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Resonances and Zeta Functions in Smooth Ergodic Theory and Geometry

Periodic Reporting for period 1 - Resonances (Resonances and Zeta Functions in Smooth Ergodic Theory and Geometry)

Reporting period: 2019-09-01 to 2021-02-28

This project relates to understanding resonances in smooth Ergodic Theory and its applications.
The study of smooth ergodic theory and dynamical systems deals with the long term behaviour of transformations or flows on manifolds. This subject has made enormous advances in recent years and remains an area of innovation and rapid progress. This is particularly true for “chaotic” systems where the instability of trajectories makes the study particularly amenable to techniques from probability theory, ergodic theory and statistical mechanics. A paradigm for these systems are Anosov or hyper- bolic systems. Such dynamical systems can be very usefully characterised and usefully quantified by numerical values including entropy, non-zero Lyapunov exponents, or exponential growth of periodic orbits.
The objective of this research project is to develop new and original methods to address a number of fundamental questions in the development of smooth ergodic theory and dynamical systems. The aim is to develop a broad theory which is both effective and entirely mathematically rigorous.
Furthermore, this theory will have important and diverse applications to many different areas of mathematics (in particular geometry, but also number theory and topology) as well as, potentially, other areas of science (such as statistical physics and computation).
The methods we will develop in this research complements the existing theory, but bypasses many tra- ditional problems by a radical approach running contrary to the prevailing school of thought amongst researchers. The generality of the approach draws heavily on the PIs experience with the ergodic theory of hyperbolic systems, an area of research in which the PI has played a leading role for many years, but the risk and ambition stems from the novelty and risk inherent in trying to apply ideas from one area to another. The questions we are interested in are specifically related to the study of resonances, which characterise statistical properties of dynamical systems via the correlation function and the zeta function. This would be particularly important in the special case of geodesic flow.
There are three major strands to this work although they are bound together by interactions both at the level of the conclusions and the methodology.
(a) The first is the development of new methods for determining numerically basic characteristic values, which are theoretically poorly understood, for classical (“chaotic”) hyperbolic systems. The important new development is a completely rigorous estimate on the approximation.
(b) The second main theme will be the establishment of a radically new theory of resonances and correlation functions for a broader classes of systems, for which there is presently no existing theory. Typically these examples arise in geometry, and will help our understanding of that area.
(c) The third main topic will be applications to specific major problems, particularly in hyperbolic geometry, and number theory.
A key ingredient in our approach is the blending of classical ideas from classical thermodynamic formalism with more progressive functional analytic techniques that continue to emerge.
These three themes are both interactive and symbiotic, and are designed to knit together to give a coherent and substantial programme of research.
I am working on the initial part of the project

(A1) Establish a fast, efficient and accurate approach to numerically compute entropy, lyapunov ex- ponents, and other characteristic values for Anosov diffeomorphisms and Gibbs measures (and their generalisations).

(A2). Establish a fast, efficient and accurate approach to numerically estimating resonances for Anosov diffeomorphisms and Gibbs measures (and their generalizations).
It is early days, but I have been involved in a radically new approach to getting lower bounds on dimension of Bernoulli convolutions