Measure spaces. Probability measures. L^p spaces. Radon-Nykodim theorem and measures with a density. Invariant measures.
Hilbert spaces. Fourier series. Regular submanifolds of R^n. Tangent space.
Introduction to information theory. Shannon entropy. Relative entropy. Mutual information. Asymptotic equipartition theory.
Topological dynamical systems. Transitivity. Minimality. Gottshalk-Hedlund theorem. Topological entropy.
Measurable dynamical systems. Poincaré recurrence theorem. Krylov-Bogoliubov Theorem. Birkhoff Ergodic Theorem. Koopman and Perron-Frobenius operators. Ergodicity, mixing. Kolmogorov-Sinai entropy.
Bernoulli schemes. Markov chains.
Continued fractions, Gauss map. Hyperbolic automorphisms of tori.
Introduction to Lagrangian and Hamiltonian dynamical systems. Constrained systems.
The goal of the course is to introduce the fundamental notions of the modern theory of dynamical systems and of information theory.
Fasano-Marmi: Meccanica Analitica
Marmi: introduzione ai sistemi dinamici (dispense che verranno distribuite agli studenti)
Cover-Thomas: Elements of Information Theory
Sternberg: Dynamical Systems