ARTICLE IN PRESS. J. Math. Anal. Appl. ( ) Note. On pairwise sensitivity. Benoît Cadre, Pierre Jacob

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1 S X /SCO AID:9973 Vol. [DTD5] P YJMAA:m1 v 1.35 Prn:15/02/2005; 16:33 yjmaa9973 by:jk p. 1 J. Math. Anal. Appl. Note On pairwise sensitivity Benoît Cadre, Pierre Jacob 12 UMR CNRS 519, Equipe de Probabilités et Statistique, Université Montpellier II, CC 051, Place E. Bataillon, Montpellier cedex 5, France 13 1 Received 8 December Submitted by H.W. Broer 17 Abstract We define the concept of pairwise sensitivity with respect to initial conditions for the endomorphisms on Lebesgue metric spaces. The idea is that the orbits of almost every pair of nearby initial points for the product of the invariant measure of a pairwise sensitive map may diverge from a positive quantity independent of the initial points. We study the relationships between pairwise sensitivity and weak mixing, pairwise sensitivity and positiveness of metric entropy and we compute the largest sensitivity constant Elsevier Inc. All rights reserved. 28 Keywords: Sensitive dependence on initial conditions; Measure-preserving transformation; Ergodicity; Mixing; Metric entropy Introduction The concept of sensitive dependence on initial conditions has attracted much attention in recent years and several authors have tried to formalize it in various ways. The phrase sensitive dependence on initial conditions was first used by Ruelle [12], to indicate some exponential rate of divergence of orbits of nearby points. More generally, it captures the idea that a very small change in the initial condition can cause a big change in the trajectory. 2 * Corresponding author address: cadre@math.univ-montp2.fr B. Cadre X/$ see front matter 2005 Elsevier Inc. All rights reserved. 5 doi: /j.jmaa

2 S X /SCO AID:9973 Vol. [DTD5] P YJMAA:m1 v 1.35 Prn:15/02/2005; 16:33 yjmaa9973 by:jk p. 2 2 B. Cadre, P. Jacob / J. Math. Anal. Appl. Following the pioneer work by Guckenheimer [10], Devaney [8] called sensitive a self-map T : X X on the metric space X, d satisfying the property: there exists δ>0 such that for all x X and all ε>0 there is some y X which is within a distance ε of x and for some n 0, dt n x,t n y δ. In the last years, several authors proposed sufficient conditions both on T and X, d to ensure the sensitivity property cf. Abraham et al. [1,2], Banks et al. [], Glasner and Weiss [5], Guckenheimer [10]. However, sensitive dependence on initial conditions was first defined in Chaos Theory to measure the divergence of orbits of nearby points, by analogy with the butterfly effect described by the meteorologist Ed Lorentz for an overview in Chaos Theory, we refer the reader to the book by Devaney [8]. From this point of view, we think that the dissymetric feature of the above definition of sensitivity should be avoided. In order to follow the sensitivity idea drawn by the butterfly effect, one could say that T is sensitive if there exists δ>0 such that for all x,y in X, one can find n 0 with dt n x,t n y δ. However, this property appears to be too strong because it is never satisfied by the non-injective maps, such as the archetype of a chaotic map, namely the quadratic one Tx = x1 x on X =[0, 1]. As an attempt to weaken the previous definition of sensitivity, we make use of tools from Ergodic Theory. From now on, we consider an endomorphism T on a probability Lebesgue space X, B,µ cf. Petersen [11, p. 16] and we fix a metric d on X. For simplicity, we assume throughout that the support of µ, denoted supp µ, is not reduced to a single point. Our sensitivity property described below is easily shown to be stronger than Guckenheimer s one. Definition. The endomorphism T is said to be pairwise sensitive with respect to initial conditions if there exists δ>0 a sensitivity constant such that for µ 2 -a.e. x, y X 2, one can find n 0 with dt n x,t n y δ. Equivalently, T is pairwise sensitive if there exists δ>0 with µ 2 T n A δ = 0, where, here and in the following, T = T T is the map on X 2 defined by T : x, y T x, T y and, for any r>0, A r stands for the set A r := { x, y X 2 : dx,y < r }. We note that this sensitivity property is the exact measure theoretic equivalent of the concept studied in Akin and Kolyada [3] in a topological dynamic context. See also the article by Blanchard et al. [6] on the abundance of Li Yorke pairs. Section 2 is devoted to the computation of the sensitivity constant and to the case where T is weakly mixing. The case where T is of positive metric entropy is studied in Section 3.

3 S X /SCO AID:9973 Vol. [DTD5] P YJMAA:m1 v 1.35 Prn:15/02/2005; 16:33 yjmaa9973 by:jk p. 3 B. Cadre, P. Jacob / J. Math. Anal. Appl Pairwise sensitivity, weak mixing and the sensitivity constant Observe that if δ is a sensitivity constant for T, then so is any positive δ δ. This leads to consider the following quantity, denoted T : T = sup{δ: δ is a sensitivity constant for T }. From now on, diama stands for the diameter of A X and for all z X, r>0, Bz,r is the open ball: Bz,r = { x X: dz,x < r }. Theorem 2.1. Assume that T is pairwise sensitive. Then, i there exists δ>0such that for µ 2 -a.e. x, y X 2, one can find a sequence n k k 0 with dt n kx,t n ky δ for all k 0; ii for µ 2 -a.e. x, y X 2, one has sup dt n x,t n y T ; iii T diamsupp µ. Notice that, since the support of µ is not reduced to a single point, there exists δ>0 with µ 2 A δ <1. Hence, the quantity aµ := sup { δ: µ 2 A δ <1 } is positive. Lemma 2.1. One has aµ = diamsupp µ. Proof. First notice that aµ D := diamsupp µ because for all ε>0, µ 2 A D+ε = 1. Moreover, let for all ε>0, { F ε = x, y supp µ 2 : dx,y D ε }, 2 where supp µ 2 denotes the support of µ 2. Then, µ 2 F ε <1 because F ε is a closed set and F ε supp µ 2 = supp µ supp µ. Since A D ε F ε, one deduces that µ 2 A D ε <1 and hence, that aµ D. Proof of Theorem 2.1. i It is a straightforward consequence of Halmos Recurrence Theorem cf. Petersen [11, p. 39]. ii If T is pairwise sensitive, then for all ε>0 small enough: µ 2 T n A T ε = 0.

4 S X /SCO AID:9973 Vol. [DTD5] P. 1-8 YJMAA:m1 v 1.35 Prn:15/02/2005; 16:33 yjmaa9973 by:jk p. B. Cadre, P. Jacob / J. Math. Anal. Appl. We get from a monotonicity argument that µ 2 T n A T ε = lim µ 2 T n A T ε = 0, 3 ɛ 0 ε>0 hence assertion ii, because { T n A T ε = x, y X 2 :supd T n x,t n y } < T. 8 ε>0 8 iii Assume that aµ < T. For any r ]aµ, T [, one has simultaneously µ 2 A r = 1 and µ 2 T n A r = 0, which is a contradiction. Therefore, aµ T and iii is now straightforward from Lemma 2.1. Theorem 2.2. Assume that T is weakly mixing. Then, T is pairwise sensitive and moreover, Remarks. T = diamsupp µ. Ergodicity is not strong enough in order to ensure the pairwise sensitivity property. Indeed, consider the case X = R/Z, µ the Haar Lebesgue measure and d the natural metric on X. The self-map T defined by Tx= x + θmod 1, where θ is an irrational number, being an isometry for d, cannot be pairwise sensitive. However, it is known to be ergodic. In the case of a Guckenheimer s type definition of sensitivity, Abraham et al. [1,2] also provide some bounds for the largest sensitivity constant. Thus, for the classical dynamical systems such as r-adic maps, tent map or quadratic map, one has T = 1. This means that the orbits of almost all pairs of nearby initial points ultimately move away one from another as far as possible. Proof of Theorem 2.2. Let δ<diamsupp µ. Since T is an ergodic endomorphism on X 2, B B,µ 2 cf. Petersen [11, p. 65] and T n A δ is a T -invariant set, one has µ 2 T n A δ = 0, because µ 2 A d δ <1 by Lemma 2.1. Hence, T is pairwise sensitive and T diamsupp µ. Apply now Theorem 2.1iii, and the theorem is proved.

5 S X /SCO AID:9973 Vol. [DTD5] P YJMAA:m1 v 1.35 Prn:15/02/2005; 16:33 yjmaa9973 by:jk p. 5 B. Cadre, P. Jacob / J. Math. Anal. Appl. 5 Observe now that for µ 2 -a.e. x, y X 2, one has sup d T n x,t n y diamsupp µ. 3 3 Corollary 2.1 below is then a straightforward consequence of Theorems 2.1ii and 2.2. Corollary 2.1. If T is weakly mixing, then for µ 2 -a.e. x, y X 2, sup d T n x,t n y = diamsupp µ Pairwise sensitivity and metric entropy For any measurable countable partition α of X, we denote by ht, α the metric entropy of the transformation T with respect to the partition α cf. Petersen [11, Chapter 5]. Whatever the definition of sensitivity, it is expected that positiveness of metric entropy implies the sensitivity property cf. Glasner and Weiss [9], Blanchard et al. [7], and Abraham et al. [1], in which positiveness of the Lyapunov exponent is considered. Theorem.1 below gives an answer to this problem in the case of pairwise sensitivity. Recall that a measurable set A in X is a µ-continuity set if its boundary A satisfies µ A = 0 see Billingsley [5, p. 11]. Theorem 3.1. Assume that T is ergodic and consider a finite measurable partition α = {P 1,...,P l } of X. IfP 1,...,P l are µ-continuity sets for d and if ht, α > 0, then T is pairwise sensitive. A very similar conclusion is obtained in Blanchard et al. [7], but these authors consider the case where T is a homeomorphism on a compact space. Proof. Without loss of generality, we can assume that ht, α <. For all D B and ε>0, denote by D ε the internal ε-boundary of D: D ε = { x D: dx,d c <ε }, and moreover, K ε = exp 2l l µ P ε. i 37 i=1 37 Since the P i s are µ-continuity sets, K ε 1asε 0. Hence, one can choose δ>0 such that K δ 2 ht,α/2 < The map x µ T n B T n x,δ

6 S X /SCO AID:9973 Vol. [DTD5] P YJMAA:m1 v 1.35 Prn:15/02/2005; 16:33 yjmaa9973 by:jk p. 6 6 B. Cadre, P. Jacob / J. Math. Anal. Appl. defined on X is T -invariant and moreover, according to the Fubini theorem, µ 2 T n A δ = µ T n B T n x,δ µdx. X Consequently, by ergodicity of T,wehaveforµ-a.e. x X: µ 2 T n A δ = µ T n B T n x,δ We deduce from the von Neumann Ergodic Theorem that for µ-a.e. x X: 1 l l n card k {0,...,n}: T k x Pi δ µ Pi δ 13 i=1 i=1 We now fix a point x X which satisfies both 3.2 and 3.3. It is associated with it a sequence i n {1,...,l} N such that T n x P in for all n 0. For all n 0, we let Q n = { k {0,...,n}: B T k x,δ P ik }. For all n 0, card Q 21 n card k {0,...,n}: T k x P δ i k card k {0,...,n}: T k x Pi δ i=1 2 By 3.3, there exists N 1 0 such that for all n N 1, l. card Q 28 n 2n µ Pi δ 28 i=1 For all n 0, we let S n = { s k k=0,...,n : s k {1,...,l} if k Q n and s k = i k if k/ Q n }. This set satisfies, for all n N 1, card S n = expcard Q n log l exp 2nl i i=1 Now denote, for n 0 and s S n, n L n,s = T k P sk. 1 k=0 1 k=0 k Q c n Q n k Q c n l l µ P δ = Kδ n. 3. For all n 0, we have the following inclusions: n T k B T k x,δ = T k B T k x,δ T k P ik L n,s. 3.5 s S n

7 S X /SCO AID:9973 Vol. [DTD5] P YJMAA:m1 v 1.35 Prn:15/02/2005; 16:33 yjmaa9973 by:jk p. 7 B. Cadre, P. Jacob / J. Math. Anal. Appl. 7 Now fix ε ]0,hT,α/2[. By the Entropy Equipartition Property cf. Petersen [11, p. 263], there exists N 2 0 such that for all n N 2, the elements of {1,...,l} n+1 can be divided into two disjoints classes, G n and B n, such that µ ε, 5 L n,s 5 6 s B 6 n and, for all s G n, µl n,s 2 nht,α ε. We then deduce from 3. and 3.5 that for all n maxn 1,N 2, n µ T k B T k x,δ µ + µl n,s s S n G n 13 L n,s 13 1 k=0 s S n B n 1 ε + card S n max µl n,s s G n 17,α ε 2 nht 17 ε + Kδ n ε + Kδ n 2 nht,α/2, where the latter inequality comes from the fact that ε<ht,α/2. Letting n,we deduce from 3.1 that for all ε>0 small enough, n lim µ T k B T k x,δ ε, n 2 k=0 2 hence, letting ε 0, µ T n B T n x,δ = Finally, we deduce from 3.2 and the choice of x that T is pairwise sensitive. References 35 [1] C. Abraham, G. Biau, B. Cadre, Chaotic properties of mappings defined on a probability space, J. Math Anal. Appl [2] C. Abraham, G. Biau, B. Cadre, On Lyapunov exponent and sensitivity, J. Math. Anal. Appl [3] E. Akin, S. Kolyada, Li Yorke sensitivity, Nonlinearity [] J. Banks, J. Brooks, G. Cairns, G. Davis, P. Stacey, On Devaney s definition of chaos, Amer. Math Monthly [5] P. Billingsley, Convergence of Probability Measures, Wiley, New York, [6] F. Blanchard, E. Glasner, S. Kolyada, A. Maas, On Li Yorke pairs, J. Reine Angew. Math [7] F. Blanchard, B. Host, S. Ruette, Asymptotic pairs in positive-entropy systems, Ergodic Theory Dynam. Systems [8] R.L. Devaney, Chaotic Dynamical Systems, second ed., Addison Wesley, New York, [9] E. Glasner, B. Weiss, Sensitive dependence on initial conditions, Nonlinearity

8 S X /SCO AID:9973 Vol. [DTD5] P YJMAA:m1 v 1.35 Prn:15/02/2005; 16:33 yjmaa9973 by:jk p. 8 8 B. Cadre, P. Jacob / J. Math. Anal. Appl. 1 [10] J. Guckenheimer, Sensitive dependence to initial conditions for one dimensional maps, Comm. Math. 1 2 Phys [11] K. Petersen, Ergodic Theory, Cambridge Univ. Press, Cambridge, [12] D. Ruelle, Dynamical systems with turbulent behavior, in: Mathematical Problems in Theoretical Physics, Lecture Notes in Phys., Springer-Verlag, Berlin, 1978.