Delivered at:: Department of Fundamental Mathematics (Faculty of Informatics, Mathematics, and Computer Science (HSE Nizhny Novgorod))

Course type:: Compulsory course

When:: 1 year, 3, 4 module

Instructor

Малкин Михаил Иосифович

Full Syllabus

Abstract

The course “Ergodic Theory” is aimed for introducing the students to the modern state and problems of dynamical systems having complicated (chaotic) behavior. Such systems usually can be described and studied with the help of Ergodic Theory.

Learning Objectives

To show the actual state of studying complicated dynamical systems using invariant measures and dynamical invariants which are responsible for complexity of the system. To explain the phenomena of limit behavior of chaotic systems and its relation to the evolution limits of probability distributions in ergodic theorems ( the results going back to the results of Poincare, Birkhoff, , Khinchin, Krylov, Bogolyubov, Kolmogorov, Sinai). To provide constructions and methods for modeling invariant measures and computing the invariants for concrete dynamical systems..

Expected Learning Outcomes

Understanding the constructions of Symbolic Dynamics.
Understanding the relationship between topological dynamics and Ergodic Theory.
Understanding the constructions of invariant measures
Knowledge of classical Ergodic theorems

Course Contents

Symbolic Dynamics
1. Application scheme of Symbolic Dynamics for specific dynamical systems. The space of symbolic sequences and its properties/ The shift map on the spaces of one-sided and two-sided symbolic sequences (Bernoulli shifts). 2. Topological Markov chains and their properties. Reducible and irreducible topological Markov chains, periodic and aperiodic chains; examples. 3. Perron-Frobenius Theorem. 4. Application of Perron-Frobenius Theorem for asymptotics of periodic points. Dynamical Artin-Mazur zeta function.
Entropic Theory of Discrete Dynamical Systems.
1. Definition of the topological entropy of a continuous map via open covers of the compact (after Adler, Conheim and MacAndrew). General properties of topological entropy. 2. Topological entropy as a topological conjugacy invariant. Generating covers and their application for computing the topological entropy. 3. Examples of computing the topological entropy for topological Markov chains and one dimensional maps. 4. Definition of topological entropy via separated and spanned sets (Bowen’s definition). Equivalence of different definitions of topological entropy. 5. Metric entropy and its properties. The Variational Principle. The measure with maximal entropy.
Classical Theorems of Ergodic Theory. Ergodic Theory of Low Dimensional Systems.
The Poincare Recurrence Theorem and their generalizations. Krylov-Bogolyubov Theorem and constructions of invariant measures. Uniquely ergodic dynamical systems. Ergodic measures and ergodic transformations. Ergodic Birhgoff-Khinchin Theorem and its applications. Strong and weak mixing, and their properties. Piecewise monotone maps: symbolic dynamics and invariant measures/. Invariant measures absolutely continuous with respect to the Lebesgue measure. Piecewise monotone maps and their models with constant slope. Symbolic constructions of conformal measures.

Assessment Elements

Реферат
Экзамен

Interim Assessment

Interim assessment (3 module)
0.5 * Реферат + 0.5 * Экзамен
Interim assessment (4 module)
0.5 * Реферат + 0.5 * Экзамен

Bibliography

Recommended Core Bibliography

Hasselblatt, Boris. Ergodic Theory and Negative Curvature [Электронный ресурс] / Boris Hasselblatt; БД springer. - Springer, Cham, 2017 - ISBN: 978-3-319-43058-4 (Print).

Recommended Additional Bibliography

. Kuznetsov, Sergey. Strange Nonchaotic Attractors : Dynamics Between Order and Chaos in Qua-siperiodically Forced Systems [Электронный ресурс] / Sergey Kuznetsov, Arkady Pikovsky, and Ul-rike Feudel. – World Scientific Publishing Co Pte Ltd, 2014, . – ISBN: 9789812566331 (Print).

Master’s Programme 'Mathematics'

Contacts

Еrgodic theory