# Applications of ergodic theory to geometry: Dynamical Zeta Functions and their applications

Lead Research Organisation:
University of Warwick

Department Name: Mathematics

### Abstract

The proposed research lies at the interface of Ergodic Theory and Dynamical Systems, geometry, number theory, partial differential operators and mathematical physics. Central to this research programme are the the application of ideas from smooth ergodic theory to problems in different areas of mathematics. As such it is a highly intra-disciplinary research program. It also seems very timely, since there has been an explosion of activity in these areas in the last year which has attracted widespread attention. The proposed research is at the cutting edge of this development. In particular, the basis for this project rests on four important inter-related strands in applications of ergodic theory and dynamical systems to other areas: zeta functions and Poincare series (with their connections to number theory and geometry); Decay of correlations and resonances (with applications to the physical sciences); Numerical algorithms (with applications to both Pure and Applied Mathematics); and Teichmuller theory and Weil-Petersson metrics (at the boundary of ergodic theory, analysis and geometry).

The study of geometric zeta functions for closed geodesics on negatively curved manifolds was initiated by Fields Medallist A. Selberg in the 1950s (following his earlier work on number theory). Selberg studied the case of constant curvature manifolds, using trace formulae and ideas from representation theory which do not generalise. However, recent work of Giulietti, Liverani and myself used a completely different viewpoint involving ideas in ergodic theory to extend the zeta function for negatively curved manifolds (and even more generally smooth Anosov flows, generalizing the geodesic flow). This provides the starting point for our proposed research on zeta functions, providing both a springboard to a whole host of significant applications and providing the scientific framework via the new ideas and techniques it initiated.

The study of geometric zeta functions for closed geodesics on negatively curved manifolds was initiated by Fields Medallist A. Selberg in the 1950s (following his earlier work on number theory). Selberg studied the case of constant curvature manifolds, using trace formulae and ideas from representation theory which do not generalise. However, recent work of Giulietti, Liverani and myself used a completely different viewpoint involving ideas in ergodic theory to extend the zeta function for negatively curved manifolds (and even more generally smooth Anosov flows, generalizing the geodesic flow). This provides the starting point for our proposed research on zeta functions, providing both a springboard to a whole host of significant applications and providing the scientific framework via the new ideas and techniques it initiated.

### Planned Impact

This project will support UK universities' drive to be among the prime generators of top-quality research. Of course this is not only a matter of prestige, as no country in the world can expect to maintain long term economic success without a strong scientific base and strong universities and, in particular, without a strong mathematical foundation.

The fields of Ergodic Theory, Dynamical Systems, geometry, number theory, analysis, partial differential operators and mathematical physics play a fundamental part in our understanding of the world and have had proven impact on every branch of science, engineering and social sciences. Probability in particular is used everywhere: medicine, the weather, demographics, economics, the stock market, communications, cell biology, particle physics, climate change, etc. This project will bolster the scientific base of this subject and ensure continued impact.

This project is based on subject areas within mathematics which are perhaps easier to popularise than many other areas. During the term of this Fellowship, I plan to liaise with the planned public lecture series run by the Mathematics and Statistics Departments at Warwick, by inviting top-quality internationally renowned speakers to give talks in this area aimed at the general public, to raise public awareness and stimulate interest in this important and exciting area of mathematics. I will also make myself available to the media for interviews through the appropriate channels at the University of Warwick.

The fields of Ergodic Theory, Dynamical Systems, geometry, number theory, analysis, partial differential operators and mathematical physics play a fundamental part in our understanding of the world and have had proven impact on every branch of science, engineering and social sciences. Probability in particular is used everywhere: medicine, the weather, demographics, economics, the stock market, communications, cell biology, particle physics, climate change, etc. This project will bolster the scientific base of this subject and ensure continued impact.

This project is based on subject areas within mathematics which are perhaps easier to popularise than many other areas. During the term of this Fellowship, I plan to liaise with the planned public lecture series run by the Mathematics and Statistics Departments at Warwick, by inviting top-quality internationally renowned speakers to give talks in this area aimed at the general public, to raise public awareness and stimulate interest in this important and exciting area of mathematics. I will also make myself available to the media for interviews through the appropriate channels at the University of Warwick.

## People |
## ORCID iD |

Mark Pollicott (Principal Investigator / Fellow) |

### Publications

Alcaraz Barrera R
(2018)

*Entropy, topological transitivity, and dimensional properties of unique $q$-expansions*in Transactions of the American Mathematical Society
Baker S
(2020)

*Equidistribution results for sequences of polynomials*in Journal of Number Theory
Baker S
(2020)

*Two bifurcation sets arising from the beta transformation with a hole at 0*in Indagationes Mathematicae
Baker S
(2019)

*Exceptional digit frequencies and expansions in non-integer bases*in Monatshefte für Mathematik
Baker S
(2020)

*On the complexity of the set of codings for self-similar sets and a variation on the construction of Champernowne*in Advances in Mathematics
Cipriano I
(2018)

*Stationary measures associated to analytic iterated function schemes*in Mathematische Nachrichten*Stationary measures associated to analytic iterated function schemes*in Mathematische Nachrichten

FRASER J
(2015)

*Micromeasure distributions and applications for conformally generated fractals*in Mathematical Proceedings of the Cambridge Philosophical Society
Fraser J
(2017)

*Uniform scaling limits for ergodic measures*in Journal of Fractal Geometry
Galatolo S
(2017)

*Controlling the statistical properties of expanding maps*in NonlinearityDescription | I have completed my goals for the first year of my fellowship. This includes developing a framework for extending the generalized dynamical zeta function. I have also developed a framework which will show exponential decay of frame flows close to constant curvature. |

Exploitation Route | There are many researchers who are now using techniques I have developed, |

Sectors | Digital/Communication/Information Technologies (including Software),Education,Transport |

URL | http://homepages.warwick.ac.uk/~masdbl/preprints.html |