Research in Ergodic Theory at UEA

Overview : Books : Web-based resources : Publications : PhD theses

Overview

Ergodic theory is the study of statistical properties of deterministic dynamical systems. By statistical properties we mean properties which are expressed through the behaviour of time averages along trajectories of dynamical systems.

Ergodic theory overlaps with smooth dynamical systems. Problems, examples, and methods in ergodic theory come from many branches of mathematics, including number theory, combinatorics, harmonic analysis, coding theory and group theory.

I am interested in dynamical properties of commuting maps and algebraic dynamical systems (actions by group automorphisms). I am also interested in dynamical realization of integer sequences and other connections between number theory and dynamical systems.

I take part in a regular Arithmetic Seminar which brings together researchers with an interest in number theory.

I am on the joint Editoral Advisory Board for the London Mathematical Society publications Proceedings, Journal and Bulletin.

The diagrams are taken from a recent paper by Manfred Einsiedler, Douglas Lind,
Richard Miles and Thomas Ward; it shows how an amoeba describes the expansive
subdynamics in an algebraic dynamical system.

Research activity at UEA in ergodic theory is focused on higher dimensional Markov shifts (dynamical systems in which the acting group is a lattice) and connections between arithmetic and ergodic theory.

The arcs show the non-expansive subdynamics in
a three-dimensional algebraic dynamical system.

A rich source of examples are provided by algebraic constructions, and the structure of these systems is studied using tools from abelian harmonic analysis, commutative algebra, and finite group theory. Of particular interest are higher-order mixing and rigidity phenomena. The entropy of higher-dimensional Markov shifts is also studied.

A slice of a three-dimensional amoeba set.

Books

For more information on ergodic theory and its connections with other areas of mathematics, have a look at the following books.

`Ergodic Theory' by Karl Petersen, Cambridge University Press (1983).
`Dynamical Systems and Ergodic Theory' by Mark Pollicott and Michiko Yuri, Cambridge University Press (1998). Text and corrections available on-line.
`An Introduction to Ergodic Theory' by Peter Walters, Springer-Verlag (1982).
`Ergodic Theory' by I.P. Cornfeld, S.V. Fomin and Ya.G. Sinai, Springer-Verlag (1981).
`Symbolic Dynamics and Coding' by Douglas Lind and Brian Marcus, Cambridge University Press (1995).
`Introduction to the Modern Theory of Dynamical Systems' by A. Katok and B. Hasselblatt, Cambridge University Press (1995).
`Dynamical Systems of Algebraic Origin' by K. Schmidt, Birkhaüser (1995).
`Recurrence in Ergodic Theory and Combinatorial Number Theory' by H. Furstenberg, Princeton University Press (1981).
`Heights of Polynomials and Entropy in Algebraic Dynamics' by G. Everest and T. Ward, Springer-Verlag (1999).

Web-based resources

Google Dynamical Systems People
Open Problems in Dynamical Systems and Ergodic Theory (Sergei Kolyada)
Math. Subject Classifications related to ergodic theory. The top-level classification 37 covers Dynamical systems and ergodic theory. Some specific related areas are:
- 11K Probabilistic theory: distribution modulo $1$; metric theory of algorithms
- 37A Ergodic theory
- 37B Topological dynamics
- 37C Smooth dynamical systems: general theory
- 28D Measure-theoretic ergodic theory
- 22D Locally compact groups and their algebras
- 47A General theory of linear operators
- 60B Probability theory on algebraic and topological structures
- 60F Limit theorems
Preprints in Dynamical Systems (from the XXX archive)

Publications

PhD theses

Periodic points in S-integer dynamical systems - PhD thesis by Vijay Chothi, 1996 (jointly supervised with G. Everest)
Dynamical constraints on group actions - PhD thesis by Gary Morris, 1998
Arithmetic Dynamical Systems - PhD thesis by Richard Miles, 2000
Arithmetic of numbers of periodic points - PhD thesis by Yash Puri, 2000
The arithmetic of realizable sequences - PhD thesis by Patrick Moss, 2003 (jointly supervised with G. Everest)
Orbit counting far from hyperbolicity - PhD thesis by Victoria Stangoe, 2004 (jointly supervised with G. Everest)
Prime appearance in elliptic divisibility sequences" - PhD thesis by Helen King, 2005 (jointly supervised with G. Everest)

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