Periodic Reporting for period 4 - Resonances (Resonances and Zeta Functions in Smooth Ergodic Theory and Geometry)
Reporting period: 2024-03-01 to 2025-02-28
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 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.
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.
At every stage of the project there has been substantial progress in pursuit of the aims of this proposal. This project has resulted in surprisingly good numerical estimates with interesting applications to problems in many other areas, particularly geometry and analysis, but most notably in number theory. This has lead to both more efficient and accurate estimates improving on established results and also given completely new results previously out of reach. This was tested on a number of classical problems whose recent analysis was based on specific rigorous and verifiable values (e.g. Hausdorff dimension of a set or Lyapunov exponents of maps or random products of matrices). Interesting applications included: the density one Zaremba Conjecture and related problems; significant advances of our understanding of the Lagrange and Markov spectra in Diophantine Approximation of irrational numbers; and insight into the Hausdorff dimension of harmonic measures in hyperbolic geometry. These methods and ideas have been widely disseminated and will provide the scientific community with valuable tools for future research.
Our thermodynamic approach involved the analysis of the zeros (prototypical examples of resonances) for a suitable complex function (called the dynamical zeta function) and the method we developed to estimate these values exploits specific features associated to this particular value. The basic ingredients in the analysis followed the planned programme of research. In the context of other resonances for partially hyperbolic settings, work of Pollicott-Zhang studied resonances and mixing rates for compact group extensions of hyperbolic flows. However, at every stage we have also taken advantage of the flexibility available to fine tune our strategy and develop a superior methodology than that originally anticipated. The final refined approach has considerable malleability and applies to a number of different interesting problems. The method is sufficiently flexible that it has been 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 allowed a connection between the classical century old Schottky-Klein prime function and the distances between geodesics in hyperbolic three manifolds. Aspects of the project related to asymptotic counting of geometric quantities using dynamical and thermodynamic quantities were covered in the monograph(s) of Pollicott-Urbanski.
Another direction in which good progress was made has been in the development (by Colognese-Pollicott) 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 by Aimino-Pollicott.
The research undertaken followed the general scheme of the original proposal while exploring new original research opportunities as they arose.
Our thermodynamic approach involved the analysis of the zeros (prototypical examples of resonances) for a suitable complex function (called the dynamical zeta function) and the method we developed to estimate these values exploits specific features associated to this particular value. The basic ingredients in the analysis followed the planned programme of research. In the context of other resonances for partially hyperbolic settings, work of Pollicott-Zhang studied resonances and mixing rates for compact group extensions of hyperbolic flows. However, at every stage we have also taken advantage of the flexibility available to fine tune our strategy and develop a superior methodology than that originally anticipated. The final refined approach has considerable malleability and applies to a number of different interesting problems. The method is sufficiently flexible that it has been 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 allowed a connection between the classical century old Schottky-Klein prime function and the distances between geodesics in hyperbolic three manifolds. Aspects of the project related to asymptotic counting of geometric quantities using dynamical and thermodynamic quantities were covered in the monograph(s) of Pollicott-Urbanski.
Another direction in which good progress was made has been in the development (by Colognese-Pollicott) 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 by Aimino-Pollicott.
The research undertaken followed the general scheme of the original proposal while exploring new original research opportunities as they arose.
A particularly interesting new direction in which the circles of ideas developed in this project was applied was to the famous setting of Bernoulli convolutions which rise to a one dimensional family of stationary probability measures on the real line and made famous by the work of Erdos in the 1930s. Kleptsyn-Pollicott-Vytnova presented 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 more challenging fractal set to study 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. Pollicott-Sewell improved the best known upper bounds on its Hausdorff Dimension to 1.7407. This also has implications to improving the bounds on the constants in the asymptotic formula for counting solutions to the Markoff-Hurwicz equation, one of the problems originally proposed at the start of the project.
The progress made during this program 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 lead to new examples of cocompact Fuchsian groups for which the harmonic is singular, supporting the famous Kaimanovich-Le Prince conjecture.
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 more challenging fractal set to study 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. Pollicott-Sewell improved the best known upper bounds on its Hausdorff Dimension to 1.7407. This also has implications to improving the bounds on the constants in the asymptotic formula for counting solutions to the Markoff-Hurwicz equation, one of the problems originally proposed at the start of the project.
The progress made during this program 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 lead to new examples of cocompact Fuchsian groups for which the harmonic is singular, supporting the famous Kaimanovich-Le Prince conjecture.