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: 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 31 7.6. Topological determinism and zero entropy 32