**MTH 623: Introduction to Ergodic Theory (4)**

**Pre-requisites**: *MTH 304 Metric Spaces and Topology, MTH 404 Measure and Integration*

*Discrete Dynamical systems:* definition and examples - maps on the circle, the doubling map, shifts of finite type, toral automorphisms.

*Topological and Symbolic dynamics:* transitivity, minimality, topological conjugacy and discrete spectrum, topological mixing, topological entropy, topological dynamical properties of shift spaces, circle maps and rotation number.

*Ergodic Theory:* invariant measures and measure-preserving transformations, ergodicity, recurrence and ergodic theorems (Poincare recurrence, Kac's lemma, Von Neumann's ergodic theorem, Birkhoff's ergodic theorem), applications of the ergodic theorem (continued fractions, Borel normal numbers, Khintchine’s recurrence theorem), ergodic measures for continuous transformations and their existence, Weyl’s equidistribution theorem, mixing and spectral properties.

*Information and entropy -* topological, measure-theoretic, and their relationship. Skew products, factors and natural extensions, induced transformations, suspensions and towers. Topological pressure and the variational principle, thermodynamic formalism and transfer operators, applications of thermodynamic formalism: (i) Bowen's formula for Hausdorff dimension, (ii) central limit theorems.

*Suggested Books*:

- P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York, 1982
- M.G. Nadkarni, Basic Ergodic Theory, Second Edition, Hindustan Book Agency, India
- M. Brin and G. Stuck, Introduction to Dynamical Systems, CUP, 2002
- M. Pollicott and M. Yuri, Dynamical systems and Ergodic theory, CUP, 1998
- P. R. Halmos, Lectures on Ergodic Theory, Chelsea, New York, 1956
- W. Parry, B. Bollobas, W. Fulton, Topics in Ergodic Theory, CUP, 2004
- A.B. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge, 1995

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