Examples and Basic concepts: Dynamical system, orbits, Circle rotations, Shifts and sub-shifts, Hyperbolic toral automorphisms, The Horseshoe, The solenoid, Flows and differential equations, Chaos and Lyapunov exponents; Topological dynamics: Limit sets, Recurrence, Mixing, Transitivity, Entropy; Symbolic Dynamics: Subshifts, Sofic shifts, codes, Perron Frobenius Theorem, Data storage; Ergodic Theory: Ergodicity and mixing, Ergodic theorems (Von Neumann Ergodic Theorem, Birkhoff Ergodic Theorem), Invariant measures for continuous maps,; Unique ergodicity and Weyl’s theorem, Discrete spectrum, Weak mixing, Internet search; Hyperbolic dynamics: Stable and unstable manifolds, Anosov diffeomorphisms, Axiom A and structural stability; Ergodicity of Anosov diffeomorphisms: Holder continuity of the stable and unstable distributions, Absolute continuity of stable and unstable foliations; |