# «Zentrum Mathematik, Technische Universit¨t M¨nchen, a u John-von-Neumann Lecture, 2013 Sergiy Kolyada Topological Dynamics: Minimality, Entropy and ...»

**Topological Dynamics:**

Minimality, Entropy and Chaos.

Sergiy Kolyada

Institute of Mathematics, NAS of Ukraine, Kyiv

Zentrum Mathematik, Technische Universit¨t M¨nchen,

a u

John-von-Neumann Lecture, 2013

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos.

Li-Yorke sensitivity and weakly mixing maps

6. Li-Yorke sensitivity and weakly mixing maps

Throughout this lecture a dynamical system (X, T ) is a pair where X is a nonvoid compact metric space with metric ρ and T : X → X is a surjective, continuous map.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos.

Li-Yorke sensitivity and weakly mixing maps

6. Li-Yorke sensitivity and weakly mixing maps Throughout this lecture a dynamical system (X, T ) is a pair where X is a nonvoid compact metric space with metric ρ and T : X → X is a surjective, continuous map. The assumption of compactness of X is vital.

The assumption of surjectivity of T is convenient and can always be obtained, given compactness, by replacing X by its largest T invariant subset if necessary.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos.

Li-Yorke sensitivity and weakly mixing maps The diﬀerent notions of chaos at least share a common motivating intuition of unpredictability due to the divergence of nearby orbits.

Diﬀerent deﬁnitions begin with diﬀerent interpretations of this divergence. We consider two popular ideas and examine a concept which bridges them.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos.

Li-Yorke sensitivity and weakly mixing maps The diﬀerent notions of chaos at least share a common motivating intuition of unpredictability due to the divergence of nearby orbits.

Diﬀerent deﬁnitions begin with diﬀerent interpretations of this divergence. We consider two popular ideas and examine a concept which bridges them.

The idea of sensitivity was introduced in Auslander and Yorke (1980) and popularized in Devaney (1989).

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos.

Li-Yorke sensitivity and weakly mixing maps The diﬀerent notions of chaos at least share a common motivating intuition of unpredictability due to the divergence of nearby orbits.

Diﬀerent deﬁnitions begin with diﬀerent interpretations of this divergence. We consider two popular ideas and examine a concept which bridges them.

The idea of sensitivity was introduced in Auslander and Yorke (1980) and popularized in Devaney (1989). For a positive ε we consider pairs of points (x, y ) whose orbits are frequently at least ε apart. That is,

The diﬀerent notions of chaos at least share a common motivating intuition of unpredictability due to the divergence of nearby orbits.

The idea of sensitivity was introduced in Auslander and Yorke (1980) and popularized in Devaney (1989). For a positive ε we consider pairs of points (x, y ) whose orbits are frequently at least ε apart. That is,

Recall that in studying maps of the interval, Li and Yorke (1975) suggested that the “divergent pairs” to consider are the pairs (x, y ) which are proximal but not asymptotic.

Recall that in studying maps of the interval, Li and Yorke (1975) suggested that the “divergent pairs” to consider are the pairs (x, y ) which are proximal but not asymptotic. That is,

Recall that in studying maps of the interval, Li and Yorke (1975) suggested that the “divergent pairs” to consider are the pairs (x, y ) which are proximal but not asymptotic. That is,

Li and Yorke call a system chaotic when it contains an uncountable scrambled set. A subset A ⊂ X is scrambled when any pair of distinct points in A satisfy this Li–Yorke condition.

Motivated by these results, we deﬁne the concept which links sensitivity with the Li–Yorke versions of chaos.

Deﬁnitions A dynamical system (X, T ) is called Li–Yorke sensitive if there exists a positive ε such that every x ∈ X is a limit of points in ProxT (x) \ AsymT,ε (x), i.e. x ∈ ProxT (x) \ AsymT,ε (x).

The proof of the following theorem is based on arguments in a paper by Huang and Ye (2001). We will refer to these as the Huang-Ye Equivalnces.

** Theorem 6.1 For a dynamical system (X, T ) the following conditions are equivalent.**

(1) The system is sensitive.

(2) There exists a positive ε such that Asymε (T ) is a ﬁrst category subset of X × X.

(3) There exists a positive ε such that for every x ∈ X AsymT,ε (x) is a ﬁrst category subset of X.

(4) There exists a positive ε such that every x ∈ X is a limit point of the complement of AsymT,ε (x), i.e. x ∈ X \ AsymT,ε (x).

(5) There exists a positive ε such that the set of pairs

This condition is strictly stronger than sensitivity. For example, any minimal system which is distal but not equicontinuous is sensitive but not Li–Yorke sensitive. On the other hand, we prove that weak mixing systems are Li–Yorke sensitive.

This condition is strictly stronger than sensitivity. For example, any minimal system which is distal but not equicontinuous is sensitive but not Li–Yorke sensitive. On the other hand, we prove that weak mixing systems are Li–Yorke sensitive.

** Theorem 6.2 Let (X, T ) be a dynamical system.**

If (X, T ) is Li–Yorke sensitive then it is sensitive.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos.

Li-Yorke sensitivity and weakly mixing maps This condition is strictly stronger than sensitivity. For example, any minimal system which is distal but not equicontinuous is sensitive but not Li–Yorke sensitive. On the other hand, we prove that weak mixing systems are Li–Yorke sensitive.

** Theorem 6.2 Let (X, T ) be a dynamical system.**

If (X, T ) is Li–Yorke sensitive then it is sensitive. If (X, T ) is sensitive and for every x ∈ X the proximal cell ProxT (x) is dense in X then (X, T ) is Li–Yorke sensitive.

Beweis.

By the Huang-Ye Equivalences, Li–Yorke sensitivity implies sensitivity. On the other hand, Prox(T ) is a Gδ subset of X × X and so each ProxT (x) is a Gδ subset of X.

Beweis.

By the Huang-Ye Equivalences, Li–Yorke sensitivity implies sensitivity. On the other hand, Prox(T ) is a Gδ subset of X × X and so each ProxT (x) is a Gδ subset of X. Hence, if ProxT (x) is dense and AsymT,ε (x) is of ﬁrst category, then ProxT (x) \ AsymT,ε (x) is dense as well by the Baire Category Theorem.

The weakly mixing systems are a good class of test systems for any topological deﬁnition of chaos. The system (X, T ) is called weakly mixing when the product system (X × X, T × T ) is transitive.

The weakly mixing systems are a good class of test systems for any topological deﬁnition of chaos. The system (X, T ) is called weakly mixing when the product system (X × X, T × T ) is transitive.

The Furstenberg Intersection Lemma implies that for weakly mixing systems the collection of subsets of Z+ : {n(U, V ) ∩ [k, ∞) : U, V opene in X and k ≥ 0} generates a ﬁlter which we will denote F).

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos.

Li-Yorke sensitivity and weakly mixing maps The weakly mixing systems are a good class of test systems for any topological deﬁnition of chaos. The system (X, T ) is called weakly mixing when the product system (X × X, T × T ) is transitive.

The Furstenberg Intersection Lemma implies that for weakly mixing systems the collection of subsets of Z+ : {n(U, V ) ∩ [k, ∞) : U, V opene in X and k ≥ 0} generates a ﬁlter which we will denote F). The dual family kF = {A : A ∩ B = ∅ for all B ∈ F} satisﬁes what Furstenberg calls the Ramsey Property: if a ﬁnite union of subsets of Z+ lies in kF then one of the subsets is in kF (this condition is just dual to the ﬁlter property for F).

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos.

Li-Yorke sensitivity and weakly mixing maps The weakly mixing systems are a good class of test systems for any topological deﬁnition of chaos. The system (X, T ) is called weakly mixing when the product system (X × X, T × T ) is transitive.

The Furstenberg Intersection Lemma implies that for weakly mixing systems the collection of subsets of Z+ : {n(U, V ) ∩ [k, ∞) : U, V opene in X and k ≥ 0} generates a ﬁlter which we will denote F). The dual family kF = {A : A ∩ B = ∅ for all B ∈ F} satisﬁes what Furstenberg calls the Ramsey Property: if a ﬁnite union of subsets of Z+ lies in kF then one of the subsets is in kF (this condition is just dual to the ﬁlter property for F). Notice that if A ∈ kF and B ∈ F then A ∩ B is inﬁnite because B ∩ [k, ∞) ∈ F for all k 0.

Beweis.

Given a point x ∈ X and an opene set U in X it suﬃces to ﬁnd a point y ∈ U which is proximal to x.

For k = 1, 2,... let Gk be a cover by opene subsets of diameter less than 1/k.

Beweis.

Given a point x ∈ X and an opene set U in X it suﬃces to ﬁnd a point y ∈ U which is proximal to x.

For k = 1, 2,... let Gk be a cover by opene subsets of diameter less than 1/k. Observe that the union of the hitting time sets n(x, G ) as G varies over Gk is all of Z+.

Beweis.

Given a point x ∈ X and an opene set U in X it suﬃces to ﬁnd a point y ∈ U which is proximal to x.

For k = 1, 2,... let Gk be a cover by opene subsets of diameter less than 1/k. Observe that the union of the hitting time sets n(x, G ) as G varies over Gk is all of Z+. So by the Ramsey Property, there exists Gk ∈ Gk an opene set of diameter less than 1/k such that n(x, Gk ) ∈ kF.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos.

Li-Yorke sensitivity and weakly mixing maps Beweis.

Given a point x ∈ X and an opene set U in X it suﬃces to ﬁnd a point y ∈ U which is proximal to x.

For k = 1, 2,... let Gk be a cover by opene subsets of diameter less than 1/k. Observe that the union of the hitting time sets n(x, G ) as G varies over Gk is all of Z+. So by the Ramsey Property, there exists Gk ∈ Gk an opene set of diameter less than 1/k such that n(x, Gk ) ∈ kF.

Now let U0 = U and deﬁne inductively opene sets U1, U2,... and positive integers nk as follows.

Beweis (Cont.) Because n(Uk−1, Gk ) ∈ F, the intersection n(Uk−1, Gk ) ∩ n(x, Gk ) is inﬁnite and so we can choose Uk an opene subset of Uk−1 and an integer nk k such that T nk (x) ∈ Gk and T nk (Uk ) ⊂ Gk.

Beweis (Cont.) Because n(Uk−1, Gk ) ∈ F, the intersection n(Uk−1, Gk ) ∩ n(x, Gk ) is inﬁnite and so we can choose Uk an opene subset of Uk−1 and an integer nk k such that T nk (x) ∈ Gk and T nk (Uk ) ⊂ Gk. If y is a point of the nonempty intersection ∩k Uk then T nk (x), T nk (y ) ∈ Gk and so ρ(T nk (x), T nk (y )) 1/k. Thus, y is a point of U which is proximal to x.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos.

Li-Yorke sensitivity and weakly mixing maps Beweis (Cont.) Because n(Uk−1, Gk ) ∈ F, the intersection n(Uk−1, Gk ) ∩ n(x, Gk ) is inﬁnite and so we can choose Uk an opene subset of Uk−1 and an integer nk k such that T nk (x) ∈ Gk and T nk (Uk ) ⊂ Gk. If y is a point of the nonempty intersection ∩k Uk then T nk (x), T nk (y ) ∈ Gk and so ρ(T nk (x), T nk (y )) 1/k. Thus, y is a point of U which is proximal to x.

Corollary 6.1 If a nontrivial system (X, T ) is weakly mixing then it is Li–Yorke sensitive.

Beweis (Cont.) Because n(Uk−1, Gk ) ∈ F, the intersection n(Uk−1, Gk ) ∩ n(x, Gk ) is inﬁnite and so we can choose Uk an opene subset of Uk−1 and an integer nk k such that T nk (x) ∈ Gk and T nk (Uk ) ⊂ Gk. If y is a point of the nonempty intersection ∩k Uk then T nk (x), T nk (y ) ∈ Gk and so ρ(T nk (x), T nk (y )) 1/k. Thus, y is a point of U which is proximal to x.

Corollary 6.1 If a nontrivial system (X, T ) is weakly mixing then it is Li–Yorke sensitive.

The following problem is still open – Are all Li–Yorke sensitive systems Li–Yorke chaotic?

6. Li-Yorke sensitivity and weakly mixing maps General references

1. F. Blanchard, E. Glasner, S. Kolyada, and A. Maass, On LiYorke pairs, J. Reine Angew. Math., 547, 51–68 (2002).

2. Wen Huang and Xiangdong Ye, Devaney’s chaos or 2–scattering implies Li–Yorke chaos, Topology Appl., 117, 259–272.

3. E. Akin and S. Kolyada, Li-Yorke sensitivity, Nonlinearity, 16 (2003), 1421–1433.

4. E. Akin, Recurrence in Topological Dynamics: Furstenberg Families and Ellis Actions, Plenum, New York, 1997.

6. Li-Yorke sensitivity and weakly mixing maps HOMEWORK The system is called proximal when every pair is proximal.

Exercise 6.1 To prove that a proximal dynamical system is Li–Yorke sensitive iﬀ it is sensitive.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos.