NMAA13031U Dynamical Systems (DynSys)

A dynamical system is a system which evolves as times goes by.
If the evolution law is fixed and deterministic,  we are dealing with iterations of a transformation from a space into itself  (discrete time) or an ordinary differential equation whose solution is a flow (continuous time). Many interesting examples arise in physics  (mechanics - in particular the solar system -  and statistical mechanics)  but also number theory, computer science, chemistry, biology... The long time evolution  is often unpredictable in practice ("chaos", "butterfly effect"), but  a qualitative description is possible for many interesting classes of systems, using e.g. methods from  probability theory  and functional analysis. Topics which will be presented include:
-Topological dynamics (coding, topological entropy...).
-Ergodic theory (Poincaré theorem, Birkhoff theorem, Kolmogorov  entropy, mixing, Shannon-McMillan-Breiman theorem, central limit  theorem and correlations functions...)
-Differentiable dynamics (one-dimensional real dynamics,  including  use of the Perron-Frobenius transfer operator;  hyperbolic dynamics in dimension two: stable and unstable manifolds, Hartman-Grobman theorem ...).
-Without proofs: Lyapunov exponents (Oseledec theorem, Ruelle inequality).