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Introduction to dynamical systems I

Schedule

Tuesday, 1 October 2019 to Saturday, 29 February 2020
Total hours: 60
Hours of lectures: 40
Hours of supplementary teaching: 20

Examination procedure

  • Written test
  • oral exam

Syllabus

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.

 

Educational goals:

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

 

 

 

Bibliographical references

Fasano-Marmi: Meccanica Analitica

Marmi: introduzione ai sistemi dinamici (dispense che verranno distribuite agli studenti)

Cover-Thomas: Elements of Information Theory

Sternberg: Dynamical Systems