proof of Theorem <ref>, we can prove that the group G_2 has to pointwise fix a neighbourhood of K_2 in F. This is not possible as K_2 is accumulated by an infinite orbit under G.


Hence any orbit of the action of G(F) on F is finite. As in the proof of Theorem <ref> and as the group G(F) is finitely generated, we obtain the following property. If a finitely generated subgroup of G(F) pointwise fixes a point of F, then it pointwise fixes a neighbourhood of this point: otherwise, the group G(F) would have an infinite orbit. Hence, using the compactness of F, we deduce that the group G(F) is finite.

Let us prove the second point of the proposition. As the group G(F) is finite, the subgroup G_3 of G consisting of elements which pointwise fix F is a finite index subgroup of G : it is the kernel of the restriction morphism G → G(F). Hence it suffices to prove that the group G_1 is a finite index subgroup of G_3.

Observe that, as elements of G_3 pointwise fix F, the derivative of any element of G_3 at each accumulation point of F is equal to one. Let us denote by F' the set of accumulation points of F. As the group G_3 is finitely generated, there exists a neighbourhood U of F' such that the derivative of any element of G_3 is positive on U. Observe that the set F ∖ U is compact and consists of isolated points : this set is finite. Moreover, the group G_1 is the kernel of the morphism

    [                    G_3                      →         { -1,1 }^F ∖ U;                      g                      ↦ (sgn(g'(x)))_x ∈ F ∖ U ],

where, for any real number λ≠ 0, sgn(λ)=1 if λ >0 and sgn(λ)=-1 if λ <0. As the group { -1,1 }^F ∖ U is finite, the group G_1 is a finite index subgroup of G_3.




§ GROUPS WITHOUT FREE SUBSEMIGROUPS ON TWO GENERATORS


In this section, we prove Theorem <ref>. As in the proof of Theorem <ref>, we will use Definition <ref> of the group 𝔡𝔦𝔣𝔣^1+Lip(K). Hence we see our Cantor set K as a subset of ℝ.

We fix a group G satisfying the hypothesis of Theorem <ref>: the subgroup G of 𝔡𝔦𝔣𝔣^1+Lip(K) does not contain any free subsemigroup on two generators.

The proof is divided in three steps which correspond to subsections <ref>, <ref> and <ref>.


  * First, we find a finite index subgroup G_1 of G such that any minimal invariant set for the action of G_1 on K is a fixed point.

  * Then we prove that any element of the derived subgroup G'_1 of G_1 pointwise fixes a neighbourhood of Fix(G_1). This is the main step of the proof which heavily relies on distortion estimates.

  * We deduce the theorem from the two above steps.


The following subsection is devoted to a useful preliminary result.



 §.§ A preliminary result


We will often need the following result. For any element g in 𝔡𝔦𝔣𝔣^1(K), we denote by Per(g) the set of periodic points of g, i.e. the set of points p of K such that there exists an integer n ≥ 1 such that g^n(p)=p.

 
For any element g in 𝔡𝔦𝔣𝔣^1(K), there exists N ≥ 1 such that

    Per(g)= { p ∈ K,  g^N(p)=p }.




Fix an element g in 𝔡𝔦𝔣𝔣^1(K) and define


    [                             T:                         Per(g)                              →                            ℝ_+;                                                             p                              ↦ T(p)= min{ T ≥ 1,  g^T(p)= p } ]
    .


This lemma is a consequence of the two following claims.

 
For any point p in Per(g), there exists an open neighbourhood U of the point p such that
U ∩Per(g)= U ∩Per(g) and T_|U ∩Per(g) is bounded.


 
Per(g)=Per(g).


By Claim <ref> and Claim <ref>, the set Per(g) is compact and the function T is locally bounded on Per(g). Hence the function T is bounded by an integer M. It suffices to take N= M! to prove the lemma.



By definition of T, for any point p of Per(g), g^T(p)(p)=p. Recall that the set K is contained in ℝ. Fix a point p in Per(g). By Definition <ref>, there exists an open interval I' of ℝ which contains the point p and a homeomorphism h̃_1: I' →h̃_1(I) such that h̃_1|I' ∩ K=g^T(p)_|I ∩ K. The homeomorphism h̃_1 is not necessarily orientation-preserving but there exists an open interval I of ℝ which contains the point p and an orientation-preserving homeomorphism h̃:I →h̃(I) such that g^2T(p)_|I ∩ K= h̃_|I ∩ K. 

Let U= I ∩ K. We will prove that, for any point x in U, either g^2T(p)(x)=x or x ∉Per(g). This proves the claim as

    Per(g) ∩ U = { x ∈ U,  g^2T(p)(x)=x }

is closed in U and T is bounded by 2T(p) on U.

Take a point x in U and suppose that g^2T(p)(x) ≠ x. Then, as h̃ is an increasing map which fixes the point p, either

    { g^2nT(p)(x), n >0 }= {h̃^n(x),  n>0 }

is infinite and contained in U or

    { g^2nT(p)(x), n <0 }={h̃^n(x),  n<0 }

is infinite and contained in U. In either cases x ∉Per(g).



Let p be a point of Per(g). Then the closure of the orbit of p under the action of the diffeomorphism g contains a a minimal set F. By Proposition <ref>, this set F  has to be a periodic orbit of g and F ⊂Per(g): there is no periodic point of g in a neighbourhood of a hyperbolic fixed point of g. Moreover, as, by Claim <ref>, the set Per(g) is open in Per(g), there exists n>0 such that the point g^n(p) belongs to Per(g). Hence the point p belongs to Per(g). 




 §.§ Definition of the finite index subgroup G_1
 

This section is devoted to the proof of the following proposition. For any subgroup H of 𝔡𝔦𝔣𝔣^1+Lip(K), we denote by Fix(H) the subset of K consisting of points which are fixed under all the elements of the group H.

 
There exists a finite index subgroup G_1 of G such that the two following properties hold.


  * Any minimal invariant subset for the action of G_1 on K is a point of Fix(G_1).

  * For any diffeomorphism g in the group G_1 and any point p of Fix(G_1), we have

    g'(p)>0.




We start the proof of this proposition by the following lemma, which is more or less a consequence of Sacksteder's Theorem.

 
Any minimal invariant subset for the action of the group G on K is finite.




If one looks closely at the proof of Sacksteder's theorem, one can see that it implies directly that, if the action of the group G on the Cantor set K has an infinite minimal invariant set, then the group G contains a free subsemigroup on two generators. We provide here a proof which uses Sacksteder's theorem.

Suppose that the action of the group G on K has an infinite minimal subset K_min. We want to prove that the group G contains a free subsemigroup on two generators. This will give a contradiction and will complete the proof of the lemma. To do this, we use the following classical lemma (see <cit.> for a proof).

 
Let H be a group acting on a set E. Assume there exist elements h_1 and h_2 of H as well as disjoint nonempty subsets A and B of E such that

    {[ h_1(A ∪ B) ⊂ A; h_2(A ∪ B) ⊂ B ].
    .

Then the subsemigroup of G generated by h_1 and h_2 is free.


First, as the set K_min is infinite, it contains accumulation points. As the set of accumulation points of K_min is closed and invariant under the action of the group G and as K_min is a minimal invariant set, we deduce that the set K_min is a Cantor set. Then, by Proposition <ref>, there exists an element h of G with a hyperbolic fixed point p ∈ K_min. Taking h^-1 instead of h if necessary, we can suppose that | h'(p) | <1. Taking the definition of a diffeomorphism of K, we know that  there exists a C^1+Lip-diffeomorphism h̃ from an open interval I of ℝ which contains the point p to an open interval h̃(I) such that h̃_| I ∩ K=h_| I ∩ K. Moreover, choose the interval I sufficiently small so that sup_x ∈ I | h̃'(x) | <1. Hence the sequence of sets (h̃^n(I))_n ≥ 0 is decreasing and

    ∩_n ∈ℕh̃^n(I)= { p }.

As K_min is a minimal Cantor set, there exists a point p' in G.p ∩ (I ∖{ p }). Fix an element g of the group G such that g(p)=p'. Let g̃ be a diffeomorphism between two intervals of ℝ which coincides with g on a neighbourhood of p. Take an integer N_1 sufficiently large so that g̃∘h̃^N_1 is defined on I, sup_x ∈ I |(g̃∘h̃^N_1)'(x) | <1 and g ∘ h^N_1(I ∩ K) ⊂ I ∩ K. The map h_1=g ∘ h^N_1_| I ∩ K has a unique fixed point p': indeed, the map g̃∘h̃^N_1 has a fixed point on I which is the limit of any positive orbit under the action of g̃∘h̃^N_1. Hence this fixed point belongs to K. Observe that h_1(p)=g(p) ≠ p. Hence there exists an open interval J_1 which contains the point p such that h_1(J_1) is disjoint from J_1. Finally take an open interval J_2⊂ I which contains the point p' and is disjoint from the interval J_1. Then there exist integers N_2 >0 and N_3>0 such that

    h^N_2(J_1∪ J_2) ⊂ J_1

and

    h_1^N_3(J_1∪ J_2) ⊂ J_2.

By the positive ping-pong lemma, the group G contains a free semigroup on two generators.


Now, we can finish the proof of Proposition <ref>


By Lemma <ref>, any minimal subset for the action of the group G on the Cantor set K is contained in the set

    F= ⋂_g ∈ GPer(g).

By Lemma <ref>, the set F is a closed subset of K. Moreover, it is invariant under the action of G. Let us denote by G(F) the group of restrictions to F of elements of G. By definition of F and by Lemma <ref>, the group G(F) consists of finite order elements. By Proposition <ref>, the group G(F) is finite. Moreover, let G_1  be the subgroup of G consisting of elements which pointwise fix F and whose derivative at each point of F is positive. Then the group G_1 is a finite index subgroup of G by the same proposition. 

Let us check that this group G_1 satisfies the wanted property. Let K_min be a minimal invariant subset of the action of the group G_1 on the Cantor set K. Then the set

    M=⋃_g ∈ G g(K_min)

is a closed G-invariant subset of K which consists of a finite number of copies of K_min. Any G-orbit of a point in this set M is dense in M: it is a minimal subset for the action of G on K. Hence, by Lemma <ref>,

    K_min⊂ M ⊂ F ⊂Fix(G_1).





 §.§ Behaviour of individual elements of G'_1
 

Let us fix a subgroup G_1 of G which satisfies Proposition <ref> for the rest of this section. We prove the following result. Recall that the derived subgroup H' of a group H is the subgroup of H generated by the commutators of elements of H, i.e. elements of the form [h_1,h_2]=h_1h_2h_1^-1h_2^-1 with h_1,  h_2∈ H.

 
Any element of the derived subgroup G'_1 of the group G_1 pointwise fixes a neighbourhood of Fix(G_1).  


In this proposition, the neighbourhood can a priori depend on the chosen element of the group G'_1.

We split the proof of Proposition <ref> into two steps.


  * We first prove that any point of Fix(G_1) is accumulated by points of Fix(G'_1).

  * Then we use this first step to prove Proposition <ref>.


Throughout the proof of the proposition, we will use the two following definitions.


Let I be an interval of ℝ⊃ K and f be an element of 𝔡𝔦𝔣𝔣^1+Lip(K). We say that


  * f is monotonous on I if there exists a C^1+Lip-diffeomorphism f̃: I →f̃(I) such that f_|I ∩ K=f̃_I ∩ K.

  * f is increasing on I if there exists an orientation-preserving (i.e. increasing) C^1+Lip-diffeomorphism f̃: I →f̃(I) such that f_|I ∩ K=f̃_I ∩ K.



Let A be a subset of ℝ and p be a point of A. We call left-neighbourhood (respectively right-neighbourhood) of the point p in A any subset of ℝ which contains a set of the form [p-α,p) ∩ A (resp. (p,p+α] ∩ A), for some α >0. 

We say that the point p is accumulated on the left (respectively accumulated on the right) by the set A if, for any α>0, [p-α,p) ∩ A ≠∅ (resp. (p,p+α] ∩ A ≠∅. Equivalently, a point p is accumulated on the left (resp. right) by the set A if any left-neighbourhood (resp. right-neighbourhood) of the point p is nonempty. 

We say that the point p is isolated on the left (resp. on the right) in A if it is not accumulated on the left (resp. on the right) by the set A or, equivalently, if it has a left-neighbourhood (resp. right-neighbourhood) in A which is empty.



  §.§.§ First step


We formulate this first step as a proposition.

 
Let p be a point of Fix(G_1).

If the point p is accumulated on the left (respectively on the right) by points of K, then the point p is accumulated on the left (resp. on the right) by points of Fix(G'_1).



Observe that it suffices to prove the following property.
Any point of Fix(G_1) which is isolated on the left (respectively on the right) in Fix(G_1) but not in K is accumulated on the left (resp. on the right) by points of Fix(G'_1).
As the proof of the property for points which are isolated on the right is similar, we will prove the property only for points which are isolated on the left. Hence let p be a point of Fix(G_1) which is isolated on the left in Fix(G_1) but not in K. 

To start the proof, we will define a “local minimal invariant subset” K_1 in a left-neighbourhood of p. The subset K_1 accumulates to p and we will see later that any element of G'_1 fixes the points of K_1 in a left-neighbourhood of p.

As G_1 is a finite index subgroup of G and the group G is finitely generated, then the subgroup G_1 is finitely generated. Fix a finite symmetric generating set S of G_1 and α_0 >0 small enough so that Fix(G_1) ∩ [p-α_0,p)= ∅ and, for any element s of S, the diffeomorphism s is monotonous on [p-α_0,p]. Then any element of S is increasing on [p-α_0,p] as, by the choice of G_1, any element of G_1 has a positive derivative at the point p.

Consider the set ℳ of closed nonempty subsets A of (p-α_0,p)∩ K such that, for any element s of the generating set S

    s(A) ∩ (p-α_0,p) ⊂ A.

This last property is an analogue of a “local invariance” property. Of course, the set ℳ is nonempty as the subset (p-α_0,p) ∩ K belongs to ℳ. 

Take a point p_1 in (p-α_0,p) ∩ K. Let 

    p_2= max{ s(p_1),  s ∈ S }.

Observe that, necessarily, p_2>p_1: otherwise, the point p_2 would be fixed under any element of S hence any element of G_1, in contradiction with the definition of α_0. To define the subset K_1, we need the following lemma.

 
For any A in ℳ,

    A ∩ [p_1,p_2] ≠∅

and the point p is accumulated by points of A. 


Before proving Lemma <ref>, let us see how to construct the subset K_1. 

The set ℳ is partially ordered by the set inclusion relation. Moreover, by compactness and Lemma <ref>, for any totally ordered family (A_i)_i ∈ I of elements of ℳ, the set

    ⋂_i ∈ I A_i

is nonempty and belongs to ℳ: this set is a lower bound for this totally ordered family. By Zorn's lemma, the set ℳ contains a minimal element for the inclusion relation. We denote by K_1 this element of ℳ. We can see it as a minimal invariant set for the left-germ of G_1 at p.

Now, let us prove Lemma <ref>.


Denote by p' a point in A ∩ (p-α_0,p).

If the point p' belongs to [p_1,p_2], there is nothing to prove.

Suppose now that the point p' belongs to the interval (p_2,p). Denote by ℬ the set of elements g of G_1   with the following property. There exists a family (s_i)_1 ≤ i ≤ p of elements of S such that g=s_1s_2… s_p and 

    ∀ 1 ≤ i ≤ p,   s_is_i+1… s_p([p',p] ∩ K) ⊂ (p-α_0,p].

Now let B be the subset of K ∩ (p-α_0,p) defined by 

    B= { b(p'), b ∈ℬ}.

Observe that, as the subset A belongs to ℳ, the subset B is contained in A. 

Let m=inf(B ∩ [p_1,p]). It suffices to prove that the element m belongs to [p_1,p_2). We will do it by contradiction. Suppose that p_2≤ m. Then, as the point m does not belong to Fix(G_1), there exists an element s of S such that s(m) < m (recall that the set S is symmetric). As p_2≤ m and by construction of the point p_2, we have p_1≤ s(m) and p-α_0 < s(m). Hence, if x is a point of B close to m, then s(x) is also a point of B which belongs to [s(m),m), in contradiction with the definition of m.

If p-α_0 < p'<p_1, the proof is analogous. Namely look at the same set B but look at the supremum of B ∩ [p_1,p_2] instead of the infimum and prove that it belongs to (p_1,p_2].

Observe that we can take the point p_1 as close as we want to the point p. Hence the point p is accumulated by points of A.


Let H be any subgroup of 𝔡𝔦𝔣𝔣^1(K), F be a closed subset of K and q be a point of Fix(H) which is accumulated on the left by F such that any element of H has a positive derivative at q. Define

    H_F,q,-= { g ∈ H  |  ∃α>0,  g_F∩ [q-α,q]= Id_F∩ [q-α,q]}.

Observe that H_F,q,- is a normal subgroup of H. We denote by ℋ(F,q) the group H /  H_F,q,-. This is the “group of left-germs at q of elements of H restricted to F”. In particular we set 𝒢_1(K_1,p)=G_1 /  G_1  K_1,p,-. This is the “group of left-germs at p of elements of G_1 restricted to K_1”.

The following proposition completes the proof of the first step. 

 


  * Any element of G_1 either fixes all the points of K_1 or has no fixed point on a left-neighbourhood of p (which depends on the element of G_1).

  * The group 𝒢_1(K_1,p) is abelian.



 Actually, the first statement of this proposition implies the second one, as we will see during the proof of this proposition. 

We obtain immediately the following corollary which will be useful later.

  
Let p be a point of Fix(G_1) which is isolated on the left in Fix(G_1) but accumulated on the left by K. Then there exists an element h of G_1 and α >0 such that


  * the diffeomorphism h is increasing on [p-α,p].

  * ∀ x ∈ [p-α,p] ∩ K, h(x)>x.



Of course, we have analogous statements for points which are isolated on the right in Fix(G_1).

To prove Proposition <ref>, we need the following lemma.

 
Let H be a subgroup of 𝔡𝔦𝔣𝔣^1(K), F be a closed subset of K and q be a point of Fix(H) which is accumulated on the left by F such that any element of H has a positive derivative at q..
Suppose that there exists α_0>0 such that, for any diffeomorphism h in H, the following property holds.

For any element h of the group H and any α∈ (0,α_0), if the diffeomorphism h is increasing on [q-α,q] and if the diffeomorphism h has a fixed point x in [q-α,q] ∩ F then the diffeomorphism h pointwise fixes [x,q] ∩ F.

Then the group ℋ(F,q) is abelian.


To prove this lemma, we use the same techniques as in the proof of a famous theorem by Hölder, which states that any group of fixed-point-free homeomorphisms of the real line is abelian. We need the following definition for this proof.

Observe that we could have thought of a weaker and more natural hypothesis for Lemma <ref>: any element of ℋ(F,q) has a representative with no fixed point in V ∖{ q }, where V is a left-neighbourhood of q in F. However, our proof does not work with this weaker hypothesis. This is due to the various speeds of convergence of the orbits to the point q that can exist. The hypothesis of Lemma <ref> that we took avoids this problem.


Let H be a group and ≼ be an order on H. The order ≼ is called


  * total if, for any elements h_1 and h_2 of H, either h_1≼ h_2 or h_2≼ h_1.

  * biinvariant if, for any elements h_1, h_2 and h_3 of H,

    h_1≼ h_2⇒ h_3h_1≼ h_3h_2

and

    h_1≼ h_2⇒ h_1h_3≼ h_2h_3.


  * Archimedean if, for any element h_1 and h_2 of H with 1 ≺ h_2 (i.e. 1 ≼ h_2 and h_2≠ 1), there exists an integer n ≥ 0 such that h_1≼ h_2^n. 



The main idea of the proof of Lemma <ref> is to apply the following lemma (see <cit.> for a proof of this lemma).

 
Any group which admits a bi-invariant total Archimedean order is a subgroup of (ℝ,+).



We define an order ≼ on the group ℋ(F,q) in the following way. For any elements ξ and η which are respectively represented by elements g_ξ and g_η of H, we have ξ≼η if and only if there exists α >0 such that, for any point x of F ∩ (q-α,q), we have g_ξ(x) ≤ g_η(x). We will prove that this defines a biinvariant total Archimedean order on ℋ(F,q). By Lemma <ref>, this implies that the group ℋ(F,q) is abelian and proves Lemma <ref>.

Let ξ_1, ξ_2 and ξ_3 be elements of ℋ(F,q) with ξ_1≼ξ_2. Let us prove that ξ_1ξ_3≼ξ_2ξ_3 and that ξ_3ξ_1≼ξ_3ξ_1.

Take elements g_1, g_2 and g_3 of the group H which respectively represent the elements ξ_1, ξ_2 and ξ_3. Take α >0 small enough such that


  * The diffeomorphisms g_1, g_2 and g_3 are increasing on [q-α,q].

  * For any point x in [q-α,q) ∩ F, g_1(x) ≤ g_2(x).
 
Take α' >0 small enough so that g_3([q-α',q] ∩ F) ⊂ [q-α,q] ∩ F. Then, for any point x in [q-α',q] ∩ F,

    g_1(g_3(x)) ≤ g_2(g_3(x))

and ξ_1ξ_3≼ξ_2ξ_3.
Now, take α”>0 small enough such that  g_1([q-α”,q] ∩ F) ⊂ [q-α,q] ∩ F and  g_2([q-α”,q] ∩ F) ⊂ [q-α,q] ∩ F. Then, as the diffeomorphism g_3 is increasing on [q-α,q], for any point x of [q-α”,q] ∩ F,

    g_3(g_1(x)) ≤ g_3(g_2(x))

and ξ_3ξ_1≼ξ_3ξ_2.

We have just proved that the order ≼ is biinvariant. Let us explain why this order is total. Let ξ and η be elements of ℋ(F,q) which are respectively represented by elements g_ξ and g_η of H. By hypothesis of Lemma <ref>, either the diffeomorphism g_ξ∘ g_η^-1 is equal to the identity on F ∩ (q-α,q) for α>0 small enough or this diffeomorphism displaces all the points of F in a left-neighbourhood of q. Hence either ξη^-1≽ 1 or ξη^-1≼ 1. By invariance of the order ≼ under right-multiplication, we deduce that either ξ≽η or ξ≼η. The order ≼ is total. 

Now, let us prove that it is Archimedean. Let ξ_1 and ξ_2 be elements of ℋ(F,q) such that ξ_1≻ 1. If ξ_2≼ 1, then ξ_2≼ 1 ≺ξ_1 so we suppose that ξ_2≻ 1 in what follows.

Take respective representatives g_1 and g_2 of ξ_1 and ξ_2 in H.
Take α∈ (0,α_0) small enough such that


  * The diffeomorphisms g_1 and g_2 are increasing on [q-α,q].

  * For any point x in [q-α,q) ∩ F, g_1(x) >x and g_2(x) >x.

Fix a point x_0 in F ∩ [q-α,q). Observe that any positive orbit under g_1 of points of F ∩ [q-α,q) converges to the point q. Hence there exists k>0 such that
g_2(x_0) < g_1^k(x_0), which can be rewritten

    g_2(x_0) < g_1^kg_2^-1(g_2(x_0)).

As the diffeomorphism g_1^kg_2^-1 is increasing on [g_2(x_0),q] then, by hypothesis of Lemma <ref>, one of the following occurs.


  * Either the diffeomorphism g_1^kg_2^-1 pointwise fixes a left-neighbourhood of the point q in F. In this case ξ_1^kξ_2^-1=1.

  * Or it has no fixed point in [g_2(x_0),q] ∩ F in which case

    ∀ x ∈ F ∩ [g_2(x_0),p],  g_1^k∘ g_2^-1(x) >x.


In either case, ξ_1^kξ_2^-1≽ 1 and, by invariance of the relation ≼ under right-multiplication, ξ_1^k≽ξ_2. The order ≼ is Archimedean.



We denote by K'_1 the set of accumulation points of K_1. By minimality of K_1, observe that either K'_1∩ (p-α_0, p)= ∅ or K'_1∩ (p-α_0,p)= K_1. Indeed, if the set K'_1∩ (p-α_0,p) is nonempty, then K'_1∩ (p-α_0,p) is an element of ℳ which is contained in K_1, hence which is equal to K_1 by minimality of K_1. This remark splits the proof into two cases.

First case: K'_1∩ (p-α_0,p)= ∅. As any point of the set K_1 is isolated, any element of G_1 either pointwise fixes all the points of K_1 or displaces all the points of K_1 on a left neighbourhood of the point p (neighbourhood which a priori depends on the element of G_1). Lemma <ref> implies that the group 𝒢_1(K_1,p) is abelian.

Second case: K'_1∩ (p-α_0, p)=K_1. In this second case, it also suffices to prove that any element of G_1 satisfies the hypothesis of Lemma <ref>. However, proving this fact is harder than in the first case.

Suppose for a contradiction that there exists α>0 and a diffeomorphism g_1 in G_1 such that


  * The diffeomorphism g_1 is increasing on [p-α,p].

  * There exists a point x in K_1∩ [p-α,p) such that g_1(x)=x.

  * The diffeomorphism g_1 does not fix all the points of [x,p] ∩ K_1.
 

Let p_1<p_2 be two points of [x,p] ∩Fix(g_1) such that K_1∩ (p_1,p_2) ≠∅ and, for any point x in K_1∩ (p_1,p_2), g_1(x) ≠ x. In this second case, Proposition <ref> is a consequence of the two following lemmas.

 
There exists an element h of G_1 such that p_1 < h(p_1) < p_2 and the diffeomorphism h is increasing on [p_1,p].


Before stating the second lemma, we need a definition which is a generalization of a standard definition for pseudogroups or groups of homeomorphisms of the real line (see <cit.>).


Two elements g and h of 𝔡𝔦𝔣𝔣^1+Lip(K) are crossed if there exists points p_1<p_2 of the Cantor set K such that


  * The diffeomorphisms g and h are increasing on [p_1,p_2].

  * The points p_1 and p_2 are fixed under g but the diffeomorphism g has no fixed point in (p_1,p_2).

  * Either we have p_1<h(p_1)<p_2 or p_1<h(p_2)<p_2.



Observe that the element g_1 which is defined above and the element h which is given by Lemma <ref> are crossed. Now, it suffices to use Lemma <ref> below to obtain the wanted contradiction.

 
If two elements g and h of 𝔡𝔦𝔣𝔣^1+Lip(K) are crossed, then the subgroup generated by g and h contains a free semi-group on two generators.




Now, let us prove Lemmas <ref> and <ref>.

 Denote by ℬ the set of elements g of G_1  with the following property. There exists a family (s_i)_1 ≤ i ≤ p of elements of S such that g=s_1s_2… s_p and 

    ∀ 1 ≤ i ≤ p,   s_is_i+1… s_p([p_1,p] ∩ K) ⊂ (p-α_0,p].

Now let A be the closure in K_1 of the subset of K ∩ (p-α_0,p) defined by 

    B={ b(p_1), b ∈ℬ}.

Observe that, as the subset K_1 belongs to the collection ℳ, we have A ⊂ K_1. Moreover, the set A belongs to the collection ℳ so that A=K_1, by minimality of K_1. Hence the set B is dense in K_1 and B ∩(p_1,p_2) ≠∅ as K_1∩ (p_1,p_2) ≠∅. This proves Lemma <ref>.


The proof of Lemma <ref> is similar to <cit.>.


Suppose for instance that p_1< h(p_1) < p_2. Moreover, as the diffeomorphism g is increasing on [p_1,p_2], taking g^-1 instead of g if necessary, we can suppose that

    ∀ x ∈ (p_1,p_2) ∩ K,  g(x) < x.

Observe that, for any point x in (p_1,p_2) ∩ K, the sequence g^n(x) converges to the point p_1. In particular, the point p_1 is accumulated on the right by points of K. Take α>0 small enough so that


  * p_1 + α∈ K and the diffeomorphism h is monotonous on [p_1,p_1+α].

  * h(p_1+ α) < p_2.

  * The sets [p_1,p_1+α] and h([p_1,p_1+α] ∩ K) are disjoint.

Take a sufficiently large integer n such that g^n(h(p_1+α)) <p_1+α. Let f_1=g^n and f_2=h ∘ g^n. Observe that

    f_1([p_1,p_1+α]∩ K ∪ h([p_1,p_1+α]∩ K)) ⊂ [p_1,p_1+α] ∩ K

and that

    f_2([p_1,p_1+α] ∩ K ∪ h([p_1,p_1+α] ∩ K)) ⊂ h([p_1,p_1+α] ∩ K).

Now, by the positive ping-pong lemma (Lemma <ref>), the semigroup generated by f_1 and f_2 is free.




  §.§.§ Second step


Now, we are ready for the second step of the proof of Proposition <ref>. We will reformulate this second step as two propositions. Fix a point p in Fix(G_1) which is accumulated on the left by the set K. Denote by 𝒢_1,p the group of left-germs at p of elements of G_1: this the quotient of the group G_1 by the normal subgroup

    G_1, p,-= { g ∈ G_1  |  ∃α>0,  g_K∩ [p-α,p]= Id_F∩ [p-α,p]}.

 
 
Suppose that the point p is isolated on the left in Fix(G_1). Then the group 𝒢_1,p is abelian.


Of course, we have an analogous statement in the case where the point p is isolated on the right in Fix(G_1).

In the case where the point p is accumulated on the left by points of Fix(G_1), we will prove the following stronger proposition.

 
Suppose the point p is accumulated on the left by points of Fix(G_1). Then there exists a left-neighbourhood L_p of the point p with the following properties.


  * L_p∩ K is invariant under the action of G_1.

  * The action of G'_1 on L_p∩ K is trivial.



Once again, there is an analogous statement for points which are accumulated on the right by points of Fix(G_1).

The two above propositions and their variants for right-neighbourhoods imply Proposition <ref>.

We will start by proving Proposition <ref>. Then we will prove Proposition <ref>. 

To prove Proposition <ref>, we need the following lemma, which is a variant of Kopell lemma for diffeomorphisms of the half-line (see <cit.> to see this lemma and its proof).

 
Let g_1 and g_2 be elements of 𝔡𝔦𝔣𝔣^1+Lip(K). Let p ∈Fix(g_1) ∩Fix(g_2) be a point accumulated on the left by points of K such that the elements g_1 and g_2 have a positive derivative at p.
Suppose that


  * There exists α>0 such that (p-α,p) ∩Fix(g_1)= ∅ and such that g_2 is increasing on (p-α,p).

  * For any point x in (p-α,p)∩ K, g_1g_2(x)=g_2g_1(x).

Then (p-α,p) ∩Fix(g_2) = ∅ or the diffeomorphism g_2 fixes all the points of K in a left-neighbourhood of the point p.


Of course, we have an analogous lemma for fixed points of g_1 which are isolated on the right. In the classical Kopell lemma, we only need that g_1 is C^1+bv and g_2 is C^1. In contrast, in our case, we can lower the regularity to C^1+bv but we still need both elements g_1 and g_2 to have a C^1+bv regularity. The proof of this lemma is closely related to the proof in the standard case.


Take α>0 such that g_1 has no fixed point on [p-α,p) and such that there exist C^1+Lip-diffeomorphisms g̃_1 and g̃_2 defined on [p-α,p] such that g_1  | [p-α,p] ∩ K= g̃_1  | [p-α,p] ∩ K and g_2  | [p-α,p] ∩ K= g̃_2  | [p-α,p] ∩ K. For i=1,2, denote by k_i the Lipschitz constant of log(| g̃_i'|)_|[p-α,p] and by D the diameter of the compact subset K of ℝ.

Suppose that g_2 has a fixed point p' in [p-α,p). We will prove that there exists M>0 such that, for any k>0, sup_x ∈ [p',p] ∩ K |( g_2^k)'(x) | ≤ M. From this, we will deduce that g_2 fixes the points of [p',p] ∩ K.

Taking g_1^-1 instead of g_1 if necessary, we can suppose that, for any point x in [p-α,p) ∩ K, g̃_1(x)>x. As the diffeomorphisms g_1 and g_2 commute the points g_1^n(p'), for n ≥ 0 are fixed points of g_2 and form a sequence which converges to the point p. Hence g_2'(p)=1.

For any k ≥ 0 and any n ≥ 0, g_2^k=g_1^-ng_2^kg_1^n. Hence, for any point x of K ∩ [p',p], 
    (*)   (g_2^k)'(x)= (g_1^n)'(x)/(g_1^n)'(g_2^k(x)).(g_2^k)'(g_1^n(x)).

Observe that the sequence ((g_2^k)'(g_1^n(x)))_n converges to 1=(g_2^k)'(p) as n → + ∞. Let us prove that

    (g_1^n)'(x)/(g_1^n)'(g_2^k(x))≤ e^k_1D.

Indeed,

    [                        log(|(g_1^n)'(x)/(g_1^n)'(g_2^k(x))|)                                                            = ∑_i=0^n-1log(|g_1'(g_1^i(x))|)-log(|g_1'(g_1^i(g_2^k(x)))|). ]

 For any index i ≥ 0, denote by I_i the closed interval of ℝ whose ends are g_1^i(x) and g_1^i(g_2^k(x)) and denote by n_0≥ 0 the integer such that the point x belongs to the interval [g_1^n_0(p'),g_1^n_0+1(p')). Observe that, for any i ≥ 0, I_i⊂ [g_1^n_0+i(p'),g_1^n_0+i+1(p')). Hence the intervals I_i, for i ≥ 0 are pairwise disjoint and
 
    [ log(|(g_1^n)'(x)/(g_1^n)'(g_2^k(x))|)                                     ≤                    k_1∑_i=0^n-1 |I_i|                                     ≤                                k_1 D. ]
 
Then, by (*), for any k>0, sup_x ∈ [p',p] ∩ K |( g_2^k)'(x) | ≤ M, where M=e^k_1D.

Now let us prove that the diffeomorphism g_2 fixes the points in [p',p] ∩ K. Suppose for a contradiction that there exists 
 a point x_0 in [p',p] ∩ K which is not fixed under g_2. Let I be the connected component of [p',p] - Fix(g_2) which contains x_0. Take a point y in I-K and let (y_-,y_+) be the connected component of I-K which contains y. Then, for any k>0,

    (g̃_2^k)'(y)/(g̃_2^k)'(y_-)≤ e^k_2D.

Hence (g̃_2^k)'(y) ≤ M e^k_2D. This implies that, for any k ≥ 0, sup_x ∈ I |( g̃_2^k)'(x) | ≤ Me^k_2D. Hence, using the mean value theorem, we see that the diffeomorphism g̃_2 has to fix the points of I, a contradiction.


Propostion <ref> is a consequence of the following weaker Proposition.

 
Suppose that the point p is isolated on the left in Fix(G_1). Then the group 𝒢_1,p is metabelian, i.e. the group 𝒢'_1 is abelian.



By Corollary <ref>, there exists an element f of G_1 and α'_0>0 such that the diffeomorphism f has no fixed point on (p-α'_0,p) and is increasing on the interval [p-α'_0,p]. Taking f^-1 instead of f if necessary, we can suppose that, for any point x in (p-α'_0,p) ∩ K, f(x) >x.

Fix two nontrivial elements ξ_1 and ξ_2 of the group 𝒢'_1. We want to prove that ξ_1ξ_2= ξ_2ξ_1. Denote by 𝒢_2 the subgroup of 𝒢_1,p generated by the elements ξ_1 and ξ_2. The advantage of considering this subgroup instead of 𝒢'_1 is that the group 𝒢_2 is finitely generated whereas the group 𝒢'_1 might not be finitely generated.

Fix respective representative g_1 and g_2 of ξ_1 and ξ_2 in the group G'_1 and let K_2⊂ K the set of fixed points of the group G_2 generated by g_1 and g_2. Observe that, by Proposition <ref>, the set Fix(G'_1) ⊂ K_2 accumulates on the left of the point p. Take a point p-α'_0<p' < p such that


  * The diffeomorphisms g_1 and g_2 are increasing on [p',p].

  * The point p' belongs to the set Fix(G'_1).
 
An easy induction on wordlength proves that any element of G_2 is increasing on [p',p].

We will distinguish two cases depending on the existence of a fixed point which is outside K_2 for an element of G_2. As the elements ξ_1 and ξ_2 are supposed to be nontrivial, the set (K ∖ K_2) ∩ [p',p] is nonempty.

First case: Suppose that, for any element g in the group G_2 and any connected component (p_1,p_2) of [p',p] ∖ K_2 which meets K, either

    Fix(g) ∩ (p_1,p_2) = ∅

or g is equal to the identity on (p_1,p_2) ∩ K.
 Take a connected component (p_1,p_2) of [p',p] ∖ K_2 which meets K. Denote by G_2|(p_1,p_2) be the group of restrictions to (p_1,p_2) ∩ K of elements of the group G_2. We now use the following lemma which is once again a straightforward consequence of Lemma <ref>.

 
Let (p'_1,p'_2) be an open interval of ℝ which meets K and whose endpoints belong to the Cantor set K. Let G be a subgroup of 𝔡𝔦𝔣𝔣^1(K). Suppose that


  * Any diffeomorphism in the group G preserves (p'_1,p'_2) ∩ K and is increasing on the interval (p'_1,p'_2).

  * Any element of G which has a fixed point in (p'_1,p'_2) ∩ K is equal to the identity on (p'_1,p'_2) ∩ K.

Then the group G_|(p'_1,p'_2) of restrictions to (p'_1,p'_2) ∩ K of elements of G is abelian.



As any nontrivial element of the group G is increasing on (p'_1,p'_2) and has no fixed point on (p'_1,p'_2) ∩ K, then, for any nontrivial element g of the group G_|(p'_1,p'_2), either

    ∀ x ∈ (p'_1,p'_2) ∩ K,  g(x)>x

or

    ∀ x ∈ (p'_1,p'_2) ∩ K,  g(x)<x.

Let g and h be two elements of the group G. Hence, if there exists a point x_0 in (p'_1,p'_2) ∩ K such that g(x_0) <h(x_0) then, for any point x of (p'_1,p'_2) ∩ K, g(x) < h(x): otherwise the diffeomorphism g^-1h would be nontrivial and would have a fixed point in (p'_1,p'_2) ∩ K, which is not possible.
We define then an order ≼ on G_|(p'_1,p'_2) by setting g ≼ h if and only if there exists a point x_0 of (p'_1,p'_2) ∩ K such that g(x_0) ≤ h(x_0). The above remark proves that this defines a total order on the group G_|(p'_1,p'_2). With a proof similar to the proof of Lemma <ref> (and even easier!), we can show that this defines a biinvariant Archimedean order on the group G_|(p'_1,p'_2). By Lemma <ref>, the group G_|(p'_1,p'_2) is abelian.

 
By Lemma <ref>, the group G_2|(p_1,p_2) is abelian. As this lemma is true for any such connected component (p_1,p_2), we deduce that 
 
    ∀ x ∈ [p',p] ∩ K, g_1g_2(x)=g_2g_1(x).

Hence ξ_1ξ_2= ξ_2ξ_1.

Second case: There exists an element g in the group G_2 and a point p_0 in (K ∖ K_2) ∩ [p',p] with the following properties.


  * g(p_0)=p_0.

  * If we denote by (p_1,p_2) the connected component of [p',p] ∖ K_2 which contains the point p_0,

    g_|(p_1,p_2) ∩ K≠ Id_(p_1,p_2) ∩ K.


 Without loss of generality, we can suppose that the point p_0 is one of the endpoints of a connected component of [p',p] ∖Fix(g) which meets the Cantor set K. We will find a contradiction, namely we will construct a free subsemigroup on two generators of the group G_1. Hence the first case always holds.

As the point p_0 does not belong to the set K_2= Fix(G_2), there exists a diffeomorphism h in G_2 such that h(p_0)>p_0. 

Let (a,b) be the connected component of ℝ∖Fix(G'_1) which contains the points p_0 and h(p_0). Recall we fixed an element f at the beginning of the proof of the proposition. Take N'>0 sufficiently large so that f^N'(a)>b and let f_1=f^N'. Observe that the sets f_1^n([a,b] ∩ K), for n ≥ 0, are pairwise disjoint. Moreover, as the set Fix(G'_1) is invariant under the action of the group G_1, for any n ≥ 0, the points f_1^n(a) and f_1^n(b) are fixed under the elements of G_2 < G'_1. 

The proof uses the following lemma which relies on distortion estimates.  

 
There exists an integer N ≥ 0 such that, for any n ≥ N,

    h(f_1^n(p_0))=f_1^n(p_0).



Before proving this claim, let us see how we can obtain a contradiction from this claim. More precisely, we want to prove that the semigroup generated by f_1 and h is free, a contradiction. We will use the following lemma to do so.

 
Let p'<p be points of K. Suppose that the point p is accumulated on the left by points of K. Let f and h be diffeomorphisms in 𝔡𝔦𝔣𝔣^1(K) such that


  * The diffeomorphism f is increasing on [p',p] and 

    {[                    f(p)=p; ∀ x ∈ [p',p) ∩ K,  f(x)>x ].
    .


  * There exists a point p_*∈ (p',p) ∩ K such that

    ∀ n ≥ 0, h(f^n(p_*))=f^n(p_*).


  * There exists a point p_0 in [p',p_*) ∩ K and an integer N > 0 such that

    {[                   h(p_0) ≠ p_0; ∀ n ≥ N, h(f^n(p_0))=f^n(p_0). ].


Then there exists N'>0 such that the semigroup generated by f^N' and f^N'h is free.


To obtain the wanted contradiction, use the above lemma with f_1, h and p_*=b. The last hypothesis is satisfied thanks to Claim <ref>. To complete the proof of Proposition <ref>, it suffices to prove those two lemmas.


Recall that the point p_0 is the endpoint of a connected component of ℝ∖Fix(g)  which meets K. Suppose that this connected component is of the form (p_0,p_1) (the case where it is of the form (p_1,p_0) is analogous). Observe that the interval (f_1^n(p_0), f_1^n(p_1)) is a connected component of ℝ∖Fix(f_1^ngf_1^-n). Then, as the elements h and f_1^ngf_1^-n, as well as the elements h^-1 and f_1^ngf_1^-n, are not crossed by Lemma <ref>, either h(f_1^n(p_0))=f_1^n(p_0) or h(f_1^n(p_0))≥ f_1^n(p_1) or h(f_1^n(p_1)) ≤ f_1^n(p_0): if none of those statements occur,


  * either h(f_1^n(p_0)) ∈ (f_1^n(p_0),f_1^n(p_1)) and the elements h and f_1^ngf_1^-n are crossed,

  * or h(f_1^n(p_1)) ∈  (f_1^n(p_0),f_1^n(p_1)) and the elements h and f_1^ngf_1^-n are crossed,
 
  * or h(f_1^n(p_0)) ≤ f_1^n(p_0) and h(f_1^n(p_1)) ≥ f_1^n(p_1). In this case, h^-1(f_1^n(p_0)) ≥ f_1^n(p_0) and h^-1(f_1^n(p_1)) ≤ f_1^n(p_1) and the elements h^-1 and f_1^ngf_1^-n are crossed.

Suppose for a contradiction that there exists a sequence of integer n_k→ +∞ as k → +∞ such that, for any k ≥ 0, h(f_1^n_k(p_0)) ≥ f_1^n_k(p_1). Then, for any k ≥ 0,


    h(f_1^n_k(p_0))-h(f_1^n_k(a))/f_1^n_k(p_0)-f_1^n_k(a)= h(f_1^n_k(p_0))-f_1^n_k(a)/f_1^n_k(p_0)-f_1^n_k(a)

as the point f_1^n_k(a) belongs to Fix(G'_1) and the diffeomorphism h belongs to G'_1.
Hence

    [ h(f_1^n_k(p_0))-h(f_1^n_k(a))/f_1^n_k(p_0)-f_1^n_k(a)                                                     ≥       f_1^n_k(p_1)-f_1^n_k(a)/f_1^n_k(p_0)-f_1^n_k(a);                                                                                                           ≥  f_1^n_k(p_1)-f_1^n_k(p_0)/f_1^n_k(p_0)-f_1^n_k(a)+1. ]

Denote by f̃_1:[a,p] → [a,p] a C^1+Lip-diffeomorphism such that f̃_1|[a,p] ∩ K=f_1|[a,p] ∩ K. By the mean value theorem, there exist points c_1 and c_2 of the interval (a,b) such that

    {[ f̃_1^n_k(p_1)-f̃_1^n_k(p_0)=(f̃_1^n_k)'(c_1)(p_1-p_0);    f̃_1^n_k(p_0)-f̃_1^n_k(a)=(f̃_1^n_k)'(c_2)(p_0-a). ].


Moreover, if we denote by K the Lipschitz constant of log(f') then

    [ | log((f̃_1^n_k)'(c_1))- log((f̃_1^n_k)'(c_2)) |                                                ≤                   K ∑_i=0^n_k-1| f̃_1^i([a,b]) |;                                                                                                 ≤                                     K | p-a |=M, ]

because the intervals f̃_1^i([a,b]) are pairwise disjoint and contained in the interval (a,p). Hence

    h(f_1^n_k(p_0))-h(f_1^n_k(a))/f_1^n_k(p_0)-f_1^n_k(a)≥   e^-Mp_1-p_0/p_0-a +1 >1.

However, recall that fixed points of h accumulate to p because h ∈ G'_1. Hence h'(p)=1, in contradiction with the above inequality and the continuity of h'.

In the case where there exists a sequence of integers n_k which tends to +∞ as k → + ∞ such that, for any k, h(f^n_k(p_1)) ≤ f^n_k(p_0) (or f^n_k(p_1) ≤ h^-1(f^n_k(p_0))), we find a similar contradiction by using h^-1 instead of h.



Let N'-1 be the largest integer n such that h(f^n(p_0)) ≠ f^n(p_0). Let f_1=f^N'. Hence, for any integer n > 0, the point f_1^n(p_0) is fixed under h (but the point p_0 is not). We want to prove that the semigroup generated by f_1 and g=f_1h is free. Let

    w_1= f_1^n_k g^m_k… f_1^n_2 g^m_2 f_1^n_1 g^m_1,

with n_k≥ 0, m_1≥  0 and  n_i>0, m_i>0 otherwise, and  

    w_2= f_1^n'_k' g^m'_k'… f_1^n'_2 g^m'_2 f_1^n'_1 g^m'_1,

with n'_k'≥ 0, m'_1≥  0 and  n'_i>0, m'_i>0 otherwise, be two distinct words on f_1 and g. We see each of those words as a diffeomorphism in 𝔡𝔦𝔣𝔣^1(K). Suppose for a contradiction that, as elements of the group 𝔡𝔦𝔣𝔣^1(K), w_1=w_2. Then, simplifying these words on the right and exchanging the roles of w_1 and w_2 if necessary, we can suppose that m_1>0 and m'_1=0. Now, let us look at the image of the point p_* under those two diffeomorphisms. We have

    {[         w_1(p_*)= f_1^n_1+m_1+n_2+m_2+ … +n_k+m_k(p_*); w_2(p_*)= f_1^n'_1+m'_1+n'_2+m'_2+ … +n'_k'+m'_k'(p_*) ].
    .


Hence ∑_i=1^k (n_i+m_i)=∑_i=1^k' (n'_i+m'_i). We denote by l the common value of these sums. Now, we will prove that w_1(p_0) ≠ w_2(p_0), in contradiction with the equality w_1=w_2 as elements of the group 𝔡𝔦𝔣𝔣^1(K).

We write

    w_1=w_3h

where 

    w_3=f_1^n_k g^m_k… f_1^n_2 g^m_2 f_1^n_1 g^m_1-1f_1.

Observe that w_2(p_0)=w_3(p_0)=f_1^l(p_0). As we supposed that w_1=w_2 as elements of 𝔡𝔦𝔣𝔣^1(K), we have w_3h(p_0)=w_1(p_0)=w_2(p_0) hence h(p_0)=p_0, a contradiction.



Now, let us deduce Proposition <ref> from Proposition <ref>.


 By Proposition <ref>, the group 𝒢_1,p is metabelian, meaning that its derived subgroup is abelian. A theorem by Rosenblatt (see <cit.>) states that any metabelian group without free subsemigroups on two generators is nilpotent. Hence the group 𝒢_1,p is nilpotent.

Suppose for a contradiction that the group 𝒢_1,p is not abelian and take a nontrivial element ξ in the center of the group 𝒢_1,p which belongs to the derived subgroup 𝒢'_1. 

Recall that, by Corollary <ref>, the group 𝒢_1,p contains an element whose representative has no fixed point in a left-neighbouhood of the point p. Hence, by Lemma <ref>, the element ξ has a representative g_ξ in G_1 with no fixed point on a left-neighbourhood of the point p.

We want to apply Lemma <ref> to prove that the group 𝒢_1,p is abelian and finish the proof of Proposition <ref>. In the rest of this proof, we make sure that the hypothesis of Lemma <ref> are satisfied.

Take α_0>0 such that the diffeomorphisms g_ξ and g_ξ^-1 are increasing on [p-α_0,p] and have no fixed point in [p-α_0,p) ∩ K. Taking ξ^-1 instead of ξ if necessary, we can further suppose that, for any point x of [p-α_0,p] ∩ K,

    g_ξ(x) >x.


Suppose for a contradiction that there exist a real number α∈ (0,α_0) and a diffeomorphism g ∈ G_1 with the following properties.


  * The diffeomorphism g is increasing on [p-α,p].

  * The diffeomorphism g has a fixed point p_1 in [p-α,p) ∩ K.

  * There exists a point x_0 in [p_1,p] ∩ K such that g(x_0) ≠ x_0.

As g([p_1,p])=[p_1,p] and g_ξ([p_1,p]) ⊂ [p_1,p], we have 

    g^-1g_ξg([p_1,p]) ⊂ [p_1,p] ⊂ (p-α_0,p)

and the diffeomorphism h=[g_ξ,g]=g_ξ^-1g^-1g_ξg is increasing on [p_1,p]. Take α' >0 small enough so that p-α_0 <p_1-α' and the diffeomorphism h is increasing on (p_1-α',p].

 
For any point x in (p_1-α',p] ∩ K, h(x) = x



Suppose that there exists a point p_0 in (p_1-α',p]∩ K such that h(p_0) ≠ p_0. As the element ξ of 𝒢_1,p lies in the center of 𝒢_1,p, the diffeomorphism h pointwise fixes a left-neighbourhood of p in K. Let

    p_*= inf{ x ∈ [p_0,p], ∀ y ∈ [x,p] ∩ K, h(y)=y }.

Then apply Lemma <ref> to h and f=g_ξ to find a free subsemigroup of G_1 on two generators, a contradiction.


By Lemma <ref>, for any point x in (p_1-α',p]∩ K,

    gg_ξ(x)=g_ξg(x).

Hence, by Lemma <ref>, the diffeomorphism g has no fixed point in (p_1-α',p], a contradiction.

Therefore, we can apply Lemma <ref> and the group 𝒢_1,p is abelian.



Fix a finite generating set S of G_1. As the point p is accumulated on the left by points of Fix(G_1), for any diffeomorphism s in S, there exists a point p_s<p in Fix(G_1) such that s is increasing on the interval [p_s,p]. Let

    p' = max{ p_s| s ∈ S }.

Then any element of S is increasing on L_p=[p',p] and preserves L_p∩ K. Hence any element of G_1= ⟨ S ⟩ is increasing on L_p and preserves L_p∩ K.

Now, let us prove by contradiction that the group G'_1 acts trivially on L_p∩ K. Suppose there exists a point x_0∈ (p',p) ∩ K which is displaced by some element of G'_1. Let (p_1,p_2) be the connected component of (p',p) ∖Fix(G_1) which contains the point x_0. Then the points p_1 and p_2 belong to Fix(G_1). Moreover, the point p_1 is accumulated on the right by points of K: otherwise, if I is connected component of ℝ∖ K whose left-end is the point p_1, then its right-end is a fixed point of G_1, in contradiction with (p_1,p_2) ∩Fix(G_1) = ∅. Likewise, the point p_2 is accumulated on the left by points of K. By Corollary <ref>, there exists a point p'_2<p_2 of K and an element f of G_1 such that 

    ∀ x ∈ [p'_2,p_2) ∩ K,  f(x) >x.

We then need the following lemma.

 

    [p'_2,p_2] ∩ K ⊂Fix(G'_1).



Of course, we can likewise prove that there exists a point p'_1>p_1 of K such that [p_1,p'_1] ∩ K ⊂Fix(G'_1).

Before proving this lemma, let us see why it gives us the wanted contradiction. The set

    A = { x ∈ [p_1,p_2] ∩ K |∃ g_1∈ G'_1,  g_1(x) ≠ x }

is a closed G_1-invariant set as the group G'_1 is a normal subgroup of G_1 and any element of G_1 preserves [p_1,p_2] ∩ K. Moreover, by Lemma <ref>, this set A is contained in [p'_1,p'_2]. Also, this set contains a minimal invariant set for the action of G_1 on K, hence a fixed point for G_1 by Proposition <ref>. Hence

    ∅≠ A ∩Fix(G_1) ⊂ [p'_1,p'_2] ∩Fix(G_1) ⊂ (p_1,p_2) ∩Fix(G_1) = ∅,

a contradiction.



Suppose for a contradiction that there exists a point x_0∈ [p'_2,p_2] ∩ K and an element h in G'_1 such that h(x_0) ≠ x_0. Then let

    p_max= sup{ x ∈ [p'_2,p_2] ∩ K, h(x) ≠ x }.

By Proposition <ref>, p_max < p_2. Observe that this point p_max is fixed under the diffeomorphism h. 

Finally, take a point p_0 < p_max of K such that h(p_0) ≠ p_0 and f(p_0) > p_max. Taking h^-1 instead of h if necessary, we can suppose that h(p_0) >p_0. Then, by Lemma <ref>, the group generated by h and f contains a free semigroup on two generators, a contradiction.





 §.§ End of the proof of Theorem <ref>
 

Now, we finish the proof of Theorem <ref>, namely we prove the following proposition.

 
Let G_1 be a subgroup of G which satisfies Proposition <ref>. Then
its derived subgroup G'_1 is trivial.



For any point x in Fix(G_1), we want to define a left neighbourhood L_x and a right-neighbourhood R_x of the point x in K which will be useful for the proof. To define those, we have to distinguish cases.


  * If the point x is accumulated on the left (respectively the right) by points of Fix(G_1), take a left-neighbourhood L_x (resp. a right-neighbourhood R_x) of x such that the set L_x (resp. R_x) is pointwise fixed under the elements of G'_1 (such a neighbourhood exists by Proposition <ref>).

  * If the point x is accumulated on the left (resp. the right) by points of K but isolated on the left (resp. the right) in Fix(G_1) then take a left-neighbourhood L_x (resp. a right-neighbourhood R_x) of x such that there exists a diffeomorphism f in G_1 such that


  * The diffeomorphism f is increasing on L_x (resp. R_x).

  * For any point y in L_x∖{ x }, f(y)>y (resp. for any point y in R_x∖{ x }, f(y)<y). 

Such a diffeomorphism f exists by Corollary <ref>.

  * If the point x is isolated on the left (resp. on the right) in K, take L_x= { x } (resp. R_x= { x }).

Let

    U= ⋃_x ∈Fix(G_1) (L_x∪ R_x)

and choose the neighbourhoods L_x and R_x in such a way that the set U is open in K.
Suppose for a contradiction that the group G'_1 contains anontrivial element h. Let A be a subset of K consisting of points x of K which are displaced under some element of the group G'_1, i.e. there exists an element h of G'_1 such that h(x) ≠ x. Of course, this set is disjoint from the set Fix(G_1). We can say even more by the following claim.

 
A ∩ U = ∅.


Before proving the claim, let us see how we can finish the proof of Proposition <ref>. As the set U is open, A∩ U = ∅. 

As the group G'_1 is a normal subgroup of the group G_1, the set A is a closed G_1-invariant subset. Hence there exists a minimal set M ⊂A for the action of G_1 on A. By Proposition <ref>, M ⊂Fix(G_1) ⊂ U, a contradiction with Claim <ref>.



Suppose for a contradiction that A ∩ U ≠∅ and take a point p_0 in the intersection A ∩ U. By definition of the set A, there exists an element h in G'_1 such that h(p_0) ≠ p_0.

By definition of U, there exists a point p in Fix(G_1) such that either p_0∈ L_p or p_0∈ R_p. Suppose for instance that the point p_0 belongs to the left-neighbourhood L_p of p. Necessarily, the point p is isolated on the left in Fix(G_1): otherwise, the diffeomorphisms in G'_1 pointwise fix L_p. Moreover, by construction of L_p there exists an element f in G_1 such that


  * For any point y in L_p∖{ p }, f(y)>y.

  * The diffeomorphism f is increasing on L_p.


Finally, we use Lemma <ref> (recall that the diffeomorphism h pointwise fixes a neighbourhood of p to find the point p_*). By this lemma, the group G_1 contains a free subsemigroup on two generators, a contradiction.





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