Dynamics and Ergodic Theory

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The proposed research concerns a number of inter-related aspects of dynamical systems (the study of evolving processes) and ergodic theory (evolving processes with randomness). It also touches on probability theory. The principal themes are (1) the behaviour of piecewise isometries; (2) thermodynamic formalism; (3) Operator-valued Multiplicative ergodic theorems. Piecewise isometries are amongst the simplest examples of discontinuous dynamical systems. A map is defined by taking a partition of a subset of a Euclidean space and applying an isometry to each part. Due to the simplicity of the map on each partition element, whatever complexity occurs in these systems happens because of the discontinuous nature of the map. The goal here is to develop a broadly-based toolkit and theory for this class of mappings. In thermodynamic formalism, Gibbs measures are defined to be invariant measures maximizing a quantity analogous to those arising in thermodynamics. A limiting case of this is ergodic optimization in which one studies measures maximizing a potential. This corresponds thermodynamically to 'lowering the temperature to 0'. The project will study general properties of these maximizing measures. A second topic in this area is understanding the relationship between Gibbs measures on a subsystem and Gibbs measures on the original system. This can be described in two ways: combinatorially and analytically and the goal is to relate the two. Oseledets' famous multiplicative ergodic theorem gives a structural description of the action of a composition of a stationary sequence of random matrices on a Euclidean space. The space is decomposed into subspaces, each of which has a different asymptotic expansion rate. This work has been further developed and applied to certain families of quasi-compact operators in place of matrices. In ongoing collaborative work I propose to further develop this theory with a view to applications, in particular the application of the theory to environmental data sets.