Course Overview

Learning Achievement

Competence

Course prerequisites

Grading Philosophy

Course schedule

I. HOMEOMORPHISMS OF THE CIRCLE: ROTATION NUMBERS, 

1. Poincaré Theorem 
2. Denjoy Theorem 

II. SOME EXAMPLES:

1. Gradient flow (in particular of a Morse function),
2. Geodesic flow on hyperbolic surfaces,
3. Linear Anosov diffeomorphisms.

III. SYMBOLIC DYNAMIC.
1.Sub shifts of finite type,
2.Coding of  rotations, expansions on the circle and  Smale's
horseshoe,
3.Markov's partitions for the linear automorphisms of the torus.

IV. DYNAMIC COMPLEXITY.
1. Topological entropy,
2. Explicit computation for the above examples,
3. Entropyand action in homology.

V. ELEMENTS OF ERGODIC THEORY.
1.Invariant measures (Krylov-Bogolubov Theorem, Birkhoff Theorem).
2.Recurrence Theorems (Poincaré and others).
3.Ergodicity of linear Anosov maps and of geodesic flow for
the Liouville measure (Hopf Argument).

VI. HYPERBOLIC DYNAMICS
1. Stable Manifold Theorem, hyperbolic sets,
2. Examples: Plykinattractor, Smale solenoid,
3. Shadowing lemma and specification,
4. Structural stability,
5. Conley Theoryand global hyperbolic Theory.

VII. DYNAMICS ON HOMOGENOUS SPACES(HOWE-MOORE THEOREM, ETC?) OR
PESIN'S THEORY.

1.Homeomorphisms of the circle: rotation numbers, Poincaré Theorem.
Denjoy Theorem.

Course type

Lectures

Online Course Requirement

Instructor

Other information

Thisis a course of the M2ALGANT(Algebra,eometry and Number Theory). 

Formore information about this Master visit the
webpage: https://uf-mi.u-bordeaux.fr/algant/
[https://uf-mi.u-bordeaux.fr/algant/]

Duration: 12 weeks (spring semester)

Language of instruction: English
Mode of delivery: Face-to-face teaching

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