MULTI PING-PONG AND AN ENTROPY ESTIMATE IN GROUPS. Katarzyna Tarchała, Paweł Walczak. 1. Introduction

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1 Annales Mathematicae Silesianae 32 (208, DOI: 0.55/amsil MULTI PING-PONG AND AN ENTROPY ESTIMATE IN GROUPS Katarzyna Tarchała, Paweł Walczak Abstract. We provide an entropy estimate from below for a finitely generated group of transformation of a compact metric space which contains a ping-pong game with several players located anywhere in the group.. Introduction In [9], we provided entropy estimates for a finitely generated group G of transformations of a compact metric space X which contains two maps (pingpong players which transform a subset A of X into two disjoint subsets A and A 2 of A. The players are located anywhere in G. Here, we improve that estimate in the more general case: G contains an arbitrary finite number of ping-pong players located anywhere in G. The notion of entropy for finitely generated groups of transformations of compact metric spaces has been introduced (in the wider context of pseudogroups and foliations by Ghys, Langevin and the second author [4] (see either [2, Chapter 3] or [0] for more detailed expositions. It corresponds to the topological entropy of single transformations, depends on the choice of a generating set but its vanishing (or, non-vanishing is independent of such a choice. Received: Accepted: Published online: (200 Mathematics Subject Classification: 37B40, 54C70. Key words and phrases: topological entropy, transformation group, ping-pong.

2 34 Katarzyna Tarchała, Paweł Walczak Ping-pong in transformation groups is attributed (see [5, Chapter II.B] to Feliks Klein who used it to study Kleinian groups. It implies some complexity of the dynamics, in particular, positive entropy and in -dimensional dynamics arises always when the dynamics of the system is complicated enough (see, for example, [8] and the bibliography therein. In some sense, in one dimensional dynamics, ping-pong is related to horseshoes which can be used to estimate (or even, to calculate entropies of the systems (see [7] and, again, the bibliography therein. It is known (see, for example, Prop in [0] that ping-pong in a group (with two players implies the entropy estimate from below: entropy is greater or equal to the product of log 2 by the inverse of the maximum of distances (in the metric determined by a given generating set of ping-pong players from the identity. In the same way, ping-pong with N players provides entropy greater or equal to log N divided again by the maximum of their distances from the identity. Here, we produce a better estimate: we replace the denominator in the above by a quantity which arises from a well known lower bound for binomial distribution (see, for example, [] and is strictly larger than the quantity (maximal distance of the estimate mentioned above. Note that our estimates can be adapted to pseudogroups and foliations to relate the value of entropy with the strength of a resilient orbit (or, of a resilient leaf which can be defined and related to the entropy and expressed in terms of the length of a piece of the orbit (or, of a leaf curve providing ping-pong in the corresponding space (for foliations, via holonomy, on a transversal, see [6]. We expect (and try to get a similar estimate from above in the case of -dynamical dynamics, that is when our space X coincides with a segment, a circle or, more generally, a graph (see [7] again, also when a foliation has codimension. The work in this direction is in progress. A reader interested in such topics is referred also to Chapters 2 and 3 of [0]. 2. Preliminaries Throughout the paper, X is a compact metric space with distance d, G a finitely generated group of continuous transformations of X and G a fixed finite symmetric (i.e., such that e G and G = {g ; g G } G set of generators for G. For any n N, we put G n = {g... g n ; g,..., g n G }. Note that since e G, G G 2 G Definition 2. (Ping-pong. Let G be a group acting on a compact metric space X. We say that f, f 2 G are playing ping-pong if there exist sets

3 Multi ping-pong and an entropy estimate in groups 35 A, A, A 2 X such that A, A 2 A, dist(a, A 2 > 0, f (A A and f 2 (A A 2. Definition 2.2 (Multi Ping-pong. Let G be a group acting on a compact metric space X. We say that f,..., f N G are playing multi ping-pong if there exist sets A, A,..., A N X such that A,..., A N A, and for all i j, i, j N dist(a i, A j > 0 and f i (A A i. Definition 2.3 (Entropy. Let ε > 0 and n > 0, n N. We say that points x, y X are (n, ε-separated if there exists a continuous map f G n such that d(f(x, f(y ε. A set A X is (n, ε-separated if all the pairs of points x, y A, x y, have this property. Since X is compact, every (n, ε-separated set is finite and we may put and s(n, ε, G := max{#a; A X is (n, ε-separated} s(ε, G := lim sup n n log s(n, ε, G. The number h(g, G := lim ε 0 s(ε, G is called the (topological entropy of G with respect to G. For simplicity, in the sequel we avoid writing G in all these formulae because we are interested in only one, fixed, set of generators. In our calculations, we shall use the following (see, [3, Chapter ] lower bound for the binomial distribution. where Lemma 2.4. If k k N = n, k i = np i and P = (p, p 2,..., p N, then ( n := k, k 2,..., k N ( n k, k 2,..., k N and H(P := N i= p i log p i. n! k!k 2!... k N! = (n + N enh(p, ( ( n n k k k 2 ( N n... ξ= k ξ k N

4 36 Katarzyna Tarchała, Paweł Walczak 3. Multi ping-pong Theorem 3.. Let G be a group of transformations of X containing N continuous maps f,..., f N playing multi ping-pong. If f i G mi, m m 2... m N, then the entropy h(g satisfies where p (0, and N i= pm i =. h(g log p, Proof. Let X be a compact metric space. Take f,..., f N, A,..., A N as in Definition 2.2 and choose ε such that for all i j dist(a i, A j > ε. Choose any c X. Define the set E n,k := {f i... f in (c; i j {,..., N}, ξ<n #{j : i j = ξ} k ξ }, where k = (k,..., k N, 0 k ξ n, N ξ= N k ξ n, k N := n ξ= k ξ. From A i A j = for all i j we gain that points f i... f in (c and f j... f jn (c are different when {i,..., i n } {j,..., j n }. Therefore, we obtain the inequality #E n,k k N ( n i =0 k + i kn i i 2 =0 ( n i k 2 + i 2... k N (i +...+i N 2 i N =0 ( n (i i N 2. k N + i N Moreover, if x = f i... f in (c and y = f j... f jn (c are different points of E n,k, then d((f i... f im (x, (f j... f jm (y ε, where m is the largest number satisfying the condition i = j,..., i m = j m. Furthermore, (f i... f im G (m k +...+m N k N. Thus the set E n,k is (m k m N k N, ε-separated and N s( k i m i, ε #E n,k i= N j= k N ( j ξ= i ξ i j =0 ( j n ( ξ= i ξ. k j + i j Remember that for any n N we have N ξ= k ξ = n, so our sequences k consist of numbers depending on n.

5 Multi ping-pong and an entropy estimate in groups 37 We obtain the estimate h(g lim n N i= k im i log N j= k N ( j ξ= i ξ i j =0 ( j n ( ξ= i ξ. k j + i j Of course, the whole sum in the above is larger than its first term. So by Lemma 2.4 and putting k i := np i we obtain ( n h(g lim n N i= k log im i k, k 2,..., k N N i= p i log p i N i= p im i =: φ(p,..., p N, where N p i =, p i 0. The best estimate is obtained for the maximal value of our function φ. One can check by the method of Lagrange multipliers that the best value φ(p,..., p N = log p is attained for p i = p m i, where p (0, and N i= pm i =. Corollary 3.2. For N continuous maps f,..., f N playing multi pingpong we have the following. ( If f i G for all i N, the entropy satisfies h(g log N. (2 If f i G m for all i N, the entropy satisfies h(g log N m. Finally, we present some computer aided numerical estimates: Example 3.3. For (m,..., m N we assume that there exist N continuous maps f i such that f i G mi for i N whose are playing multi ping-pong. Then, we receive the following: ( for (m ; m 2 ; m 3 = (; 5; 5, p 0, and h(g 0, 37232, (2 for (m ; m 2 ; m 3 = (; 5; 7, p 0, and h(g 0, , (3 for (m ; m 2 ; m 3 = (; 7; 7, p 0, and h(g 0, , (4 for (m ; m 2 ; m 3 = (2; 5; 7, p 0, and h(g 0, , (5 for (m ; m 2 ; m 3 = (7; 7; 7, p 0, and h(g 0, , (6 for (m ; m 2 ; m 3 ; m 4 = (; 5; 5; 5, p 0, and h(g 0, , (7 for (m ; m 2 ; m 3 ; m 4 = (; 7; 7; 7, p 0, 7458 and h(g 0, , (8 for (m ;... ; m 5 = (; 2; 3; 4; 5, p 0, and h(g 0, , and so on. As we can see, if one of the ping-pong player is of shorter length (smaller value of m i, we gain more entropy. Bigger entropy is also received for larger amount of players in a ping-pong game. i=

6 38 Katarzyna Tarchała, Paweł Walczak References [] Ash R.B., Information theory, Dover Publ., New York, 990. [2] Candel A., Conlon L., Foliations. I, Amer. Math. Soc., Providence, [3] Cover T.M., Thomas J.A., Elements of information theory, John Wiley & Sons Publ., Hoboken, [4] Ghys É., Langevin R., Walczak P., Entropie géométrique des feuilletages, Acta Math. 60 (988, [5] Harpe P. de la, Topics in geometric group theory, Chicago Lect. in Math., Univ. of Chicago Press, Chicago, [6] Langevin R., Walczak P., Some invariants measuring dynamics of codimension-one foliations, in: T. Mizutani et al. (Eds., Geometric study of foliations, World Sci. Publ., Singapore, 994, pp [7] Llibre J., Misiurewicz M., Horseshoes, entropy and periods for graph maps, Topology 32 (993, [8] Shi E., Wang S., The ping-pong game, geometric entropy and expansiveness for group actions on Peano continua having free dendrites, Fund. Math. 203 (2009, [9] Tarchała K., Walczak P., Ping-pong and an entropy estimate in groups, Preprint 207. [0] Walczak P., Dynamics of foliations, groups and pseudogroups, Monografie Matematyczne, Vol. 64, Birkhäuser, Basel, Department of Geometry Faculty of Mathematics and Computer Science University of Łódź Banacha Łódź Poland k.tarchala@vp.pl pawelwal@math.uni.lodz.pl