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

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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.

7. On Lyapunov Numbers

7. On Lyapunov Numbers

Throughout this lecture (X, f ) denotes a topological dynamical system,

where X is a compact metric space with metric d and f : X → X is a continuous map.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos.

7. On Lyapunov Numbers

7. On Lyapunov Numbers Throughout this lecture (X, f ) denotes a topological dynamical system, where X is a compact metric space with metric d and f : X → X is a continuous map.

Recall that a dynamical system (X, f ) is called sensitive if there exists a positive ε such that for every x ∈ X and every neighborhood Ux of x, there exist y ∈ Ux and a nonnegative integer n with d(f n (x), f n (y )) ε.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos.

7. On Lyapunov Numbers Recently several authors studied the different properties related to sensitivity. The following proposition holds.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos.

7. On Lyapunov Numbers Recently several authors studied the different properties related to sensitivity. The following proposition holds.

Proposition 1.1 Let (X, f ) be a topological dynamical system. The following conditions are equivalent.

1. (X, f ) is sensitive.

2. There exists a positive ε such that for every x ∈ X and every neighborhood Ux of x, there exists y ∈ Ux with lim supn→∞ d(f n (x), f n (y )) ε.

3. There exists a positive ε such that in any opene U in X there are x, y ∈ U and a nonnegative integer n with d(f n (x), f n (y )) ε.

4. There exists a positive ε such that in any opene U ⊂ X there are x, y ∈ U with lim supn→∞ d(f n (x), f n (y )) ε.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos.

7. On Lyapunov Numbers For a dynamical system (X, f ) a point x ∈ X is Lyapunov stable if the dependence of the orbit upon the initial position is continuous at x. This

is most easily defined using the f −extension of the metric d:

df (x, y ) = sup {d(f n (x), f n (y )) : n ≥ 0} for x, y ∈ X. Clearly, df is a metric on X and df (x, y ) = max[ d(x, y ), df (f (x), f (y )) ].

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos.

–  –  –

Using these metrics we define the diameter and f-diameter for A ⊂ X, the radius and f-radius for a neighborhood Ux of a point x ∈ X diam(A) = sup {d(x, y ) : x, y ∈ A}, diamf (A) = sup {df (x, y ) : x, y ∈ A}, radius(Ux ) = sup {d(x, y ) : y ∈ Ux }, radiusf (Ux ) = sup {df (x, y ) : y ∈ Ux }.

–  –  –

The topology obtained from the metric df is usually strictly coarser than the original d topology. When we use a term like “open”, we refer exclusively to the original topology.

–  –  –

The topology obtained from the metric df is usually strictly coarser than the original d topology. When we use a term like “open”, we refer exclusively to the original topology.

A point x ∈ X is called Lyapunov stable if for every ε 0 there exists a δ 0 such that radius(Ux ) δ implies radiusf (Ux ) ≤ ε. This condition says exactly that the sequence of iterates {f n : n ≥ 0} is equicontinuous at x. Hence, such a point is also called an equicontinuity point.

–  –  –

As the label suggests, Eq(f ) is the set of equicontinuity points. If Eq(f ) = X, i.e. every point is equicontinuous, then the two metrics d and df are topologically equivalent and so, by compactness, they are uniformly equivalent. Such a system is called equicontinuous. Thus, (X, f ) is equicontinuous exactly when the sequence {f n : n ≥ 0} is uniformly equicontinuous.

–  –  –

If the Gδ set Eq(f ) is dense in X then the system is called almost equicontinuous. On the other hand, if Eqε (f ) = ∅ for some ε 0 then it is the same that the system shows sensitive dependence upon initial conditions or, more simply, (X, f ) is sensitive.

–  –  –

If the Gδ set Eq(f ) is dense in X then the system is called almost equicontinuous. On the other hand, if Eqε (f ) = ∅ for some ε 0 then it is the same that the system shows sensitive dependence upon initial conditions or, more simply, (X, f ) is sensitive.

We define Lr := sup{ε : for every x ∈ X and every neighborhood Ux of x there exist y ∈ Ux and a nonnegative integer n with d(f n (x), f n (y )) ε} and call it the (first) Lyapunov number.

–  –  –

If the Gδ set Eq(f ) is dense in X then the system is called almost equicontinuous. On the other hand, if Eqε (f ) = ∅ for some ε 0 then it is the same that the system shows sensitive dependence upon initial conditions or, more simply, (X, f ) is sensitive.





We define Lr := sup{ε : for every x ∈ X and every neighborhood Ux of x there exist y ∈ Ux and a nonnegative integer n with d(f n (x), f n (y )) ε} and call it the (first) Lyapunov number.

It can happen that Eqε (f ) = ∅ for all positive ε and yet still the intersection, Eq(f ), is empty.

–  –  –

This cannot happen when the system is transitive.

Theorem 1.1 Let (X, f ) be a topologically transitive dynamical system.

Exactly one of the following two cases holds.

Case i (Eq(f ) = ∅) Assume there exists an equicontinuity point for the system. The equicontinuity points are exactly the transitive points, i.e. Eq(f ) = Trans(f ), and the system is almost equicontinuous. The map f is a homeomorphism and the inverse system (X, f −1 ) is almost equicontinuous. Furthermore, the system is uniformly rigid meaning that some subsequence of {f n : n = 0, 1,...} converges uniformly to the identity.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos.

7. On Lyapunov Numbers This cannot happen when the system is transitive.

Theorem 1.1 Let (X, f ) be a topologically transitive dynamical system.

Exactly one of the following two cases holds.

Case i (Eq(f ) = ∅) Assume there exists an equicontinuity point for the system. The equicontinuity points are exactly the transitive points, i.e. Eq(f ) = Trans(f ), and the system is almost equicontinuous. The map f is a homeomorphism and the inverse system (X, f −1 ) is almost equicontinuous. Furthermore, the system is uniformly rigid meaning that some subsequence of {f n : n = 0, 1,...} converges uniformly to the identity.

Case ii (Eq(f ) = ∅) Assume the system has no equicontinuity points. The system is sensitive, i.e. there exists ε 0 such that Eqε (f ) = ∅.

–  –  –

So, various definitions of sensitivity, formally give us different Lyapunov numbers – quantitative measures of these sensitivities.

Directly from the definitions, the following inequalities hold

–  –  –

So, various definitions of sensitivity, formally give us different Lyapunov numbers – quantitative measures of these sensitivities.

Directly from the definitions, the following inequalities hold

–  –  –

Beweis.

Let Ld be the second Lyapunov number of (X, f ). Fix a (small enough) δ 0, a point x ∈ X and a neighborhood Ux of x.Let U0 = Ux and n0 be the first positive integer, for which diam(f n0 (U0 )) Ld − δ. There exists a point y0 ∈ U0 such that d(f n0 (x), f n0 (y0 )) (Ld − δ)/2.

–  –  –

Beweis.

Let Ld be the second Lyapunov number of (X, f ). Fix a (small enough) δ 0, a point x ∈ X and a neighborhood Ux of x.Let U0 = Ux and n0 be the first positive integer, for which diam(f n0 (U0 )) Ld − δ. There exists a point y0 ∈ U0 such that d(f n0 (x), f n0 (y0 )) (Ld − δ)/2.Choose an opene U1 with its closure contained in U0 such that y0 ∈ U1 and diam(f m (U1 )) ≤ δ/2 for every non-negative integer m ≤ n0. Let n1 be the first positive integer, for which diam(f n1 (U1 )) Ld − δ.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos.

7. On Lyapunov Numbers Beweis.

Let Ld be the second Lyapunov number of (X, f ). Fix a (small enough) δ 0, a point x ∈ X and a neighborhood Ux of x.Let U0 = Ux and n0 be the first positive integer, for which diam(f n0 (U0 )) Ld − δ. There exists a point y0 ∈ U0 such that d(f n0 (x), f n0 (y0 )) (Ld − δ)/2.Choose an opene U1 with its closure contained in U0 such that y0 ∈ U1 and diam(f m (U1 )) ≤ δ/2 for every non-negative integer m ≤ n0. Let n1 be the first positive integer, for which diam(f n1 (U1 )) Ld − δ. By the definition of U1, we clearly have n1 n0.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos.

7. On Lyapunov Numbers Beweis.

Let Ld be the second Lyapunov number of (X, f ). Fix a (small enough) δ 0, a point x ∈ X and a neighborhood Ux of x.Let U0 = Ux and n0 be the first positive integer, for which diam(f n0 (U0 )) Ld − δ. There exists a point y0 ∈ U0 such that d(f n0 (x), f n0 (y0 )) (Ld − δ)/2.Choose an opene U1 with its closure contained in U0 such that y0 ∈ U1 and diam(f m (U1 )) ≤ δ/2 for every non-negative integer m ≤ n0. Let n1 be the first positive integer, for which diam(f n1 (U1 )) Ld − δ. By the definition of U1, we clearly have n1 n0.

We define recursively opene sets U2, U3,... and positive integers n2, n3,...

as follows.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos.

7. On Lyapunov Numbers Beweis.

Let Ld be the second Lyapunov number of (X, f ). Fix a (small enough) δ 0, a point x ∈ X and a neighborhood Ux of x.Let U0 = Ux and n0 be the first positive integer, for which diam(f n0 (U0 )) Ld − δ. There exists a point y0 ∈ U0 such that d(f n0 (x), f n0 (y0 )) (Ld − δ)/2.Choose an opene U1 with its closure contained in U0 such that y0 ∈ U1 and diam(f m (U1 )) ≤ δ/2 for every non-negative integer m ≤ n0. Let n1 be the first positive integer, for which diam(f n1 (U1 )) Ld − δ. By the definition of U1, we clearly have n1 n0.

We define recursively opene sets U2, U3,... and positive integers n2, n3,...

as follows. Since nk−1 is defined, there exists a point yk−1 ∈ Uk−1 such that d(f nk−1 (x), f nk−1 (yk−1 )) (Ld − δ)/2.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos.

7. On Lyapunov Numbers Beweis.

Let Ld be the second Lyapunov number of (X, f ). Fix a (small enough) δ 0, a point x ∈ X and a neighborhood Ux of x.Let U0 = Ux and n0 be the first positive integer, for which diam(f n0 (U0 )) Ld − δ. There exists a point y0 ∈ U0 such that d(f n0 (x), f n0 (y0 )) (Ld − δ)/2.Choose an opene U1 with its closure contained in U0 such that y0 ∈ U1 and diam(f m (U1 )) ≤ δ/2 for every non-negative integer m ≤ n0. Let n1 be the first positive integer, for which diam(f n1 (U1 )) Ld − δ. By the definition of U1, we clearly have n1 n0.

We define recursively opene sets U2, U3,... and positive integers n2, n3,...

as follows. Since nk−1 is defined, there exists a point yk−1 ∈ Uk−1 such that d(f nk−1 (x), f nk−1 (yk−1 )) (Ld − δ)/2. So we can choose an opene Uk in Uk−1 such that yk−1 ∈ Uk and diam(f m (Unk )) ≤ δ/2 for every non-negative integer m ≤ n0. Let nk be the first positive integer, for which diam(f nk (Uk )) Ld − δ.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos.

7. On Lyapunov Numbers Beweis.

Let Ld be the second Lyapunov number of (X, f ). Fix a (small enough) δ 0, a point x ∈ X and a neighborhood Ux of x.Let U0 = Ux and n0 be the first positive integer, for which diam(f n0 (U0 )) Ld − δ. There exists a point y0 ∈ U0 such that d(f n0 (x), f n0 (y0 )) (Ld − δ)/2.Choose an opene U1 with its closure contained in U0 such that y0 ∈ U1 and diam(f m (U1 )) ≤ δ/2 for every non-negative integer m ≤ n0. Let n1 be the first positive integer, for which diam(f n1 (U1 )) Ld − δ. By the definition of U1, we clearly have n1 n0.

We define recursively opene sets U2, U3,... and positive integers n2, n3,...

as follows. Since nk−1 is defined, there exists a point yk−1 ∈ Uk−1 such that d(f nk−1 (x), f nk−1 (yk−1 )) (Ld − δ)/2. So we can choose an opene Uk in Uk−1 such that yk−1 ∈ Uk and diam(f m (Unk )) ≤ δ/2 for every non-negative integer m ≤ n0. Let nk be the first positive integer, for which diam(f nk (Uk )) Ld − δ.As in the previous step, by the definition of Uk we clearly have nk nk−1.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos.

7. On Lyapunov Numbers Beweis.

Let Ld be the second Lyapunov number of (X, f ). Fix a (small enough) δ 0, a point x ∈ X and a neighborhood Ux of x.Let U0 = Ux and n0 be the first positive integer, for which diam(f n0 (U0 )) Ld − δ. There exists a point y0 ∈ U0 such that d(f n0 (x), f n0 (y0 )) (Ld − δ)/2.Choose an opene U1 with its closure contained in U0 such that y0 ∈ U1 and diam(f m (U1 )) ≤ δ/2 for every non-negative integer m ≤ n0. Let n1 be the first positive integer, for which diam(f n1 (U1 )) Ld − δ. By the definition of U1, we clearly have n1 n0.

We define recursively opene sets U2, U3,... and positive integers n2, n3,...

as follows. Since nk−1 is defined, there exists a point yk−1 ∈ Uk−1 such that d(f nk−1 (x), f nk−1 (yk−1 )) (Ld − δ)/2. So we can choose an opene Uk in Uk−1 such that yk−1 ∈ Uk and diam(f m (Unk )) ≤ δ/2 for every non-negative integer m ≤ n0. Let nk be the first positive integer, for which diam(f nk (Uk )) Ld − δ.As in the previous step, by the definition of Uk we clearly have nk nk−1.



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