Thermodynamic Formalism, Dynamics and Geometry of Moduli Spaces

Lead Research Organisation: University of Warwick
Department Name: Mathematics

Abstract

This project is on the interface between ergodic theory and branch of the theory of dynamical systems that uses mathematical analysis to understand statistical properties , and the geometry of so called moduli spaces. Spaces that parametrise certain geometric structures. A recent link between these areas is that the pressure function that arises in thermodynamic formalism may be used to define a metric that describes the geometry of various moduli spaces.This point of view is particularly associated with the work of McMullen for the Teichmueller space of hyperbolic metrics but has been generalized to, for example, spaces of metric growth and spaces of representations of into higher rank Lie Groups. in more classical work from the 1980s Fried studies the space of cross sections on the flow obtained by suspending a pseudo-Anosov surface diffeomorphism. He shows that there is a geometrically natural entropy function. It seems plausible that one can interpret this in terms of the thermodynamic formalism and hence obtain an analogue of the Weil-Peterson metric to describe the geometry. This is the aim of the project. in addition to using techniques from thermodynamic formulism, it will involve the theory of dynamics on surfaces. The research is in the areas of geometry and topology and Mathematical analysis and is wholly writhing the Mathematical Sciences themer.

Publications

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Studentship Projects

Project Reference Relationship Related To Start End Student Name
EP/R513374/1 01/10/2018 30/09/2023
2105821 Studentship EP/R513374/1 01/10/2018 31/03/2022 Solly Coles