Ergodic Theory

Recommended Prerequisites

Advanced knowledge of topology, manifolds and analysis. Proficiency in English.

Teaching Methods

The content of the syllabus is presented in the lectures, where examples are given to illustrate the concepts and several exercises and problems are solved and discussed. Reading suggestions, additional bibliography and other resources are available for students at the unit’s web page or the FCUP library.

Learning Outcomes

Students should become acquainted with the methods for qualitative analysis of the statistical properties of a dynamical system with respect to an invariant measure, on a complement to the classical approach which highlights generic topological aspects. Emphasis is given to the context of physical systems with a high degree of unstability.

Work Placement(s)

No

Syllabus

Convergence in L^p, in measure and a.e.; integrability; Fatou Lemma, Monotone Convergence and Dominated Convergence; isomorphism in measure. Periodic orbit, recurrent, non-wandering, dense; conjugacy, stability. Invariant measures by a dynamic. Invariants by isomorphism. Existence of invariant probability measures for continuous dynamics on compact metric spaces. Ergodicity, mixing, uniquely ergodicity; dynamical and spectral interpretations of these metrical properties. Examples: homeomorphism of R; qx(mod1); Gauss map; shifts; powers of z and rotation in S^1; linear maps in R^n; Anosov diffeomorphisms; Smale horseshoe; pedal triangle; coupled systems; Markov chains; search engines. Poincaré, Birkhoff and Kac Theorems; connection with the second law of Thermodynamics. Applications: Borel Theorem on normal numbers; Multiple Recurrence Theorem; van der Waerden Theorem. Ergodic decomposition. Topological and measure-theoretic entropy. Variational principle and equilibrium states.

Assessment Methods

Assessment
Exam: 40.0%
Research work: 60.0%

Bibliography

P. Halmos, Measure theory, Springer New York, 1974.

P. Halmos, Lectures on Ergodic Theory, Chelsea New York, 1956.

R. Mané, Introdução à Teoria Ergódica, IMPA, 1983.

K.R. Parthasarathy, Probability Measures on Metric Spaces, Academic Press, 1967.

K. Petersen, Ergodic Theory, Cambridge University Press, 1995.

P. Walters, An Introduction to Ergodic Theory, Springer New York, 1982.