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.
| 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 | |
| 8 | 07.04. | Ergodicity of irrational rotation and \(\times p\)-map, statement of Birkhoff's individual ergodic theorem, unique ergodicity of irrational rotation; Sections 2.4, 2.6, 4.3, and 6.1 in [EW1], Section 2.1 in [ES]. Notes |
| 09.04. | Proof of Birkhoff's pointwise ergodic theorem, ergodicity of Bernoulli shifts; Section 2.1 in [ES] and Section 2.3 in [EW1]. Notes | |
| 9 | 14.04. | Definition of finite state (coordinate process) Markov chains, of \((\mathbf{p},P)\)-Markov chains, stationarity of transition probabilities, and shift-invariance; cf. Section 1.1 in [N]. Notes (updated April 28) |
| 16.04. | Stopping times, (strong) Markov property, recurrence and transience; cf. Sections 1.4 and 1.5 in [N] and Section 15.3 in [B]. Notes (updated April 28) | |
| 10 | 30.04. | Communicating classes, irreducible \((\mathbf{p},P)\)-Markov chains, integrability of first visit, formula for the expected value, pointwise almost sure convergence to stationary distribution; cf. Sections 1.2, 1.5, 1.10 in [N]. Notes |
| 11 | 05.05. | Equivalent characterizations of ergodicity of irreducible \((\mathbf{p},P)\)-Markov chains; cf. Section 1.7 in [W]. Notes |
| 07.05. | Mixing, weak mixing, and equivalent characterizations of the latter; cf. Sections 2.7 and 2.8 in [EW1]. Notes | |
| 12 | 12.05. | Equivalence of weak-mixing and continuous spectrum, equivalent characterizations of mixing for Markov shifts; cf. Section 2.8 in [EW1] and Theorem 1.31 in [W]. Notes |
| 14.05. | Invariant measures for continuous maps: The weak-∗ topology on the set of Borel probability measures, compactness of (invariant) probability measures, existence of invariant probability measures, unique ergodicity, Furstenberg's conjugacy of homeomorphisms of the torus without periodic points; cf. Section 6 in [W]. Notes |
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 \(n+2\). Note that there is no exercise sheet posted for week 9, hence the formula has to be adapted afterwards.