$Δ$-transitivity for several transformations and an application to the coboundary problem
Italo Cipriano Ryo Moore

   
   



   


   



   

   Δ-transitivity for several transformations]Δ-transitivity for several transformations and an application to the coboundary problem

[2010]37B05, 54H20




Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Santiago, Chile
icipriano@gmail.com
http://www.icipriano.org/ 



Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Santiago, Chile
rymoore@mat.puc.cl
https://sites.google.com/view/ryomoore/ 



I.C. was partially supported by CONICYT PIA ACT172001.



	R.M. was partially supported by  FONDECYT-CONICYT Postdoctorado 3170279.




Given a compact and complete metric space X with several continuous transformations T_1, T_2, … T_H: X → X, we find sufficient conditions for the existence of a point x∈ X such that  (x,x,…,x)∈ X^H has dense orbit for the transformation 
    𝒯:=T_1× T_2×⋯× T_H.


We use these conditions together with Livšic theorem, to obtain that for α-Hölder maps f_1,f_2,…,f_H: X→, the product ∏_i=1^H f_i(x_i) is a smooth coboundary with respect to 𝒯 is equivalent to the existence of a non-empty open subset U ⊂ X such that
		
    sup_Nsup_x∈ U| ∑_j=0^N∏_i=1^H f_i (T_i^j x) | < ∞.

       
   
   [
    Ryo Moore
    December 30, 2023
=====================





§ INTRODUCTION

Let (X, d) be a compact metric space, and T_i: X → X a continuous transformation (not necessarily all homeomorphisms) for i = 1, 2, … H for some integer H ≥ 2. There are two items of interests in this note. Firstly, we will be interested in the properties of these maps for which the set

    E_Δ := {x ∈ X:  The set {(T_1^nx, T_2^nx, …, T_H^nx): n ∈} is dense in  X }

is non-empty; in such case, we say the product map T_1 × T_2 ×⋯× T_H to be Δ-transitive. This topological multiple recurrence problem was originally studied by E. Glasner in <cit.> when each transformation is a power of a single transformation (i.e. T_i = T^i, where T is a continuous transformation on X). Later, T. K. S. Moothathu simplified the methods of Glasner <cit.>, and it was studied further by D. Kwietniak and P. Oprocha <cit.>. 

Generally, some variant of transitivity properties are assumed for each function, such as weak mixing, topological mixing, and/or minimality. It was shown by Moothathu that there exists a topologically mixing system for which the set E_Δ is empty <cit.>. Many of the previous studies on this topic has focused for the power of single transformations. In 2016, W. Huang, S. Shao, and X. Ye worked on this problem for several minimal and topologically weak mixing homeomorphisms that generate a nilpotent group <cit.>. We also note that I. Assani showed that if the maps T_i's are commuting (measure theoretically) weakly mixing homeomorphisms that preserve a common probability measure μ for which the measure of every non-empty open set is positive, then μ(E_Δ) equals to one; see the proof of <cit.>. One of the purpose of this note is to identify a sufficient condition for which the set E_Δ is nonempty without assuming every transformation is homeomorphism; in doing so, we consider additional topological conditions.

The second interest of this note is to obtain a Livšic type result for nonconventional ergodic sums (i.e. the sum that appears in the item (1) of Theorem <ref>). In 2018, I. Assani announced a preprint with the following result:


	Let (X, μ, T_1, T_2, …, T_H) be a measure preserving system, where  T_i is bi-measurable for each 1 ≤ i ≤ H, and f_1, f_2, …, f_H ∈ L^∞(μ). The following statements are equivalent.
	
		
  * We have
		
    sup_N∑_n=1^N ∏_i=1^H f_i ∘ T_i^n _L^∞(μ) < ∞.

		
  * There exists a real-valued function V on X^H such that the map x ↦ V ∘Φ^j(x, x, …, x) is essentially bounded and μ-measurable for every j ∈, and
		
    ∏_i=1^H f_i(x) = V(x, x, …, x) - V ∘Φ(x, x, …, x)  for μ-a.e.  x ∈ X

	


We note that Assani's result does not have any restrictions on each transformation aside from the fact that they preserve a common probability measure. We will show a topological analogue of such result, where we show the existence of continuous coboundary when a common invariant probability measure is absent. In particular, the Δ-transitivity condition allows one to show that the boundedness of the nonconventional sum is equivalent of showing the product of the functions is a coboundary with respect to the product transformation.



 §.§ Organization of the paper

In section <ref> we discuss the conditions that guarantee Δ-transitivity for the case we have two continuous transformations. The same is discussed in section <ref>, but for arbitrary many transformations. In section <ref>, we will discuss the coboundary problem.



§  Δ-TRANSITIVITY I


Let X be a compact metric space and T:X→ X,S:X→ X two continuous maps. We define the map T× S:X^2→ X^2, (x,y)↦ (Tx,Sy). A natural problem is to characterise topologically Δ-transitivity as property of T and S. The following theorem answer this.


Suppose that T and S are both weakly mixing and syndetically transitive, T is homeomorphism and T^-1S=ST^-1 is transitive. If there exists a homeomorphism A:X→ X, strongly transitive and that commutes with T and S. Then T× S is Δ-transitive.  



 Observe the following.


  * The theorem is still true if T is not necessarily an homeomorphism, but S=T^kS̃ for some k≥ 1, S̃:X→ X is a continuous map that commutes with T, and T^k-1S̃ is transitive.

  * The theorem is still true if A is not necessarily an homeomorphism, but A=T.



The main ingredients in the proof are the following theorem (similar to <cit.>, Prop. 1) and lemma (similar to <cit.>, Lem. 3).


Let T_i: X → X be continuous maps for 1 ≤ i ≤ H, where H≥ 2. The following statements are equivalent.

	
  * Given any non-empty sets U_0, U_1, U_2, …, U_H, there exists n for which
	
    U_0 ∩ T_1^-nU_1  ∩ T_2^-nU_2 ∩⋯∩ T_H^-nU_H ≠∅.

	
  * T_1 × T_2 ×⋯ T_H is Δ-transitive, i.e. there exists a G_δ set Y ⊂ X such that for any y ∈ Y, we have {(T_1^ny, T_2^ny, …, T_H^ny) : n ∈ℕ} is dense in X^H.

In addition, if one of the transformation commutes with every other transformation, then the following statement is also equivalent to the ones above:

	
  (3) There exists x ∈ X for which the set {(T_1^nx, T_2^nx, …, T_H^nx) : n ∈ℕ} is dense in X^H.




	Assume that (1) is satisfied. Let {B_k:k∈} be a countable base of open balls of X. If
	
    Y=⋂_(k_1,…,k_H)∈^H⋃_n∈⋂_i=1^H T_i^-n(B_k_i).

	Then, by the Baire Category Theorem, Y is a dense G_δ subset of X, and by construction, every x∈ Y satisfies (2).
	
	Now assume that (2) is satisfied. Then, for every x∈ Y∩ U_0 there exists n∈ such that (T_1^nx, T_2^nx, … , T_H^nx)∈ U_1× U_2×⋯× U_H, then ∩_i=0^H T_i^-n(U_i)≠∅.
	
	To show that (3) implies (1), suppose that there exists i such that T_i commutes with all T_j for 1≤ j≤ H and A=. Choose k∈ such that y=T_i^k(x)∈ U ∩ Y, then 
    {(T_1^ny, T_2^ny, …, T_H^ny): n ∈}
 is dense in X^H, because T_i commutes with all T_j for 1≤ j≤ H. In particular, there exists n∈ such that (T_1^ny, T_2^ny, ⋯, T_H^ny)∈ U_1× U_2×⋯× U_H, then ∩_i=0^H T_i^-n(U_i)≠∅.


 It is clear that (2) implies (1). Under no commutative assumptions (1) implies (2) and (2) implies (3).



Suppose that T:X→ X and S:X→ X  are continuous, A: X→ X is a homeomorphism and that T× S:X^2→ X^2 is topologically transitive. If V_T and V_S are nonempty open subsets of X. Then there exists a sequence of integers {k_n}_n=0^∞ such that k_n>n for n≥ 0 and there exist sequences of nonempty open sets {V_T^(n)}_n=0^∞ and {V_S^(n)}_n=0^∞ such that 
for every n≥ 0 

    V_T^(n)⊂ V_T, V_S^(n)⊂ V_S,


    T^k_j A^-j (V_T^(n))   ⊂ V_T j=0,…,n, 
    
    S^k_j A^-j(V_S^(n))   ⊂ V_S j=0,…,n.
 



Let V_T and V_S be nonempty open subsets of X and let W:=V_T× V_S. The proof follows by induction on n. Base case (n=0): By transitivity of T× S there exists k_0∈ such that (T× S)^k_0W∩ W≠∅, i.e., V_T^(0):=V_T∩ T^-k_0 V_T⊂ V_T and V_S^(0):=V_S∩ T^-k_0 V_S⊂ V_S are nonempty open sets with the properties needed. Indeed, 

    ∅≠ T^k_0(V_T^(0))⊂ T^k_0(V_T) ∩ T^k_0(T^-k_0 V_T)⊂ T^k_0(V_T^(0))⊂ T^k_0(V_T) ∩  V_T⊂ V_T

and similarly for V_S^(0). Inductive step: Assume there exists {k_0,…,k_n-1} and {V_T^i}_i=0^n-1, {V_S^i}_i=0^n-1 with the properties of the lemma. We define U_T:=A^-n(V_T^(n-1)), U_S:=A^-n(V_S^(n-1)) and U=U_T× U_S. By transitivity of T× S there exists k_n>n such that (T× S)^k_n U ∩ W ≠∅, i.e., Ũ_T:=U_T∩ T^-k_n V_T and Ũ_S:=U_S∩ T^-k_n V_S are nonempty open sets. We define V_T^(n):=V_T^(n-1)∩ A^n T^-k_n  (V_T) and  V_S^(n):=V_S^(n-1)∩ A^n S^-k_n (V_T). Observe that

    ∅≠ A^n(Ũ_T)⊂ A^n (U_T)∩ A^n(T^-k_n V_T)⊂ V_T^(n-1)∩ A^nT^-k_n (V_T)=V_T^(n)

and similarly 

    ∅≠ A^n(Ũ_S)⊂ A^n (U_S)∩ A^n(S^-k_n V_S)⊂ V_S^(n-1)∩ A^nS^-k_n (V_S)=V_S^(n).

Also,

    T^k_nA^-n(V_T^(n))   ⊂ T^k_nA^-n(V_T^(n-1)) ∩ T^k_nA^-n(A^n T^-k_n  (V_T))
       = T^k_nA^-n(V_T^(n-1)) ∩ T^k_n T^-k_n (V_T)
       ⊂ T^k_nA^-n(V_T^(n-1)) ∩ V_T
       ⊂ V_T,

similarly for V_S^(n) (replacing T by S).
Finally, by definition of V_T^(n) we have that V_T^(n)⊂ V_T^(n-1), then for j=0,…, n-1 

    T^k_jA^-j(V_T^(n))⊂ T^k_jA^-j(V_T^(n-1)),

using the inductive hypothesis we obtain that 

    T^k_jA^-j(V_T^(n))⊂ V_T,

similarly for V_S^(n) (replacing T by S). This concludes the proof.


Now we proceed to the prove of Theorem <ref>.


We use the Theorem <ref> to charactetise Δ-transitivity. Let U,V_T,U_S nonempty open subsets of X. We use the strong transitivity of A to find M such that ∪_i=0^M A^i U=X. We use proposition 4 in <cit.> to obtain that T× S is transitive. We use Lemma <ref> to find  {k_j}_j=0^M,  V_T^(M) and V_S^(M). We use the transitivity of T^-1S to obtain n∈ such that (T^-1S)^n(V_T^(M))∩ V_S^(M)≠∅. We fix an a∈ V_T^(M) and b∈ V_S^(M) such that (T^-1S)^n a=b. We find a y∈ X such that T^ny=a. We use strong transitivity of A to find an l∈{0,…, M} and x∈ U such that A^l x=y. We use Lemma <ref> to prove that T^k_lA^-la∈ V_T and S^k_lA^-lb∈ V_S. By construction, we have x∈ U such that

    T^l A^-l  T^n A^l x∈ V_T

and 

    T^l A^-l (T^-1S)^n  T^n A^l x∈ V_S.

By the hypotheses that A and T are homeomorphism, A commutes with T and S, and that ST^-1=T^-1S, we conclude the result.


We find other sufficient condition for Δ-transitivity motivated by the following property that appears without name in <cit.>.


We say that a dynamical systems (X,T) satisfies the density condition if for any non empty open subsets U, V ⊂ X, there exists a nonempty open set W ⊂ V and n_0 ∈ such that W ⊂ T^n(U) for any n ≥ n_0.



Suppose that (X,T) is a homeomorphism that satisfies the density condition and T^-1S=ST^-1 is syndetically transitive. Then T× S is Δ-transitive.


 The proposition is still true if T is not necessarily an homeomorphism, but S=T^kS̃ for some k≥ 1, S̃:X→ X is a continuous map that commutes with T, and T^k-1S̃ is syndetically transitive.



We use the Theorem <ref> to charactetise Δ - transitivity. Let U,V_T,U_S nonempty open subsets of X. We use the density condition to find M and W⊂ V_T such that W⊂ T^n(U) for n≥ M. We use the syndetic transitivity of T^-1S to obtain n≥ M such that (T^-1S)^n(W)∩ V_S≠∅. We fix an a∈ W and b∈ V_S such that (T^-1S)^n a=b. We find x∈ U such that T^nx=a. By construction, we have x∈ U such that

    T^n x∈ V_T

and 

    T^n (T^-1S)^n x∈ V_S.

By the hypotheses that ST^-1=T^-1S, we conclude the result.





 §.§ Examples


The following examples are easy to prove.


Let (Σ,σ) be a mixing subshift of finite type and (p,q) a pair of different positive integers. If T=σ^p and S=σ^q. Then T× S is Δ-transitive.



Let X=[0,1], f:X→ X be a continuous and mixing map, and (p,q) a pair of different positive integers. If T=f^p and S=f^q. Then T× S is Δ-transitive.



Let  X={1,2,…,N}^, (X,F) be a topologically mixing cellular automata with anticipation a and σ:X→ X be the shift. If b>a, T=σ^b and S=F. Then T× S is -transitive.




§  Δ-TRANSITIVITY II 


Let X be a compact metric space and T_1,T_2,…,T_H:X→ X be continuous maps for H≥ 2. In this section we characterise topologically Δ-transitivity for the product transformation 
    T_1× T_2×⋯× T_H:X^H→ X^H.

The main theorem of this section is the following.


Suppose that T_1,T_2,…,T_H are weakly mixing and syndetically transitive, T_1,…, T_H-1 are homeomorphism, T_i^-1T_i+1=T_i+1T_i^-1 satisfies the density condition for i=1,…,H-2 and T_H-1^-1T_H=T_HT_H-1^-1 is transitive. If there exists a homeomorphism A:X→ X, strongly transitive and that commutes with T_i for every i=1,2,…,H. Then T_1× T_2×⋯× T_H is Δ-transitive. 


Immediately we obtain the following corollary.


Suppose that T_1,T_2,…,T_H are minimal homeomorphisms, weakly mixing and commuting such that T_i+1T_i^-1 satisfies the density condition for i=1,…,H-2 and T_H-1^-1T_H=T_HT_H-1^-1 is transitive. Then T_1× T_2×⋯× T_H is Δ-transitive.




We use Theorem <ref> to characterise Δ-transitivity. Let U,V_T_1, V_T_2, …, V_T_H be nonempty open subsets of X. We use the strong transitivity of A to find M such that ∪_i=0^M A^i U=X. We use a direct generalization of proposition 4 in <cit.> to obtain that T_1× T_2×⋯× T_H is transitive. We use a direct generalisation of Lemma <ref> to find {k_j}_j=0^M,  V_T_1^(M), …, V_T_H^(M) with the properties that 

    V_T_i^(M)⊂ V_T_i

and

    T_i^k_jA^-jV_T_i^(M)⊂ V_T_ij=0,…,M,

for every i=1,…,H.

We use the density condition of T_1^-1T_2 to find W_2⊂ V_T_2^(M) such that 
W_2⊂ T_1^m (V_T_1^(M)) for every m≥ M_1. Similarly, we find W_3⊂ V_T_3^(M) such that 
W_3⊂ T_2^m (W_2) for every m≥ M_2. We repeat the argument up to finding W_H-1⊂ V_T_H-1^(M) such that 
W_H-1⊂ T_H-2^m (W_H-2) for every m≥ M_H-2.


We use that T_H-1^-1T_H is transitive to find 
    n≥max{M_i: i=1,…,H-2}
 such that 
    (T_H-1^-1T_H)^n W_H-1∩ V_T_H^(M)≠∅.


We find a_H∈ V_T_H^(M), a_H-1∈ W_H-1, …,  a_2∈ W_2, a_1∈ V_T_1^(M) such that 

    (T_1^-1 T_2)^n a_1= a_2, (T_2^-1 T_3)^n a_2= a_3,…, (T_H-1^-1 T_H)^n a_H-1= a_H.


We find a y∈ X such that T^ny=a_1. We use strong transitivity of A to find an l∈{0,…, M} and x∈ U such that A^l x=y. We use a direct generalization of Lemma <ref> to prove that T_i^k_lA^-la_i∈ V_T_i for every i=1,…,H. By construction and the assumptions on A we conclude the result.




 Observe the following.


  * The theorem is still true if T_i are not necessarily homeomorphisms, but

    T_1   =T
    
    T_2   =S_1T_1^a_1
    
    T_3   =S_2T_2^a_2
       ⋮
    
    T_H   =S_H-1T_H-1^a_H-1,

where a_i>1, S_i S_j=S_j S_i and S_i T= T S_i for every i,j=1,…,H.

  * The theorem is still true if A is not necessarily an homeomorphism, but A=T and 

    T_1   =S_1T^a_1
    
    T_2   =S_2T^a_2
       ⋮
    
    T_H   =S_HT^a_H,

where a_i≥ 1, S_i S_j=S_j S_i and S_i T= T S_i for every i,j=1,…,H.





We can state a lemma analog to Proposition 3 (iv) in <cit.> that gives an example of a product transformation that is not Δ - transitive. 


Let t≥ 2, (p_1,…,p_t)∈^t with p_i≠ p_j for i≠ j, and T_i= T^p_i for i=1,…,t. There exists a shift space X such that for T the shift action, T_1× T_2×⋯× T_H is not -transitive.



We can assume that p_1<⋯<p_t. Let consider X to be the subset of {0,1}^_0 such that for any x∈ X, 11 does not appear in x and if 

    1u_1 1 u_21⋯ 1 u_t 1

is a subwords of x for u_1,…,u_t∈{0,1}^*:=∪_n∈{0,1}^n with |u_i|:= (u_i) =n_i for i∈ [t], we have that for 
    n_0:=min{n∈: min_i∈ [t){n(p_i+1-p_i)}-1≥ 1}
 the vector 

    (n_1,…,n_t)∉∪_n≥ n_0{n(p_1,p_2-p_1,…, p_t-p_t-1)-(1,…,1)}.

Clearly X is T invariant for T:X→ X the action of the shift (that is continuous). By contradiction, assume that T_1×⋯× T_H is -transitive. Then there exists n∈ such that C(1) ∩_i∈ [t] T^-p_i nC(1) ≠∅, where C(1):={w∈ X: w_0=1}. Therefore any sequence x∈ C(1) ∩_i∈ [t] T^-p_i nC(1) is of the form x=1u_1 1u_21⋯ 1 u_t 1… , where u_1=0^n p_1-1 and u_i+1=0^n (p_i+1- p_i) for every 1≤ i <t, and by definition of x∉ X, which is a contradiction.





§ CONTINUOUS COBOUNDARIES FOR THE PRODUCT OF SMOOTH FUNCTIONS


Let (X, T) be a topological dynamical system, where X is a compact metric space and T: X → X is a continuous map. We say a real-valued continuous function f ∈𝒞(X):= 𝒞(X, ) is a coboundary if there exists g ∈𝒞(X) such that f = g - g∘ T. There are two well studied topological conditions under which a continuous function f is a coboundary: the first are provided by Livšic theorem for transitive, surjective continuous maps and the second by Gottschalk-Hedlund theorem for minimal systems. We use here only Livšic theorem. 


	Let T:X→ X be a transitive, surjective continuous map that satisfies the closing property (see Definition <ref>). Let f:X→ be a α-Hölder map (see Definition <ref>) such that for any p in X with T^k p=p for some k in , the sum  ∑_j=0^k-1f(T^j p) is equal to zero. Then there exists g:X→ such that it is α-Hölder and g - g∘ T =f.



In this section we apply the results on Δ - transitivity of 

    𝒯:=T_1× T_2 ×⋯× T_H
 to provide a version of Livšic that only requires to check a condition on the diagonal subspace Δ := {(x, x, …, x) ∈ X^H: x ∈ X } of 𝒳:= X^H.




 §.§ Livšic Theorem for Δ - transitive products


Let (X,d) be a compact metric space, that we assume to be separable, complete, and without isolated points, and let T:X→ X be a homeomorphism. We recall the closing property. 
 We say that a dynamical systems (X,T) satisfies the closing property if there exists D,δ,δ_0>0 such that for all x in X and k in  with d(x,T^k x)<δ_0 there exists p in X such that T^k p=p and such that 
    d(T^i x, T^i p)≤ D d(x,T^k x)e^-δmin{i,k-i} 0≤ i ≤ k.


This conditions (and the density condition) are satisfied by many well studied dynamical systems, for example, subshifts of finite type and hyperbolic diffeomorphisms on compact manifolds.


We define the space of α-Hölder functions on X.


 We say that a function f:X→ is α-Hölder for α∈(0,1] if there exists a constant C>0 such that for every pair of points x,y in X,
 
    |f(x)-f(y)|≤ C d (x,y)^α.



Given H in  and the continuous maps T_i:X→ X for 1≤ i ≤ H, we obtain a Livšic Theorem for multiple ergodic averages, that is, we consider the metric space (𝒳:=X^H,d_H) with 
    d_H(x,y):=max_1≤ i  ≤ Hd(x_i,y_i),
 and the continuous map 
    𝒯:=T_1× T_2 ×⋯× T_H : 𝒳→𝒳.


Under the assumption that 𝒯 is Δ - transitive, one can use Theorem <ref> to obtain the following nonconventional Livšic theorem.

 Let f:𝒳→ be an α-Hölder map. If 𝒯 is Δ - transitive and satisfies the closing property.
Then the following statements are equivalent.
	
		
  * There exists a non empty open subset U⊂ X such that 
    sup_nsup_x∈ U| ∑_j=0^n f(𝒯^jx̂) | < ∞ where x̂ := (x, x, …, x)∈Δ.
 		
		
  *  For any p ∈𝒳 such that T^k p = p, we have
		
    ∑_j=0^k-1 f(𝒯^j p) = 0.

		
  *  There exists V:𝒳→ such that it is α-Hölder and V - V ∘𝒯=f.
		
  *  There exists D > 0 such that
		
    sup_nsup_x ∈𝒳| ∑_j=0^n f(𝒯^j x)  | < D.

	



We note that statement (i) only requires one to check the boundedness of the sums along Δ, as opposed to the entire space 𝒳, that would be the case of applying directly Livšic Theorem to the map 𝒯 : 𝒳→𝒳. This may be an easier property to check than the others, and leaves a possibility for a simulation to find if such open set U exists.



We notice that the theorem can be written in terms of (X,T_1,…,T_H) instead of (𝒳,𝒯). Indeed, Theorem <ref> gives sufficient conditions on T_1,…,T_H such that 𝒯 is Δ - transitive It is also possible to prove that  𝒯 satisfies the closing property if each T_i satisfies it for every i=1,…, H. Indeed, we have the following lemma.


If (X, T_i) satisfies the closing property for every 1≤ i ≤ H. Then (𝒳, 𝒯=T_1×⋯× T_H) also satisfies the closing property.



Assume that (X, T_i) satisfies the closing property for every 1≤ i ≤ H. By the definition, for every 1≤ i ≤ H, there exists D_i,δ_i,δ_0(i)>0 such that for all x in X and k in  with d(x,T_i^k x)<δ_0(i) there exists p_i in X such that T_i^k p_i=p_i and such that d(T_i^j x_i, T_i^j p_i)≤ D_i d(x_i,T_i^k x_i)e^-δ_i min{j,k-j}  for all  j=0,…,k. Therefore, choosing D=max_1≤ i ≤ HD_i,δ=min_1≤ i ≤ Hδ_i and δ_0=min_1≤ i ≤ Hδ_0(i), we obtain that for all x=(x_1,…,x_H) in 𝒳 and k in  with d_H(x,𝒯^k x)<δ_0 there exists p=(p_1,…,p_H) in 𝒳 such that 𝒯^k p=p and such that d_H(𝒯^j x, 𝒯^j p)≤ D d_H(x,𝒯^k x)e^-δmin{j,k-j}  for all  j=0,…,k.




 §.§ Topological analog of a result by Assani


Let consider (𝒳, 𝒯), where 𝒳= X^H and 𝒯 = T_1 × T_2 ×⋯× T_H, for some H≥ 2. Assume that each T_i:X→ X is continuous for any i=1,…,H.

We say that F = ⊗_i=1^H f_i defined by ⊗_i=1^H f_i (x_1,…, x_H)=∏_i=1^H f_i (x_i) is a measure-theoretic coboundary if there exists a real-valued function V on 𝒳 such that the map x ↦ V ∘𝒯 (x, x, …, x) is essentially bounded and μ-measurable, and 
    F(x, x, …, x) = V(x, x, …, x) - V ∘𝒯(x, x, …, x)
 for μ-a.e x ∈ X.

A natural problem is finding conditions on (X,μ,T_1,…,T_H) so that F is a measure-theoretic coboundary. Recently, I. Assani <cit.> obtained a necessary and sufficient condition for this case, see Theorem <ref>.


	The proof uses the diagonal-orbit measure of μ (cf. <cit.>), which is a tool introduced in <cit.> to study the pointwise convergence of nonconventional ergodic averages. Diagonal-orbit measures were used to describe the behavior of the nonconventional ergodic sums and averages along the orbit of the diagonal space Δ⊂𝒳 iterated by the map 𝒯.


	By contrast to Assani's work, we show that for a certain class of topological system (X, T_1, T_2, …, T_H) and under certain smoothness assumption of the functions f_1, f_2, …, f_H, one can find necessary and sufficient condition so that ⊗_i=1^H f_i is a topologically smooth coboundary with respect to the map 𝒯. This answers a question that was raised during a discussion between the second author and S. Donoso: When is the product function ⊗_i=1^H f_i a continuous coboundary?
	
 Let f_1,…,f_H be α-Hölder maps. Suppose that (X,T_1,T_2,…,T_H) satisfies the hypotheses of Theorem <ref>. Then the following statements are equivalent.
	
		
  *  There exists a non-empty open subset U ⊂ X such that		
		
    sup_Nsup_x∈ U| ∑_j=0^N∏_i=1^H f_i (T_i^j x) | < ∞.

		
		
  *  The product of the function is a α-Hölder coboundary, i.e. if 𝒯=T_1× T_2 ×⋯× T_H, there exists V α-Hölder such that
		
    ⊗_i=1^H f_i = V - V∘𝒯.

	



The proof is direct from Theorem <ref> and <ref>, and the following lemma.


	If f_1,…,f_H are α-Hölder. Then ⊗_i=1^H f_i is α-Hölder.


In the proof we use the following notation. Given H ≥ 2, x=(x_1,x_2,…,x_H) and f := ⊗_i=1^H f_i , we define f^* := ⊗_i=1^H-1 f_i and x^* := (x_1,x_2,…,x_H-1).


	We proceed by an induction on H. The base case H=1 is trivial. Assume that f_1,…,f_H are α-Hölder and the property is valid for H-1 (i.e. f^* is α-Hölder). Let x,y∈𝒳.  We have that
	
    |f(x)-f(y)|
    	   = |f^*(x^*)f_H(x_H) - f^*(y^*)f_H(y_H)| 
       ≤ |f^*(x^*)||f_H(x_H) - f_H(y_H)| + |f_H(y_H)||f^*(x^*) - f^*(y^*)|.

	Using that f_H is  α-Hölder we obtain that |f_H(x_H) - f_H(y_H)|≤ C_1 d(x_H,y_H)^α for some C_1>0. Using that f^* is α-Hölder we obtain that |f^*(x^*) - f^*(y^*)|≤ C_2 d_m-1(x^*,y^*)^α for some C_2>0. Using the compactness of the space X (therefore also of X^H-1) and the continuity of f_H and f^*, we have that there exist constants C_3,C_4>0 such that C_3:=sup{ |f_H(x)|:x∈ X} and C_4:=sup{ |f^*(x)|:x∈ X^H-1}. Therefore, for C:=max{C_4 C_1,C_3 C_2}, we have that
	
    |f(x)-f(y)|≤ C_4 C_1 d(x_H,y_H)^α+ C_3 C_2 d_H-1(x^*,y^*)^α≤ 2 C d_H(x,y)^α.

	







§ ACKNOWLEDGMENTS

  We are grateful for Mario Ponce, Sebastián Donoso, and Godofredo Iommi for having fruitful discussions with us. We also thank Idris Assani for his comments.

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