Topological Dynamics: Minimality, Entropy and Chaos. Sergiy Kolyada Institute of Mathematics, NAS of Ukraine, Kyiv Zentrum Mathematik, Technische Universit¨at Munchen,¨ John-von-Neumann Lecture, 2013 Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. Throughout this part of the lecture (X ; f ) denotes a topological dynamical system, where X is a (compact) metric space with a metric d and f : X ! X is a continuous map. A point x 2 X 'moves', its trajectory being the sequence x; f (x); f 2(x); f 3(x);::: , where f n is the n-th iteration of f . The point f n(x) is the position of x after n units of time. The set of points of the trajectory of x under f is called the orbit of x, denoted by orbf (x). Topological transitivity is a global characteristic of a dynamical system. The concept of topological transitivity goes back to G. D. Birkhoff, he used him in 1920. 1.Transitive maps 1. Topologically Transitive Maps Introduction Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. A point x 2 X 'moves', its trajectory being the sequence x; f (x); f 2(x); f 3(x);::: , where f n is the n-th iteration of f . The point f n(x) is the position of x after n units of time. The set of points of the trajectory of x under f is called the orbit of x, denoted by orbf (x). Topological transitivity is a global characteristic of a dynamical system. The concept of topological transitivity goes back to G. D. Birkhoff, he used him in 1920. 1.Transitive maps 1. Topologically Transitive Maps Introduction Throughout this part of the lecture (X ; f ) denotes a topological dynamical system, where X is a (compact) metric space with a metric d and f : X ! X is a continuous map. Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. Topological transitivity is a global characteristic of a dynamical system. The concept of topological transitivity goes back to G. D. Birkhoff, he used him in 1920. 1.Transitive maps 1. Topologically Transitive Maps Introduction Throughout this part of the lecture (X ; f ) denotes a topological dynamical system, where X is a (compact) metric space with a metric d and f : X ! X is a continuous map. A point x 2 X 'moves', its trajectory being the sequence x; f (x); f 2(x); f 3(x);::: , where f n is the n-th iteration of f . The point f n(x) is the position of x after n units of time. The set of points of the trajectory of x under f is called the orbit of x, denoted by orbf (x). Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. 1.Transitive maps 1. Topologically Transitive Maps Introduction Throughout this part of the lecture (X ; f ) denotes a topological dynamical system, where X is a (compact) metric space with a metric d and f : X ! X is a continuous map. A point x 2 X 'moves', its trajectory being the sequence x; f (x); f 2(x); f 3(x);::: , where f n is the n-th iteration of f . The point f n(x) is the position of x after n units of time. The set of points of the trajectory of x under f is called the orbit of x, denoted by orbf (x). Topological transitivity is a global characteristic of a dynamical system. The concept of topological transitivity goes back to G. D. Birkhoff, he used him in 1920. Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. 1.Transitive maps 1. Topologically Transitive Maps Introduction Intuitively, a topologically transitive map f has points which eventually move under iteration from one arbitrarily small neighbourhood to any other. Consequently, the dynamical system cannot be broken down or decomposed into two subsystems (disjoint sets with nonempty interiors) which do not interact under f , i.e., are invariant under the map (A ⊂ X is invariant if f (A) ⊂ A). Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. As usual, we adopt the condition (TT) as the definition of topological transitivity, but note that some authors take (DO) instead. 1.Transitive maps 1. Topologically Transitive Maps Introduction Let X be a metric space and f : X ! X continuous. Consider the following two conditions: (TT) for every pair of nonempty open (opene) sets U and V in X , there is a positive integer n such that f n(U) \ V 6= ;, (DO) there is a point x0 2 X such that the orbit of x0 is dense in X . Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. 1.Transitive maps 1. Topologically Transitive Maps Introduction Let X be a metric space and f : X ! X continuous. Consider the following two conditions: (TT) for every pair of nonempty open (opene) sets U and V in X , there is a positive integer n such that f n(U) \ V 6= ;, (DO) there is a point x0 2 X such that the orbit of x0 is dense in X . As usual, we adopt the condition (TT) as the definition of topological transitivity, but note that some authors take (DO) instead. Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. 1.Transitive maps 1. Topologically Transitive Maps Introduction Any point with dense orbit is called a transitive point. A point which is not transitive is called intransitive. The set of transitive or intransitive points of (X ; f ) will be denoted by tr(f ) or intr(f ) respectively, provided no misunderstanding can arise concerning the phase space. In such a case we will also speak on transitive or intransitive points of f . Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. 1.Transitive maps 1. Topologically Transitive Maps On the equivalent formulations of the definition The two conditions (TT) and (TT) are independent in general. In fact, 1 take X = f0g [ f n : n 2 Ng endowed with the usual metric and 1 1 f : X ! X defined by f (0) = 0 and f ( n ) = n+1 , n = 1; 2;::: . Clearly, f is continuous. The point x0 = 1 is (the only) transitive point for (X ; f ) 1 but the system is not topologically transitive (take, say, U = f 2 g, V = f1g). So, (DO) does not imply (TT). Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. 1.Transitive maps 1. Topologically Transitive Maps On the equivalent formulations of the definition We show that neither (TT) implies (DO). To this end take I = [0; 1] and the standard tent map g(x) = 1 − j2x − 1j from I to itself. Let X be the set of all periodic points of g and f = gjX (a point x is periodic for g if g n(x) = x for some positive integer n; the least such n is called the period of x). Then the system (X ; f ) does not satisfy the condition (DO), since X is infinite (dense in I ) while the orbit of any periodic point is finite. But the condition (TT) is fulfilled. This follows from the fact that for any nondegenerate subinterval J of I there is a positive integer k k with g (J) = I . Hence, whenever J1 and J2 are nonempty open subintervals of I , there is a periodic orbit of g which intersects both J1 and J2. This gives (TT) for (X ; f ). Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. 1.Transitive maps 1. Topologically Transitive Maps On the equivalent formulations of the definition Nevertheless, under some additional assumptions on the phase space (or on the map) the two definitions (TT) and (DO) are equivalent. In fact, we have the following Silverman theorem: Theorem 1.1 If X has no isolated point then (DO) implies (TT). If X is separable and second category then (TT) implies (DO). Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. 1.Transitive maps 1. Topologically Transitive Maps On the equivalent formulations of the definition Beweis. 1. If X has no isolated point and fx0; x1; :::g is a dense orbit (where n xn := f (x0); n ≥ 0), then any opene sets U and V (in X ) there are xk 2 U and xm 2 V n fx0; x1; :::; xk g. But if m > k then f m−k (U) \ V 6= ;. Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. It is known that a metric space is separable iff it is second countable (means has a countable base). 1.Transitive maps 1. Topologically Transitive Maps On the equivalent formulations of the definition Let's recall some definitions. Definitions A set A ⊆ X is the first category (sometimes it's called meager) if it the union of countably many nowhere dense subset of X . A set B ⊆ X is the second category if it is not the first one. A space X is called separable if it contains a countable dense subset. Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. 1.Transitive maps 1. Topologically Transitive Maps On the equivalent formulations of the definition Let's recall some definitions. Definitions A set A ⊆ X is the first category (sometimes it's called meager) if it the union of countably many nowhere dense subset of X . A set B ⊆ X is the second category if it is not the first one. A space X is called separable if it contains a countable dense subset. It is known that a metric space is separable iff it is second countable (means has a countable base). Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. Let us also remark that in general compact metric spaces, (TT) and (DO) are equivalent for onto maps.