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Alexander Bufetov

Elementary introduction to dynamical systems

Course syllabus

Basic definitions, constructions, and examples

1. Interval homeomorphisms. Circle rotations. Torus translations.
2. Symbolic dynamical systems: sequence spaces, Markov chains, adding machines.
3. Birkhoff recurrence theorem. Transitivity and minimality. Group translations. Topological mixing.
4. Conjugacy and semiconjugacy. Factors. Symbolic coding.
5. Tent map. Quadratic family. Smale horseshoe. Expanding maps of a circle. Linear automorphisms of a torus.
6. Attractors of dynamical systems. Smale attractor. Henon family.

Introduction to one-dimensional dynamics

1. Circle homeomorphisms. Rotation number. Poincare classification. Denjoy theorem. Denjoy example.
2. Interval self-maps: period three implies chaos. Sharkovsky theorem.

Topological entropy

1. Topological entropy. Elementary properties. Finiteness of entropy for Lipschitz self-maps.
2. Entropy for symbolical dynamical systems.
3. Expansive maps. Entropy and periodic growth.
4. Entropy for one-dimensional maps.
5. Entropy for expanding maps and the Misiurewicz-Przytycki theorem.

Structural stability and an introduction to hyperbolic theory

1. Morse-Smale systems. Andronov-Pontryagin theorem.
2. Structural stability of expanding circle maps.
3. Structural stability of hyperbolic toral automorphisms and the Grobman-Hartman theorem.
4. Stable and unstable subspaces for a hyperbolic linear map. Hadamard-Perron theorem.
5. Structural stability for the Smale horseshoe.
6. Hyperbolic sets. Anosov diffeomorphisms.
7. Anosov closing lemma.
8. Spectral decomposition.

Fundamentals of ergodic theory

1. Invariant measures. Poincare recurrence. Krylov-Bogoliouboff theorem.
2. Ergodic theorems: Von Neumann and Birkhoff-Khintchine.
3. Ergodicity. Ergodic decomposition.
4. First return map. Kakutani skyscraper. Kac theorem. Kakutani-Rokhlin-Halmos lemma.
5. Unique ergodicity. Ergodicity and recurrence.
6. Spectre. Von Neumann spectral theorem.
7. Mixing. Weak mixing.
8. Absolutely continuous invariant measures. SRB measures. SRB measures for expanding maps.
9. Minimal attractors.

Entropy for measure-preserving transformations

1. Partitions and conditioning. Entropy.
2. Generators. Kolmogorov-Sinai theorem.
3. Abramov-Rokhlin formula.
4. Variational principle. Bowen measure.

Introduction to topological dynamics

1. Equicontinuous and distal cascades. Bebutov systems.
2. Furstenberg classification of distal cascades.
3. Multiple recurrence. Furstenberg-Weiss theorem.