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 hyperbolic 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 was to develop a broad theory which is both effective and entirely mathematically rigorous. Furthermore, this theory continues to have important and diverse applications to many different areas of mathematics (in particular geometry, including both surfaces of negative curvature and flat translation surfaces, but also number theory, including Lagrange spectra and Zaremba conjecture) as well as, potentially, other areas of science.
There were 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 was the development of new methods for determining completely rigorously numerically basic characteristic values for classical (“chaotic”) hyperbolic systems.
(b) The second main theme was 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. For example, frame flows and compact group extensions of geodesic flows.
(c) The third main topic was applications to specific major problems, particularly in hyperbolic geometry, and number theory which had not been anticipated.
