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
Jenkinson O
(2018)
Rigorous effective bounds on the Hausdorff dimension of continued fraction Cantor sets: A hundred decimal digits for the dimension of E2
in Advances in Mathematics
Pollicott M
(2017)
Amenable covers for surfaces and growth of closed geodesics
in Advances in Mathematics
Bárány B
(2017)
Ledrappier-Young formula and exact dimensionality of self-affine measures
in Advances in Mathematics
Kleptsyn V
(2022)
Uniform lower bounds on the dimension of Bernoulli convolutions
in Advances in Mathematics
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
Pollicott M
(2018)
Hyperbolic systems, zeta functions and other friends
in Banach Center Publications
Paulin F
(2016)
Logarithm Laws for Equilibrium States in Negative Curvature
in Communications in Mathematical Physics
Pollicott M
(2023)
Accurate Bounds on Lyapunov Exponents for Expanding Maps of the Interval.
in Communications in mathematical physics
Jenkinson O
(2021)
How Many Inflections are There in the Lyapunov Spectrum?
in Communications in Mathematical Physics
Baker S
(2017)
Root Sets of Polynomials and Power Series with Finite Choices of Coefficients
in Computational Methods and Function Theory
Kagiso D
(2015)
Computing multifractal spectra
in Dynamical Systems
JENKINSON O
(2017)
Joint spectral radius, Sturmian measures and the finiteness conjecture
in Ergodic Theory and Dynamical Systems
JOHANSSON A
(2017)
Phase transitions in long-range Ising models and an optimal condition for factors of -measures
in Ergodic Theory and Dynamical Systems
BAKER S
(2020)
Maximizing Bernoulli measures and dimension gaps for countable branched systems
in Ergodic Theory and Dynamical Systems
Pollicott M
(2018)
Zeros of the Selberg zeta function for symmetric infinite area hyperbolic surfaces
in Geometriae Dedicata
Pollicott M
(2017)
Critical points for the Hausdorff dimension of pairs of pants
in Groups, Geometry, and Dynamics
Baker S
(2020)
Two bifurcation sets arising from the beta transformation with a hole at 0
in Indagationes Mathematicae
Pollicott M
(2023)
An infinite interval version of the a-Kakutani equidistribution problem
in Israel Journal of Mathematics
Johansson A
(2017)
Ergodic theory of Kusuoka measures
in Journal of Fractal Geometry
Fraser J
(2017)
Uniform scaling limits for ergodic measures
in Journal of Fractal Geometry
Allaart P
(2019)
Bifurcation sets arising from non-integer base expansions
in Journal of Fractal Geometry
Pollicott M
(2021)
Fourier multipliers and transfer operators
in Journal of Fractal Geometry
Baker S
(2020)
Equidistribution results for sequences of polynomials
in Journal of Number Theory
Jenkinson O
(2017)
Rigorous Computation of Diffusion Coefficients for Expanding Maps
in Journal of Statistical Physics
Pollicott M
(2017)
A Nonlinear Transfer Operator Theorem.
in Journal of statistical physics
Pollicott M
(2018)
Apollonius circle counting.
in LMS Newsletter
Pollicott M
(2023)
Sierpinski Fractals and the Dimension of Their Laplacian Spectrum
in Mathematical and Computational Applications
BAKER S
(2018)
Numbers with simply normal ß-expansions
in Mathematical Proceedings of the Cambridge Philosophical Society
FRASER J
(2015)
Micromeasure distributions and applications for conformally generated fractals
in Mathematical Proceedings of the Cambridge Philosophical Society
Cipriano I
(2018)
Stationary measures associated to analytic iterated function schemes
in Mathematische Nachrichten
Pollicott M
(2015)
Weil-Petersson metrics, Manhattan curves and Hausdorff dimension
in Mathematische Zeitschrift
Pollicott M
(2024)
Zeta functions in higher Teichmüller theory
in Mathematische Zeitschrift
Baker S
(2023)
Overlapping Iterated Function Systems from the Perspective of Metric Number Theory
in Memoirs of the American Mathematical Society
Baker S
(2019)
Exceptional digit frequencies and expansions in non-integer bases
in Monatshefte für Mathematik
Pollicott M
(2021)
The Schottky-Klein prime function and counting functions for Fenchel double crosses
in Monatshefte für Mathematik
Bárány B
(2018)
Pointwise regularity of parameterized affine zipper fractal curves
in Nonlinearity
Pollicott M
(2016)
Linear response and periodic points
in Nonlinearity
Pollicott M
(2022)
Explicit examples of resonances for Anosov maps of the torus
in Nonlinearity
Pollicott M
(2021)
Effective estimates of Lyapunov exponents for random products of positive matrices
in Nonlinearity
Galatolo S
(2017)
Controlling the statistical properties of expanding maps
in Nonlinearity
Pollicott Mark
(2017)
Open Conformal Systems and Perturbations of Transfer Operators Preface
in OPEN CONFORMAL SYSTEMS AND PERTURBATIONS OF TRANSFER OPERATORS
Pollicott M
(2020)
Exact dimensional for Bernoulli measures and the Gauss map
in Proceedings of the American Mathematical Society
Pollicott M
(2015)
Analyticity of dimensions for hyperbolic surface diffeomorphisms
in Proceedings of the American Mathematical Society
Pollicott M
(2017)
A note on the shrinking sector problem for surfaces of variable negative curvature
in Proceedings of the Steklov Institute of Mathematics
Alcaraz Barrera R
(2018)
Entropy, topological transitivity, and dimensional properties of unique -expansions
in Transactions of the American Mathematical Society
Pollicott M
(2015)
Fractal Geometry and Stochastics V
Paulin F
(2015)
Equilibrium states in negative curvature
Pollicott M
(2017)
Open Conformal Systems and Perturbations of Transfer Operators
Description | 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 |