Dynamical Systems and Ergodic Theory Spring 2025

Lecturer
Manuel Lüthi, Menny Akka Ginosar
Lectures
Mon 14:15-17:00 in LEE C 114 (Lecture and Exercise session)
Wed 10:15-12:00 in HG G 26.5

Prerequisites

Basic knowledge of analysis on metric spaces, measure theory and integration is required. We will also need Fourier series and some functional analysis, but it suffices if students learn these topics simultaneously or a bit later.

Lecture Summaries

week date topic
1 17.02. Motivation: From initial value problems to dynamical systems, Poincaré's contributions to celestial dynamics, the three body problem, and the birth of modern dynamics; Liouville's theorem; Boltzman's ergodicity hypothesis. First examples: North-South dynamics and the circle rotation. Notes
19.02. More examples: \(\times p\)-map, one- and two-sided shifts, hyperbolic toral automorphism, continued fraction expansion, "Rationality dedection", Billards, Geodesic flow. Start of Topological Dynamics: Definition of a topological dynamical system, topological transitivity, transitivity of \(\times p\)-map. Chapter 1 in [PY]. Notes
2 24.02. Equivalent characterizations of topological transitivity, (a special case of the) Baire category theorem, minimality, characterizations of minimality, existence of minimal subsystems, Birkhoff recurrence, minimality of irrational rotation. Chapter 1 in [PY]. Notes
26.02. Conjugacy of topological dynamical systems. Homeomorphisms of the circle, liftings, orientation preseravation/inversion. Basic properties of liftings of homeomorphisms of the circle, definition of the rotation number as a limsup, proof of convergence of the corresponding sequence. Chapter 6 in [PY]. Notes
3 03.03. Further properties of rotation numbers, Proposition 6.2 and Lemma 6.3 in [PY]. Notes
05.03. Proof of Denjoy's theorem for minimal orientation preserving homeomorphisms of the circle, introduction of logarithmic total variation, cyclic orderings of the circle, Corollary 6.3.1 and Sublemma 6.5.1 from [PY]. Notes
4 10.03. Denjoy's theorem for homemorphisms of bounded logarithmic total variation; Theorem 6.5 in [PY]. Notes
12.03. Introduction to symbolic dynamics. Subshifts, Vertex shifts, and dynamical properties of vertex shifts with aperiodic or irreducible adjacency matrix; Section 1.2 from [ES]. Notes
5 17.03. Shifts of finite type, every shift of finite type is a vertex shift, complexity function and its growth properties for shifts of finite type, Morse--Hedlund theorem; covered ins Section 1.2 of [Br]. Notes
19.03. Sturmian shifts, definition and construction from irrational rotation; cf. Section 4.3.3 from [Br]. Introduction to Ergodic Theory; cf. Section 1.1 from [W] and Section 2.1 from [EW1]. Notes
6 24.03. Finite state Markov chains, Gauss map; Section 1.1 in [W] and Section 3.2 in [EW1]. Notes
26.03. An isomorphism between the one-sided shift on two symbols and the \(\times 2\)-map, Poincaré recurrence, isometries associated with measure preserving systems; Sections 1.3 and 1.4 in [W] and Sections 2.1 and 2.2 in [EW1]. Notes
7 31.03. von Neumann's mean ergodic theorem, definition of ergodicity, six equivalent characterizations of ergodicity; Sections 2.3 and 2.5 in [EW1]. Notes
02.04. Proof of equivalence of characterizations of ergodicity, definition and proof of existence and uniqueness of conditional expectation; Sections 2.3 and 5.1 in [EW1]. Notes

Exercises

The new exercises will be posted here on Fridays. We expect you to solve the problems in the following week on your own and during the time alloted on Mondays. Exercise sheet \(n\) can be discussed in weeks \(n+1\) and \(n+2\).

Exercise sheet Solutions/Hints
Exercise sheet 1 Solutions
Exercise sheet 2 Solutions
Exercise sheet 3 Solutions
Exercise sheet 4 Solutions
Exercise sheet 5 Solutions
Exercise sheet 6 Solutions
Exercise sheet 7

Literature

Textbooks which can be used as additional reference for some of the topics include: