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Resonances and Zeta Functions in Smooth Ergodic Theory and Geometry

Periodic Reporting for period 2 - Resonances (Resonances and Zeta Functions in Smooth Ergodic Theory and Geometry)

Reporting period: 2021-03-01 to 2022-08-31

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 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 traditional 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.
There has been substantial progress in pursuit of the original aims. In particular, with respect to the objectives in the first phase of the project this has resulted in surprisingly good numerical estimates on certain dynamical quantities with interesting applications to problems in many other areas, including geometry and analysis, but most notably notably number theory. This has lead to more efficient and accurate estimates on established results and to completely new results previously out of reach. This was first tested out on a number of classical problems where a specific rigorous and verifiable value for the Hausdorff dimension of a subset of the real line leads to solutions of interesting questions in pure mathematics, for example recovering estimates related to the density one Zaremba Conjecture. This done, the method was used to make significant advances on our understanding of the Lagrange and Markov spectra in Diophantine Approximation of irrational numbers.

The Hausdorff dimension of the sets we need to consider appears as a zero for a suitable complex function and (dynamical zeta function) and the method we used to estimate these values exploits specific features associated to this particular value.
The basic ingredients in the analysis and the ideas follow the planned programme of research. However, in the details we have taken advantage of the flexibility available to fine tune and develop a superior methodology than that originally anticipated. The approach alighted upon has considerable malleability and applies to a number of different problems. This is illustrated by the first steps into approximating Lyapunov exponents, which measure the exponential rate at which typically the orbits of nearby points in a dynamical system diverge.

The basic method is sufficiently flexible that it can be applied to estimate more geometric values. including the bottom of the spectrum of the Laplacian for infinite area surfaces of constant negative curvature. A higher dimensional setting allows a connection between the classical century old Schottky-Klein prime function and the distances between geodesics in hyperbolic three manifolds. This fits into the general programme of asymptotic counting of geometric quantities using dynamical and thermodynamic quantities as advanced in our monograph with Urbanski.

Another direction in which good progress has been made has been in the development (with Colognese) of the connection between the thermodynamic themes of the original setting and, somewhat surprisingly, another classical setting of translation surfaces (i.e. surfaces that are flat except at a finite number of cone singularities). These are traditionally thought of as "zero entropy" dynamical systems, but with a suitable new interpretation of entropy the theory for hyperbolic systems can be applied. Previously, it was understood that one approach to introducing "thermodynamic" ideas into the setting of translation surfaces was at the level of dynamics on the parameter space (or moduli space). This was explored with Aimino.

The research undertaken followed the general scheme of the original proposal while exploring new original research opportunities as they arose.
A particularly interesting direction in which the circles of ideas developed in this project was applied was to the famous setting of Bernoulli convolutions. This is an area of research in probability theory made famous by the work of Erdos in the 1930s. More precisely, this gives rise to a natural one dimensional family of stationary probability measures on the real line. We provided a significantly better uniform lower bound on the Hausdorff dimension of these measures than previously known.

Another novel example of estimates was to Fourier multipliers. These arise in the context of Fourier analysis and we were able to substantiate a numerical conjecture of Volkmer and Chen.

A very topical example of a fractal set is the Rauzy Gasket, which appears as an exceptional parameter set in a variety of different problems (from interval exchange transformations to billiards). Whereas this superficially appears similar to other simpler famous fractal sets, it is actually very difficult to establish rigorous bounds. Avila-Hubert-Skripchenko showed it had dimension strictly smaller than 2, building on work of Yoccoz. We improve (i.e. reduce) the best known upper bounds on its Hausdorff Dimension to 1.7407. This also has implications for improving the bounds on the constants in the asymptotic formula for counting solutions to the Markoff-Hurwicz equation, one of the problems we original proposed at the end of the project.

The progress made during the first half of the programme has opened the door to some exciting results in other new directions. In particular, the study of the drift associated to random walks on examples of Fuchsian groups and the nature of the associated harmonic measure on the circle at infinity. This leads to new examples of cocompact Fuchsian groups for which the harmonic is singular, supporting the famous Kaimanovich-Le Prince conjecture.

However, all of this will be encompassed within the framework of the original research plan. We are still on track to complete the project on schedule, with an emphasis on zeros for zeta functions and resonances, particularly now that the team is up to a full complement.
Resonances for the geodesic flow on a negatively curved surface with boundary