# «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.

Minimal maps

2. Minimal Maps

Introduction

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos.

Minimal maps

2. Minimal Maps

Introduction Throughout this part of the lecture (X, f ) denotes a topological dynamical system, where X is a (compact) metric (Hausdorﬀ) space with a metric d and f : X → X is a continuous map.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos.

Minimal maps

2. Minimal Maps Introduction Throughout this part of the lecture (X, f ) denotes a topological dynamical system, where X is a (compact) metric (Hausdorﬀ) space with a metric d and f : X → X is a continuous map.

A subset M of X is a (topologically) minimal set if M is closed, non-empty and invariant (i.e., f (M) ⊆ M) and if M has no proper subset with these three properties.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos.

Minimal maps

2. Minimal Maps Introduction Throughout this part of the lecture (X, f ) denotes a topological dynamical system, where X is a (compact) metric (Hausdorﬀ) space with a metric d and f : X → X is a continuous map.

A subset M of X is a (topologically) minimal set if M is closed, non-empty and invariant (i.e., f (M) ⊆ M) and if M has no proper subset with these three properties. A dynamical system (X ; f ) is called minimal if the set X is minimal.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos.

Minimal maps

2. Minimal Maps Introduction Throughout this part of the lecture (X, f ) denotes a topological dynamical system, where X is a (compact) metric (Hausdorﬀ) space with a metric d and f : X → X is a continuous map.

A subset M of X is a (topologically) minimal set if M is closed, non-empty and invariant (i.e., f (M) ⊆ M) and if M has no proper subset with these three properties. A dynamical system (X ; f ) is called minimal if the set X is minimal. In such a case we also say that the map f itself is minimal.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos.

Minimal maps

A subset M of X is a (topologically) minimal set if M is closed, non-empty and invariant (i.e., f (M) ⊆ M) and if M has no proper subset with these three properties. A dynamical system (X ; f ) is called minimal if the set X is minimal. In such a case we also say that the map f itself is minimal.

Minimal systems are natural generalizations of periodic orbits, and they are analogues of ergodic measures in topological dynamics. They were deﬁned by G. D. Birkhoﬀ in 1912.

2. Minimal Maps Introduction Given a dynamical system (X, f ), a set M ⊆ X is called a minimal set if it is non-empty, closed and invariant and if no proper subset of M has these three properties. So, M ⊆ X is a minimal set if and only if (M, f |M ) is a minimal system. A system (X, f ) is minimal if and only if X is a minimal set in (X, f ).

2. Minimal Maps Introduction Given a dynamical system (X, f ), a set M ⊆ X is called a minimal set if it is non-empty, closed and invariant and if no proper subset of M has these three properties. So, M ⊆ X is a minimal set if and only if (M, f |M ) is a minimal system. A system (X, f ) is minimal if and only if X is a minimal set in (X, f ).

The basic fact discovered by G. D. Birkhoﬀ is that in any compact system (X, f ) there are minimal sets. This follows immediately from the Zorn’s lemma.

2. Minimal Maps Introduction Given a dynamical system (X, f ), a set M ⊆ X is called a minimal set if it is non-empty, closed and invariant and if no proper subset of M has these three properties. So, M ⊆ X is a minimal set if and only if (M, f |M ) is a minimal system. A system (X, f ) is minimal if and only if X is a minimal set in (X, f ).

The basic fact discovered by G. D. Birkhoﬀ is that in any compact system (X, f ) there are minimal sets. This follows immediately from the Zorn’s lemma.

**The following conditions are equivalent:**

The basic fact discovered by G. D. Birkhoﬀ is that in any compact system (X, f ) there are minimal sets. This follows immediately from the Zorn’s lemma.

**The following conditions are equivalent:**

2. Minimal Maps Introduction Since any orbit closure is invariant, we get that any compact orbit closure contains a minimal set. In particular, this is how compact minimal sets may appear in non-compact spaces.

2. Minimal Maps Introduction Since any orbit closure is invariant, we get that any compact orbit closure contains a minimal set. In particular, this is how compact minimal sets may appear in non-compact spaces.

Two minimal sets in (X, f ) either are disjoint or coincide.

2. Minimal Maps Introduction Since any orbit closure is invariant, we get that any compact orbit closure contains a minimal set. In particular, this is how compact minimal sets may appear in non-compact spaces.

Two minimal sets in (X, f ) either are disjoint or coincide. A minimal set M is strongly f -invariant, i.e. f (M) = M, provided it is compact Hausdorﬀ.

Two minimal sets in (X, f ) either are disjoint or coincide. A minimal set M is strongly f -invariant, i.e. f (M) = M, provided it is compact Hausdorﬀ.

**Some other equivalent deﬁnitions of a minimal system:**

2. Minimal Maps More examples of minimal maps Example 2.1 Consider a homeomorphism of the 2-torus, S : T → T, of the form S(x, y ) = (x + α, y + β), where 1, α, β ∈ R are rationally independent and + : R/Z × R → R/Z is deﬁned in the obvious way. Then S is minimal (and ergodic with respect to Lebesgue measure).

2. Minimal Maps More examples of minimal maps Example 2.1 Consider a homeomorphism of the 2-torus, S : T → T, of the form S(x, y ) = (x + α, y + β), where 1, α, β ∈ R are rationally independent and + : R/Z × R → R/Z is deﬁned in the obvious way. Then S is minimal (and ergodic with respect to Lebesgue measure).

** Example 2.2 Let K = (kn )n0 be a sequence of integers kn ≥ 2.**

Let ΣK be the set of all one-sided inﬁnite sequences (in )n≥1 for which 0 ≤ in ≤ kn − 1.

2. Minimal Maps More examples of minimal maps Example 2.1 Consider a homeomorphism of the 2-torus, S : T → T, of the form S(x, y ) = (x + α, y + β), where 1, α, β ∈ R are rationally independent and + : R/Z × R → R/Z is deﬁned in the obvious way. Then S is minimal (and ergodic with respect to Lebesgue measure).

** Example 2.2 Let K = (kn )n0 be a sequence of integers kn ≥ 2.**

Let ΣK be the set of all one-sided inﬁnite sequences (in )n≥1 for which 0 ≤ in ≤ kn − 1. Think of these sequences as ’integers’ in multibase notation, the base of the nth digit in being kn. With the natural (product) topology, ΣK is homeomorphic to the Cantor set.

** Example 2.2 Let K = (kn )n0 be a sequence of integers kn ≥ 2.**

Let ΣK be the set of all one-sided inﬁnite sequences (in )n≥1 for which 0 ≤ in ≤ kn − 1. Think of these sequences as ’integers’ in multibase notation, the base of the nth digit in being kn. With the natural (product) topology, ΣK is homeomorphic to the Cantor set. Deﬁne a map αK : ΣK → ΣK which informally may be described as ’add 1 and carry’ where the addition is performed at the leftmost term i1 and the carry proceeds to the right in multibase notation. Then αK is a minimal homeomorphism and is called a ”generalized adding machine” or an ’odometer’.

** Example 2.2 Let K = (kn )n0 be a sequence of integers kn ≥ 2.**

Let ΣK be the set of all one-sided inﬁnite sequences (in )n≥1 for which 0 ≤ in ≤ kn − 1. Think of these sequences as ’integers’ in multibase notation, the base of the nth digit in being kn. With the natural (product) topology, ΣK is homeomorphic to the Cantor set. Deﬁne a map αK : ΣK → ΣK which informally may be described as ’add 1 and carry’ where the addition is performed at the leftmost term i1 and the carry proceeds to the right in multibase notation. Then αK is a minimal homeomorphism and is called a ”generalized adding machine” or an ’odometer’.

2. Minimal Maps On the equivalent formulations of the deﬁnition As a general reference see e.g. T. Downarowicz, Survey of odometers and Toeplitz ﬂows, Algebraic and topological dynamics, 7-37, Contemp.

Math., 385, Amer. Math. Soc., Providence, RI, 2005.

2. Minimal Maps Topological properties of minimal maps A continuous map f : X → Y between topological spaces is called irreducible if it is surjective and f (A) = Y for every proper closed subset A ⊂ X.

2. Minimal Maps Topological properties of minimal maps A continuous map f : X → Y between topological spaces is called irreducible if it is surjective and f (A) = Y for every proper closed subset A ⊂ X.

A set B ⊆ X is said to be a redundant open set for a map f : X → Y if B is opene and f (B) ⊆ f (X \ B) (i.e., its removal from the domain of f does not change the image of f ).

2. Minimal Maps Topological properties of minimal maps A continuous map f : X → Y between topological spaces is called irreducible if it is surjective and f (A) = Y for every proper closed subset A ⊂ X.

A set B ⊆ X is said to be a redundant open set for a map f : X → Y if B is opene and f (B) ⊆ f (X \ B) (i.e., its removal from the domain of f does not change the image of f ).

By taking B = X \ A one can have the following equivalent deﬁnition – a continuous map f : X → Y between topological spaces is irreducible if it is surjective and has no open redundant sets.

A set B ⊆ X is said to be a redundant open set for a map f : X → Y if B is opene and f (B) ⊆ f (X \ B) (i.e., its removal from the domain of f does not change the image of f ).

By taking B = X \ A one can have the following equivalent deﬁnition – a continuous map f : X → Y between topological spaces is irreducible if it is surjective and has no open redundant sets.

A map f : X → Y is called almost open if it sends opene sets to sets with non-empty interior (the terminology is not uniﬁed – instead of almost open some authors say semi-open, feebly open, somewhat open or quasi-interior).

By taking B = X \ A one can have the following equivalent deﬁnition – a continuous map f : X → Y between topological spaces is irreducible if it is surjective and has no open redundant sets.

A map f : X → Y is called almost open if it sends opene sets to sets with non-empty interior (the terminology is not uniﬁed – instead of almost open some authors say semi-open, feebly open, somewhat open or quasi-interior). It is easy to see that a map is almost open if and only if the inverse image of every dense subset is dense.