On the Interplay between Measurable and Topological Dynamics E. Glasner and B. Weiss
نویسندگان
چکیده
Part 1. Analogies 3 1. Poincaré recurrence vs. Birkhoff’s recurrence 3 1.1. Poincaré recurrence theorem and topological recurrence 3 1.2. The existence of Borel cross-sections 4 1.3. Recurrence sequences and Poincaré sequences 5 2. The equivalence of weak mixing and continuous spectrum 7 3. Disjointness: measure vs. topological 10 4. Mild mixing: measure vs. topological 12 5. Distal systems: topological vs. measure 19 6. Furstenberg-Zimmer structure theorem vs. its topological PI version 21 7. Entropy: measure and topological 22 7.1. The classical variational principle 22 7.2. Entropy pairs and UPE systems 22 7.3. A measure attaining the topological entropy of an open cover 23 7.4. The variational principle for open covers 28 7.5. Further results connecting topological and measure entropy 30 7.6. Topological determinism and zero entropy 31
منابع مشابه
On the Interplay between Measurable and Topological Dynamics
Part 1. Analogies 3 1. Poincaré recurrence vs. Birkhoff’s recurrence 3 1.1. Poincaré recurrence theorem and topological recurrence 3 1.2. The existence of Borel cross-sections 4 1.3. Recurrence sequences and Poincaré sequences 5 2. The equivalence of weak mixing and continuous spectrum 7 3. Disjointness: measure vs. topological 10 4. Mild mixing: measure vs. topological 12 5. Distal systems: to...
متن کامل2 00 3 a Geometric Approach to Voiculescu - Brown Entropy
A basic problem in dynamics is to identify systems with positive entropy, i.e., systems which are “chaotic.” While there is a vast collection of results addressing this issue in topological dynamics, the phenomenon of positive entropy remains by and large a mystery within the broader noncommutative domain of C∗-algebraic dynamics. To shed some light on the noncommutative situation we propose a ...
متن کاملAn updated version of the paper in Nonlinearity 6, (1993), 1067–1075. SENSITIVE DEPENDENCE ON INITIAL CONDITIONS
It is shown that the property of sensitive dependence on initial conditions in the sense of Guckenheimer, follows from the other two more technical parts of one of the most common recent definitions of chaotic systems. It follows that this definition applies to a broad range of dynamical systems, many of which should not be considered chaotic. We investigate the implications of sensitive depend...
متن کاملEnveloping Semigroups in Topological Dynamics
This is a survey of the theory of enveloping semigroups in topological dynamics. We review the, already classical, theory of enveloping semigroups, due mainly to Robert Ellis, and then proceed to describe some new connections which were discovered in the last few years between three seemingly unrelated theories: of enveloping semigroups, of chaotic behavior, and of representation of dynamical s...
متن کاملB -AND B - COMPLETENESS IN LOCALLY CONVEX ALGEBRAS AND THE E x THEOREM
Let E be a B-complete (B -complete) locally convex algebra and $ the topological direct sum of countably many copies of the scalar field with a jointly continuous algebra multiplication. It has been shown that E is also B-complete (B -complete) for componentwise multiplication on E . B-and Br-completeness of E , the unitization of E, and also of E x for other multiplications on E ...
متن کامل