Dynamical systems take a place in mathematics as in their applications: "It is useful to solve differential equations." according to the Newton's secret motto. It was true at the celestial mechanics and modern physics fundation, and it is still true today with the use of models which analysis is often from dynamical system theory (population evolution, crystal state...).
If functional analysis and digital analysis study existence, unicity and approximation processes of such model solutions, the dynamical system theory seeks to establish their long term properties (e.g.: long term statistical predictability despite medium term unpredictability).
In a less obvious way for the novice, dynamical systems also appear in pure mathematics. Some problems in geometry and number theory are elegantly and fruitfully translated into questions of dynamic.
The aim of this course is to present basic notions of the modern dynamical system theory linked with some questions in geometry and number theory.
Program
Ergodic theory:
- Poincaré recurrence theorem
- irreductibility notions: ergodicity, mixing, Bernoulli
- average and punctual ergodic theorems
- measured entropy
Topological dynamic:
- Birkhoff recurrence theorem
- irreductibility notions: transitivity, mixing, minimality
- simplex of invariant measures (unique ergodicity)
- tropologic entropy
Number theory:
- development in radix and in continued fraction
- equidistribution of P(n) value, n descirbing integers and P being a non-constant polynomial with an irrational coefficient
- Furstenberg's correspondance principle and Szemerédi's theorem
Dynamic of circle homeomorphisms:
- rotate number
- Denjoy theorem and counterexample
Dynamic of linear hyperbolic automorphisms of torus:
- delay of finite-type
- Markov partition
- entropy
Required level: Essential tools (only in measure theory) will be briefly reviewed or introduced. A familiarity with basic notions of topology will be a plus.
Course language: French or English according to the request
- Teaching coordinator: Burguet David