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# Alexander Bufetov

## Elementary introduction to dynamical systems

## Course syllabus

### Basic definitions, constructions, and examples

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

### Introduction to one-dimensional dynamics

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

### Topological entropy

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

### Structural stability and an introduction to hyperbolic theory

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

### Fundamentals of ergodic theory

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

### Entropy for measure-preserving transformations

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

### Introduction to topological dynamics

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