On the convergence of spherical multiergodic averages for Markov action groups

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1 On the convergence of spherical multiergodic averages for Markov action groups A.M. Mesón and F. Vericat Abstract. In this note we study the norm convergence of multiple ergodic spherical averages from actions of word hyperbolic groups or more generaly strongly Markov groups. The objective is the generalization of the results by Bufetov, Khristoforov and Klimenko and by Pollicott and Sharp. We use the techniques that Walsh introduced for proving the norm convergence of multiple ergodic averages from measure-preserving transformations and for integer valuated polynomials. M.S.C. 200: 37C85 Key words: Multiergodic averages; Markov groups; complexity of systems. Introduction The classical Birkhoff ergodic theorem on a probability space (X, B, µ) states the convergence of averages of real valuated maps on X along orbits of measure preserving transformations T : X X. One important subject is the extension of the ergodic theorem with action groups as dynamics. Among the most relevant contributions about the pointwise convergence of ergodic averages for action groups the following works can be mentioned. In [2] Nevo and Stein proved the pointwise ergodic convergence for finite measure preserving actions of the free group F r, r 2,and in [5] Fujiwara established an ergodic convergence for exponentially mixing actions of wordhyperbolic groups. Lindenstrauss[] considered actions of amenable groups Γ on a Lebesgue space (X, µ) and maps φ : X R, to analyze averages of the form φ (γx) dm Γ (γ), m Γ (E) E where E Γ and m Γ is the Haar measure on Γ. In that article was proved the convergence of these averages under the condition of the existence of adequate sequences (F n ) in Γ, called Følner sequences. Bufetov[2] proved an ergodic theorem of spherical averages for free semigroups. Later Bufetov and Series[4] applied this result to prove the ergodic convergence for a class of Fuchsian groups which includes surface groups, i.e. the fundamental groups of surfaces of genus g 2. Applied Sciences, Vol.7, 205, pp c Balkan Society of Geometers, Geometry Balkan Press 205.

2 Spherical multiergodic averages 63 Let Γ be a finitely generated Markov groups Γ and let S(n) be the sphere of radius n word metric, i.e. S(n) = {γ : γ = n}, where γ, is the minimal number of generators needed to represent γ. If γ Γ, then by T γ is denoted the transformation in X given by T = T γ (x) = γx, now for a map φ L (X), can be considered the following average S N, φ(x) := N cards(n) φ (T γ (x)) (When cards(n) = 0 then we set S N, φ(x) = 0). Recently Bufetov, Khristoforov and Klimenko[3] analyzed these spherical averages and proved the pointwise convergence of Cesáro averages for the spherical sequence {S N, φ(x)}.the word-hyperbolic groups are Markov groups, for this special case Pollicott and Sharp [5] gave a more concise proof of the main result of [3]. Multiple ergodic averages appeared as a dynamical version of the Szemeredi theorem in combinatorial number theory. The analogy was made by Furstenberg[6] who studied ergodic averages in a measure-preserving probability space (X, B, µ, T ) of the form (.) µ ( A T n A... T kn A ), N where A B and j N. Furstenberg established that if µ (A) > 0 then lim inf N N µ ( A T n A... T jn A ) > 0. This relevant result serves to prove by arguments from Ergodic Theory the Szemeredi theorem, which states that if S is a set of integers with positive upper density then S contains arithmetic progressions of arbitrary length. After this the task was the study of the convergence of averages like (.2) N i= j ( φ i T in ), where T : X X is an invertible measure preserving transformation and φ, φ 2,..., φ j L (X). Furstenberg proved the L 2 convergence for the case j = 2, of course for the case j = is obtained the L 2 Von Neumann ergodic theorem. For any j, Host and Kra have demonstrated the convergence in L 2 (X) of the multiergodic averages (.2). The machinery used was mainly the theory of factors developed by Host-Kra- Ziegler and the approximation of multiergodic averages by averages on nyl-systems. A more general problem is to consider the following averages

3 64 A.M. Mesón and F. Vericat (.3) N j i= ) φ i (T pi(n) x, where p i are polynomials valuated in Z. The average (.3) corresponds to the case of j ( ) linear polynomials. Bergelson [] proved the existence of lim φ i T p i(n) N N i= under the assumption that T ergodic. Later Leibman[0] proved convergence without assuming ergodicity. A more ambitious problem was to consider a nilpotent group G of measure preserving transformations of a probability space (X, µ) and study the norm convergence of averages (.4) N j i= T p i,(n)...t p i,,l(n) l φ i, where T,..., T l G and p i,,..., p i,l Z [n]. The average (.3) corresponds to the case of T =... = T l, i.e. when G is a cyclic group. The L 2 convergence of (.4) was recently proved by Walsh[6], who used for the demonstration an argument based doing induction on the complexity of systems of polynomial sequences in G. He also adapted results by a decomposition theory of functions developed by Gowers[8]. Another contribution in this direction is the work of Zorin-Kranich[7] who proved, using the Walsh argument, the convergence in norm of averages along Følner sequences in an amenable group. The aim of this article is to study multiple ergodic spherical averages for hyperbolic, or more generally Markov action groups, in order to obtain a multiergodic version of the results in [3] and [5]. The novelty of our work with respect to the Walsh one is that we consider nonconmutative action groups and with respect to the Zorin-Kranich article is that we have no need to impose the existence Følner sequences. The result to be proved herein is Theorem.: Let (X, µ) be a measure space and let Γ be a finitely generated Markov group. Let us consider actions h, h 2,..., h j of Γ on X, i.e. for any γ Γ are defined measure-preserving maps h i (γ) : X X, i =, 2,.., j. If the actions commute in the sense that h i (γ 2 ) h k (γ ) = h k (γ ) h i (γ 2 ), for any γ, γ 2 Γ and i k then for any φ, φ 2,..., φ j L (X), the multiergodic sequence (.5) N cards(n) φ (h (γ)x) φ 2 (h (γ)h 2 (γ)x)...φ j (h (γ)...h j (γ)x), converges in L 2 norm. In order to apply the Walsh techniques notice the following, if h i, i =, 2,..., j is a measure-preserving action of Γ on X then can be defined maps g i : Γ G, where G is a group of unitary operators on a space of maps, for our purposes we can take G a nilpotent group of unitary operators on L 2 (X). Thus each map g i, i =, 2,..., j, is defined as g i (γ) [φ] (x) = φ (h i (γ)(x)). The maps g i are antihomomorphisms, i.e. g i (γ γ 2 ) = g i (γ 2 ) g i (γ ). Thus Theorem. can be proved from this more general result:

4 Spherical multiergodic averages 65 Theorem.2. Let (X, µ) be a measure space and let Γ be a finitely generated Markov group. Let g, g 2,..., g j : Γ G, with G a nilpotent group of unitary operators on L 2 (X). If the system g = (g, g 2,..., g j ) has finite complexity then the sequence (.6) N cards(n) g (γ)φ g 2 (γ)φ 2...g j (γ)φ j converges in L 2 norm, for any φ, φ 2,..., φ j L (X). The concept of complexity of a system of polynomial sequences was given by Walsh [6] who considered the maps g, g 2,..., g j : Z G, with (.7) g i (n) = T p i,(n)...t p i,,l(n) l, where T,..., T l G and p i,,..., p i,l Z [n]. The definition of complexity can be extended for non commutative groups. The innovative argument of [6] consists in using the complexity as induction parameter, so that is necessary that the system of maps g, g 2,..., g j have finite complexity. Systems with finite complexity can be reduced to the trivial system ( G ) in a finite number of steps. The polynomial systems are special systems of maps, for instance if Γ = Z, then g (n) = T p (n)...t p,l(n) l with T,..., T l measure-preserving transformations and each p i Z [n]. Polynomial systems have finite complexity, this was proved for Γ = Z in [6], and for non commutative Γ in [7]. 2 Markov groups Let Γ be a group with a finite symmetric set of generators S. If G is a directed graph then E [G] denotes the set of edges of G and P [G, v] denotes the set of finite paths in G starting in the vertex v. The group Γ is a Markov group if there is a labelling map λ : E [G] S, and a distinguished vertex v 0, such that i) λ can be lifted to a map λ : P [G, v 0 ] Γ which is a bijection, so that for to a path of vertices v 0, v,..., v n can be assigned the value λ (v 0 v ) λ (v v 2 )...λ (v n v n ). ii) if γ is represented by the labelling λ (v 0 v ) λ (v v 2 )...λ (v n v n ) then γ = n. The word metric in a group Γ, with a finite symmetric set of generators S, is defined by d (γ, γ 2 ) = γ, γ 2, where γ = γ S is the minimal number of elements of S needed to represent γ. A group Γ is word hyperbolic, in Gromov sense, if there is a number δ > 0 if in any geodesic triangle every point in one side of it is within distance δ of the other sides. A group is hyperbolic if and only if its Cayley graph is hyperbolic. Any word-hyperbolic group is a Markov group[7]. Recall that S(n) = {γ : γ = n}, a known fact is that a word hyperbolic group has exponential growth rate with respect to the word length, i.e. if ρ = lim log cards(n) n n

5 66 A.M. Mesón and F. Vericat then there are constants C 2 > C > 0 C ρ n cards(n) C 2 ρ n. Thus for ergodic averages for word hyperbolic actions groups can be considered weights ρ n instead of cards(n). 3 Complexity of systems Let g = (g, g 2,..., g j ) be an ordered tuple of maps from Γ to G, which is called a system. If a, b Γ then derivative with respect to (a, b) of map g : Γ G at the point γ is (3.) D (a,b) (g) (γ) := g (γ) g (aγb), and D (a,b) ( g ) (γ) := g(γ)g (aγb). Let g, h : Γ G, and set (3.2) g, h (a,b) (γ) := D (a,b) ( g ) (γ)t (a,b) (h) (γ), where T (a,b) (h) (γ) = h (aγb). The (a, b) reduction of the system g = (g, g 2,..., g j ) is defined to be the system (3.3) g (a,b) = ( g, g 2,..., g j, g j, g (a,b),... g j, g j (a,b) ).. The concept of complexity is given by induction in the following way. A system constituted only by ( the identity ) G map has complexity 0, it is denoted by cplx ({ G }) 0. If cplx g(a,b) C, for any a, b Γ then cplx(g) C. The property known as cheating says that if are added constants, repeated maps or repeated maps multiplied by a constant to a system it does not change its complexity. For example cplx(g, c g, c, g, g 2, c 2 g 2 ) = cplx(g, g 2 ). A filtration on a group G is a sequence of subgroups G = G 0 G......, such that G 0 = G and [G i, G i ] G i+j. The length of the filtration is d when G d+ is the identity. Let us denote by G +t the filtration given by (G + ) t = G i+t. If G has length d then G +t has length d t. A map g : Γ G is polynomial with respect to a filtration G on G of length d N { }, or just G polynomial, if g = G, so d = or D (a,b) (g) is G + polynomial. When G is a nilpotent group a map g : Γ G is a polynomial of scalar degree d if for any a, b,..., a d+, b d+ Γ D (a,b )...D (ad+,b d+ ) (g) G. If G is not nilpotent then the set of polynomials of degree d, may be not a group.

6 Spherical multiergodic averages 67 The system g = (g, g 2,..., g j ), where g i : Γ G are polynomial maps and G is nilpotent, has finite complexity. For more details and justification of the mentioned results see[7]. The following concept was introduced by Walsh. We present herein an adapted definition for Markov groups. Let σ L (X), with σ, L N, 0 < ε <. The function σ is reducible with respect to L and to a system g = (g, g 2,..., g j ), or just (L, g) reducible if there is a number M > 0 and functions b 0, b,..., b j L (X), b i, i = 0,,..., j, such that fore any l L and for any γ S(l) holds g j(γ)σ M M r=0 cards(r) γ S(r) j g j, G (a,γ ) (γ)b 0 g j, g i (a,γ ) (γ)b i < δ, for any a Γ, and where δ is a constant depending on ε, to be specified later. For N N, let us denote i= (3.4) S g N [φ, φ 2,..., φ j ] = N cards(n) j g i (γ)φ i γ S(n) i= and for N, N N, let S g N,N [φ, φ 2,..., φ j ] = S g N [φ, φ 2,..., φ j ] S g N [φ, φ 2,..., φ j ]. Let ϕ : R + R +, η > 0, and let C η,ϕ [2η 2 ],..., Cη,ϕ be constants defined recursively as { C η,ϕ [2η 2 ] =, Cη,ϕ n = max Cn η,ϕ, 2ϕ (C η,φ n ) }. For ε > 0 let C = C (ε) := C η,ϕ with η = ε ε2 2 5 and ϕ(x) = In the definition x of L reducible of above will be taken δ = ε/6c (ε). Proposition 3.: Let g = (g, g 2,..., g j ) with cplx(g) C, for any ε > 0 there exists a K = K (ε, C) N such that for any M N and for any function F : N N there is a sequence M M ε,c,f... M ε,c,f K M = O M,δ,c,ω (), in such a way, that if φ, φ 2,..., φ j L (X), φ i, i =,..., j and φ = k λ t σ t, where σ t is (L, g) reducible for any L < F (M ) and k λ t < C then S g N,N [φ, φ 2,..., φ j, φ] < ε L2 (X) 4, for N M i, N F (M i ), for some i {.2,..., K}. Proof. Let us consider a system g = (g, g 2,..., g j ) with cplx(g) C, and ε > 0. Let M t and b 0,t, b,t,..., b j,t L (X) be the integer and the functions corresponding to the (L, g) reducibility of σ t. Thus ] k S g N [φ, φ 2,..., φ j, λ t σ t = k j λ t g i (γ) φ i g j (γ)σ t. N cards(n) i=

7 68 A.M. Mesón and F. Vericat If in this average g j (γ)σ t is replaced by M t M t r=0 cards(r) γ S(r) j g j, G (a,γ ) (γ)b 0 g j, g i (at,γ ) (γ)b i,t with a t Γ, then is obtained an average with respect to a system of lower complexity, but with an error of δ = ε/(6c ). Thus [ k φ, φ 2,..., φ j, Sg N,N k M t M t r=0 λ t cards(r) i= λ t σ t]l2 (X) Sg (a t,γ ) N,N [φ, φ 2,..., φ j, b 0,t, b,t,..., b j,t ] + ε 8, γ S(r) with N, N L. Recall that the coefficients λ t were chosen such that k λ t < C. Using the complexity as induction parameter each term of the sum can be bounded for N, N belonging to some integer interval of the form [ M, F (M) ], but the bound is not uniform in the sense that the integer M may depend on the average. By a direct application of the Walsh techniques the norms can be uniformly bounded, with λ t respect to the weights, by a given δ. cards(r) Let F, F 2,..., F r : N N, { with ( r = r (ε, )} C), be functions defined recursively by F = F and F i = max F M δ,c,f i, where the M i are given by the M i N hypothesis of induction. Let i, i 2,..., i r {.2,..., K} and define M (i ) = M δ,c,f i ( M (i,i 2,) =. M δ,c,f 2 i M (i,i2,...,is) = ( ( M δ,c,f 2 i ) δ,c,f i 2 ) δ,c,f i 2, ) δ,c,fs.... i s Let s = r and (t, γ ) {0..,..., k } S(M t ), by induction hypothesis there is some i {.2,..., K} such that for a given δ > 0 and for N, N [ M (i), F (M (i) ) ] holds S g (at,γ ) N,N [φ, φ 2,..., φ j, b 0,t, b,t,..., b j,t ] < δ, L2 (X) for at least K of the pairs (t, λ t γ ), with respect to the weights cards(r). γ S(r). Continuing this process for the pairs (t, γ λ t ) with weights cards(r).γ S(r), can be found an i 2 {.2,..., K} such that for δ > 0 and for N, N [ M (i,i 2 ), F ( M (i,i 2 ) ) ] holds S g (at,γ ) N,N [φ, φ 2,..., φ j, b 0,t, b,t,..., b j,t ] < δ, L2 (X)

8 Spherical multiergodic averages 69 ( ) 2 K for at least a proportion of C of the pairs (t, γ ), with respect to the K λ t weights cards(m t ). By repeatedly applying the process, one can find an i s {.2,..., K}, s r, such that for any N, N [ M (i,i 2,...,i s ), F ( M (i,i 2,...,i s ) ) ] holds Sg (a t,γ ) N,N [φ, φ 2,..., φ j, b 0,t, b,t,..., b j,t ] < δ, L2 (X) ( ) r K for at least a proportion of C of the (t, γ ), with respect to the weights K λ t. Thus we have cards(r) k γ S(M t ) Choosing r such that λ t cards(m t ) N,N [φ, φ 2,..., φ j, b 0,t, b,t,..., b j,t ] Sg (a t,γ ) k ( ) r K λ t δ + C + ε K 8. ( ) r K C < ε, we obtain K 6 S g N,N [φ, φ 2,..., φ j, φ] L2 (X) for any N, N [ M (i,i 2,...,i s ), F ( M (i,i 2,...,i s ) ) ] < ε 6 + ε 6 + ε 8 = ε 4, L 2 (X) Lemma 3.2. Let ε > 0, and φ, φ 2,..., φ j, with φ i. Let us assume that there is a map ψ and a constant C [, C ] such that ψ 3C. If S g N [φ, φ 2,..., φ j, ψ] L 2 (X) > ε/6, for some N, then there is a (L, g) reducible function σ, L < N, such that C(ψ, σ) := ψσ > 2ϕ(C), where ϕ is defined above Proposition 3.. Proof. Let h 0 = Sg N [φ, φ 2,..., φ j, ψ] φ ψ, and h i = φ i, i. We have C N S N [φ, φ 2,..., φ j, ψ] 2 L 2 (X) > 2ϕ(C), and ψ cards(n) j i= g i (γ)φ i g j (γ)ψ, S N [φ, φ 2,..., φ j, ψ] 2 L 2 (X) ψ =

9 70 A.M. Mesón and F. Vericat So that if C ψ N (3.5) σ := N cards(n) cards(n) j g j (γ) h 0 g i (γ)h i. i= j g j (γ) h 0 g i (γ)h i i= then holds C(ψ, σ) > 2ϕ(C). It remains to prove that σ is (L, g) reducible. L < C N, with 0 < C <, if l L and γ S (l) then σ j g j (γ γ) h 0 g i (γ γ) h i N cards(n) < ε/(6c ), i= Let since S g N [φ, φ 2,..., φ j, ψ] L 2 (X) 3C 3C. If we change γ S(),..., γ S(N) by γ S( + l),..., γ S(N + l) then the magnitude of σ changes at most in 6lC N < ε/(6c ). Thus applying g j (γ) to σ we get g j(γ)σ N cards(n) j g j, G (a,γ) (γ)h 0 g j, g i (a,γ) (γ)h i < ε/(6c ), with γ S(l), l C N, C = ε/[96(c ) 2 ]. Therefore the definition of (L, g) reducibility is verified, with M = N, b i = h i. 4 Gowers theory for decomposition of maps i= One of the main ingredients in the Walsh techniques are the Gowers results[8] which, by an adequate adaptation of the Hahn-Banach theorem, give a decomposition of map in a structured component and a random component. More specifically, the problem posed in Gowers s survey is when a real map ϕ can be written as ϕ = ϕ + ϕ 2 with ϕ with an structure enough strong such that the properties of ϕ can be explicitly analyzed and ϕ 2 such that the properties of ϕ are not affected by the perturbation by ϕ 2. In [8], as a corollary of the Hahn-Banach theorem, was obtained the following result: let K, K 2,..., K r open, convex subsets of a Hilbert space (H, ), such that 0 K 2, for any i, let c, c 2,..., c r > 0 such that any φ H cannot be written as φ = r i= c i φ i, with φ i K i. Then there is a L H such that L, φ and L, φ i c i, for any i =, 2,..., r. Let (H,, ) be a Hilbert space and let be the norm induced by,. If N is a norm in H equivalent to then for φ H set (4.) φ N = sup { φ, ψ : ψ N }.

10 Spherical multiergodic averages 7 The norm N is called the dual norm of N and is equivalent to N. It can be considered a family of norms { N } indexed by N. Walsh considered families { N } N N decreasing with N and such the family of dual norms also decreases with N. Recall the definition of the constants C η,ϕ [2η 2 ],..., Cη,ϕ by C η,ϕ [2η 2 ] =, C η,ϕ n { = max Cn η,ϕ, 2ϕ (Cn η,φ ) },with ϕ : R + R +. Proposition 4..[6] Let { N } N N be a family of norms in H with the above property, let 0 < δ, c < and ϕ : R + R + be a decreasing function. Let ω : N N be a map such that ω (N) > N for any N. Then for any M > 0 there is a sequence M M... M [2δ 2 ] M = O M,δ,c,ω (), independent of the norms, such that for any φ H can be found some i {,..., [ 2δ 2]} and some A, B N with M A M i ω (M i ) B, in such a way that φ can be decomposed as φ = φ + φ 2 + φ 3 with φ B < C η,ϕ i, φ 2 A < ϕ (Cn η,φ ), φ 3 < δ. Other result adapted by Walsh from Gowers theory is Lemma 4.2. ([6],[8]) Let H 0 be a bounded subset of a Hilbert space H and set k k φ H0 = inf λ i : φ = c j σ j, σ j 0 j=0 If it is assumed that the norm H0 is well defined and equivalent to, then its dual norm is given by φ H 0 = sup { φ, σ : σ H 0 }. j=0 5 Norm convergence of spherical averages Let us consider the following family of spaces Σ L,g = {σ : σ is (L, g) reducible}, and the norms L := ΣL and L := Σ L. If L > L 2, then Σ L Σ L2. We have: Theorem 5.. Let ε > 0 and g = (g, g 2,..., g j ) be a system with cplx(g) C, then there exists a natural K = K ε.c such that for any function F : N N and for any M N, there are natural numbers M ε,c,f,..., M ε,c,f K M such that for any maps φ, φ 2,..., φ j L (X, µ), with φ i, i =,,,.j there is a number i {, 2,..., K ε.c } with the property that for any N, N [M i, F (M i )] holds (5.) S g N,N [φ, φ 2,..., φ j ] < ε. 2 Proof. For any M N and for any function F : N N, by the Proposition 3. there is a sequence M M ε,c,f... M ε,c,f K M, thus set ω (M) = F (M ).

11 72 A.M. Mesón and F. Vericat By the decomposition theorem φ j can be written as with k k φ j = λ t σ t + ψ + ζ, λ t < C i, with C i [, C ], given by the Lemma 3.2 and σ t Σ B,g with B ω (M) = F (M ). Moreover, ψ M i < ϕ (C i ) and ζ 2 < δ, with δ = ε To apply Lemma 3.2, a control on ψ is needed. Let S = {x X : ζ(x) < C i }, The issue is to see that the points outside S may be neglected. Since we have ψ I S C k k φ j = λ t σ t + ψ + ζ, k λ t σ t I S C + ζ I S C + φ j I S C λ t σ t I S C + ζ I S C 3 ζ I S C, where I A is the characteristic function of A. Since σ t Σ ω(m),g holds ψi S C 3 ζ 2. By the Chebyshev inequality (5.2) µ (S c ) ζ2 2 C 2 i < δ2 Ci 2. Let σ Σ M,g,replacing if necessary ψ by ψi S C and ζ by ζ + ψi S C, it can be assumed that ζ < 3C i. In this case we would have ψ M i < 2ϕ (C i ) and ζ 2 < 4δ. We have ψi S C,σ < ϕ (C i ) + 3 ζ 2 µ (S c ) /2 ϕ (C i ) + 3δ 2 + δ/c i < 2ϕ (C i ). Thus, by the counter reciprocal of the lemma 3.2 for N, N M S g N,N [φ, φ 2,..., ζ] ε/3. 2 By hypothesis φ j, and therefore Thus S g N [φ, φ 2,.., ψi S C + ζ] 2 3δ + ζ 2 < 8δ, for any N. S g N,N [φ, φ 2,..., ζ] < 6δ = ε/6. Moreover, 2 Sg N,N [ ] k φ, φ 2,..., λ t σ t < 2 ε 3, for any N, N [M i, F (M i )], for some M i [M, ω (M)]. Finally [ k Sg N,N φ, φ 2,..., λ t σ t ψ + ζ] < ε 3 + ε 3 + ε 3 = ε, 2

12 Spherical multiergodic averages 73 for any N, N [M i, F (M i )], for some M i [M, ω (M)]. The result of Theorem.2 is obtained as follows: if the averages S g N [φ, φ 2,..., φ j ] were not convergent in norm, there would be and ε > 0 and a function F : N N such that for any N S g N,F (N) [φ, φ 2,..., φ j ] > ε. 2 Theorem. will be proved as an immediate consequence of the following: Proposition 5.2. Let g, g 2,..., g j be antihomorphisms from Γ to G, and such that g i (γ 2 ) g k (γ ) = g k (γ ) g i (γ 2 ), for any γ, γ 2 Γ. Then the system has finite complexity. G = (g, g g 2,..., g g 2... g j ) Proof. Let D a (g) (γ) := g (γ) g (γa) and T a (g) (γ) := g (γa), i.e., the derivative and the translation with one of the element the identity of Γ. If the map g is an antihomorphism, i.e., g (γ γ 2 ) = g (γ 2 ) g (γ ), then D a ( g ) (γ) = g(γ)g (γa) = g(γ) (g (a) g(γ)) = g (a) T a (g) (γ) := g (γa) = g (a) g(γ). Thus, following [7], let i < j and a fixed. Then g g 2,...g j, g g 2,...g i a (γ) = D a (( g g 2,...g j ) ) (γ) T a (g g 2,...g i ) (γ) = = ( g g 2,...g j ) (a) (g g 2,...g i ) (a) (g g 2,...g i ) (γ) = = ( g g 2,...g j ) (a) g (a) g (γ)g 2 (a) g 2 (γ)...g i (a) g i (γ) = = (g g 2,...g i ) (γ)g i+ (γ)...g j (γ) g i+ (a)...g j (a). Recall that if c is the constant map given c G, then cplx(g,c) = cplx(g) holds. Hence cplx(g ) = cplx(g), and the result follows by continuing the induction process on j. Thus, recall that if h i, i =, 2,..., j are measure-preserving actions of Γ on X, we can define maps g i from Γ a nilpotent group of unitary operators on L 2 (X) by g i (γ) [φ] (x) = φ (h i (γ)(x)), where each g i result antihomomorphisms. Therefore, Theorem. is proved from Proposition 5.. Acknowledgements. The support of this work by Consejo Nacional de Investigaciones Científicas y Técnicas, Universidad Nacional de La Plata and Agencia Nacional de Promoción Científica y Tecnológica of Argentina is greatly appreciated. The second author is a member of CONICET. References [] V. Bergelson, Weakly mixing PET, Ergod. Th. Dynam. Sys. 7 (987), [2] A. Bufetov, Convergence of spherical averages for actions of free groups, Ann. of Math, 55 (2) (2002),

13 74 A.M. Mesón and F. Vericat [3] Bufetov, M. Khristoforov and A. Klimenko, Césaro convergence of spherical averages for measure-preserving actions of Markov semigroups and groups, to appear in IMRN. [4] A. Bufetov and C. Series, A pointwise ergodic theorem for Fuchsian groups, Math Proc of the Cambridge Phil. Soc. 5, (20), [5] K. Fujiwara and A. Nevo, Maximal and pointwise ergodic theorems for wordhyperbolic groups, Erg. Th. & Dynam. Sys. 8 (998), [6] H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szmerédi on arithmetic progressions, J. d Analyse Math. 3 (977), [7] E. Ghys and P. de la Harpe, Sur le groupes hyperbolique d apres M. Gromov, Progress in Math. 83, Birkhauser, Zurich, 990. [8] W. T. Gowers, Decomposition, approximate structure, transfernce and the Hahn- Banach theorem, Bull London Math. Soc. 42 (200), [9] B. Host and B. Kra, Non conventional ergodical averages and nilmanifolds, Ann. of Math, 6 () (2005), [0] A. Leibman, Convergence of multiple ergodic averages along polynomials of several variables, Israel Journal of Mathematics 46 (2005), [] E. Lindenstrauss, Pointwise theorems for amenable groups, Invent. Math. 46 (200), [2] A. Nevo and E.M. Stein, A generalization of Birkhoff s pointwise ergodic theorem, Acta Math. 73 (994), [3] A. Nevo, Pointwise mergodic theorems for actions of groups, in Handsndbook of Dynamical Systems, Vol. B, , Elsevier B.V., Amsterdam [4] C. Series, Geometrical Markov coding of geodesics on surfaces with of constant negative curvature, Erg. Th. Dynam. Sys. 6 (986), [5] M. Pollicott and R. Sharp, Ergodic theorems for actions of hyperbolic groups, to appear in Proc. of the Amer. Math. Soc. [6] M. Walsh, Norm convergence of nilpotent ergodic averages, Ann. of Math. 75 (202), [7] P. Zorin-Kranich, Norm convergence of nilpotent ergodic averages along Følner nets, arxiv: Authors address: Alejandro M. Mesón and Fernando Vericat IFLYSIB CONICET UNLP and GAMEFI UNLP, Calle 59, 789 (900) La Plata, Argentina. meson@iflysib.unlp.edu.ar & vericat@iflysib.unlp.edu.ar