Introduction to dynamical systems I

Academic year 2021/2022
Lecturer Stefano Marmi

Integrative teaching

Giulia Livieri


Examination procedure

Written and oral exam


Second year undegraduate students in mathematics and physics.


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