Lecture 1: Why dynamical systems?

The main goal of the theory of dynamical systems is to understand the topological properties of orbits and long time behavior. In this lecture, we present motivation, classification and basic knowledge of dynamical systems.

Lecture 2: Fundamental concepts in topological dynamics

We recall some basic notions and fundamental results in topological dynamics, e.g., Walters’ stability theorem and Smale’s spectral decomposition theorem.

Lecture 3: Expansive measures

We introduce a notion of expansiveness for Borel probability measures with respect to dynamical systems, and discuss examples and fundamental properties of expansive measures.

Lecture 4: Shadowable measures

We introduce a notion of shadowable measures for dynamical systems, and show that a dynamical system has the shadowing property if and only if every invariant measure is shadowable.

Lecture 5: Topological stability

We introduce a notion of topologically stable measures for dynamical systems, and present a measurable version of Walters’ stability theorem.

Lecture 6: Spectral decomposition

We present a measurable version of Smale’s spectral decomposition theorem which is most important in the theory of topological dynamics.