# Ergodic Theory and Analysis

The Ergodic Theory and Analysis group at Manchester currently has four members, whose combined research interests cover a wide range of topics.

Broadly speaking, analysis is the rigorous study of `taking limits' and includes the theories of continuity, differentation, infinite sequences and series, integration and measures, etc. Although the context is often either in, or motivated by, behaviour of real and complex numbers, it is often highly profitable to consider more abstract spaces or spaces that have additional structure (e.g. metric spaces, Banach spaces, Hilbert spaces).

Ergodic theory is the study of dynamical systems in the presence of a measure. Dynamical systems is the study of iteration. A basic question in ergodic theory is: given a dynamical system acting on a phase space, what is the frequency with which a typical (with respect to the given measure) orbit of that system visits a given region of the phase space? Thus ergodic theory takes a qualitative view of dynamics and often concerns the long-term behaviour of iteration. The study of ergodic theory at Manchester concerns the probabilistic structure of iterated function systems, the dynamics of skew-product dynamical systems, thermodynamic formalism, fractal geometry, connections between ergodic theory and number theory with particular emphasis on beta-transformations, and maps with holes. Ergodic theory has many connections to other areas of mathematics, and the group works closely with the Analysis and Dynamical Systems group and Probability and Stochastic Analysis group.

Other areas of analysis studied at Manchester include Noncommutative Geometry. In Noncommutative Geometry one wants to understand the geometry of abstract topological spaces by interpreting it as an algebra of operators acting on a suitable space of functions. The Baum-Connes conjecture is a central topic of study, and sets up a correspondance between the geometry of a space, the topology of a space (in terms of homotopy theory), and purely analytic constructions such as the K-theory of an associated C*-algebra.