Introduction to dynamical systems I

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Type of exam

Written and oral exam


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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. 

Educational aims

The goal of the course is to introduce the fundamental
notions of the modern theory of dynamical systems and
of information theory.

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