Dynamical Systems and Chaos

Dynamical Systems and Chaos
The course will provide quick introduction to Dynamical Systems, Ergodic Theory and Chaos. We will start with examples of dynamical systems, with basic notions such as orbits, periodic points, phase portraits, attraction and repulsion, calculus of fixed points, invariant measures, Bernoulli shifts and ergodic theorems of various types.
Then we will study bifurcations on the example of dynamics of quadratic maps. The quadratic family will be used to demonstrate the transition to chaos and the main features of chaotic behaviour. We will touch Sarkovsii's Theorem and Newton's Method.

Elements of Symbolic Dynamics and subshifts of finite type will be considered. Then we will move to fractals and discuss fractal dimension and related topics. After that we will introduce Holomorphic Dynamics and the main objects such as Julia sets and the Mandelbrot set. Time permitting, we will consider some rational maps in dimension two and higher. Henon map will be considered, as well as some maps arising in the theory of fractal groups, and the Smale horse shoe map. We will consider also spectra and spectral measures related to such groups and to fractal sets like Sierpinski gasket or Cantor set.