Integrative teaching
Giulia Livieri
Exercises
Examination procedure
Written and oral exam
Prerequisites
Second year undegraduate students in mathematics and physics.
Syllabus
Introduction to probability theory:
Measure spaces. Lebesgue measure. Probability measures. Extension of measures and Dynkin theorem. L^p spaces. Radon-Nykyodim theorem and density of measures. Invariant measures. Discrete random variables. Indenpendence. Conditional expectation.
Brief introduction to Fourier series:
Hilbert spaces. Orthonormal systems, Hibert basis, Bessel inequality. Completeness. Fourier series. Riemann-Lebesgue Lemma. Regularity of periodic functions and decay of Fourier coefficients.
Dynamical systems:
Discrete time and continuous time dynamical systems. Ordinary differential equations, flows, equilibria. Linear differential equations. Linearization of an O.D.E. Suspension of a discrete time dynamical systems. Topological dynamical systems.
Transitivity, mininality. Irrational rotations, expansive circle endomorphisms. Topological Bernoulli schemes. Iterated function systems.
Measurable dynamical systems. Poincaré recurrence theorem. Krylov-Bogoliubov theorem (statement). Birkhoff theorem. Perron-Frobenius operator. Ergodicity, mixing. Von Neumann ergodic theorem. Hyperbolic torus automorphisms.
Introduction to information theory:
Shannon entropy. Relative entropy. Mutual information. Asymptotic equipartition property. Data compression.
Markov chains, entropy rate. Random walk on a graph. Prediction, entropy and gambling: Kelly's criterion. Information theory, coding, data compression and prediction.
Entropy and dynamical systems:
Topological entropy. Kolmogorov-Sinai entropy. Bernoulli schemes. Topological and measurable Markov chains. Perron-Frobenius theorem. Google page-rank algorithm.
Bibliographical references
Introduzione ai sistemi dinamici (dispense che verranno distribuite agli studenti)
Shlomo Sternberg: Dynamical Systems
Cover-Thomas: Elements of Information Theory
Brin-Stuck: Introduction to dynamical systems
Hirsch-Smale-Devaney: Differential equations, dynamical systems and an introduction to chaos