Download Topological Dynamics: Minimality, Entropy and Chaos.

Topological Dynamics:
Minimality, Entropy and Chaos.
Sergiy Kolyada
Institute of Mathematics, NAS of Ukraine, Kyiv
Zentrum Mathematik, Technische Universität München,
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,
lim sup ρ(T n (x), T n (y )) > ε.
n→∞
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,
lim sup ρ(T n (x), T n (y )) > ε.
n→∞
The system is sensitive for some positive ε the set of pairs which satisfy
this condition is dense in X × X .
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Li-Yorke sensitivity and weakly mixing maps
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.
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Li-Yorke sensitivity and weakly mixing maps
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,
lim inf ρ(T n (x), T n (y )) = 0
n→∞
but
lim sup ρ(T n (x), T n (y )) > 0.
n→∞
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Li-Yorke sensitivity and weakly mixing maps
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,
lim inf ρ(T n (x), T n (y )) = 0
n→∞
but
lim sup ρ(T n (x), T n (y )) > 0.
n→∞
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.
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Li-Yorke sensitivity and weakly mixing maps
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).
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Li-Yorke sensitivity and weakly mixing maps
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.
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Li-Yorke sensitivity and weakly mixing maps
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
{(x, y ) ∈ X × X : lim sup ρ(T n (x), T n (y )) > ε}
n→∞
is dense in X × X .
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
(1)
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.
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.
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.
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Li-Yorke sensitivity and weakly mixing maps
Beweis.
By the Huang-Ye Equivalences, Li–Yorke sensitivity implies sensitivity.
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Li-Yorke sensitivity and weakly mixing maps
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 .
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Li-Yorke sensitivity and weakly mixing maps
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.
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.
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.
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).
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 6= ∅ 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 6= ∅ 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.
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Li-Yorke sensitivity and weakly mixing maps
Theorem 6.3
If (X , T ) is a weak mixing dynamical system then for every x ∈ X , the
proximal cell ProxT (x) is dense in X .
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.
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.
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+ .
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.
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.
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 .
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.
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.
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.
The following problem is still open –
Are all Li–Yorke sensitive systems Li–Yorke chaotic?
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Li-Yorke sensitivity and weakly mixing maps
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.
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Li-Yorke sensitivity and weakly mixing maps
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.