# Thermodynamic formalism and flows on moduli space

Lead Research Organisation:
University of Warwick

Department Name: Mathematics

### Abstract

In the broadest sense, Ergodic theory is the branch of analysis which has developed most rapidly in the last century, and which has had many striking achievements, particularly in the past few decades. This is noticable, in particular, in terms of applications to number theory. Notable important highlights were Wolf prize winner Furstenberg's proof of Szemerdi's theorem on arithmetic progressions; Fields' medallist Margulis' proof of the Oppenheim conjecture and the Einsideler-Katok-Lindenstrauss (another Fields' medallist) contribution to the classical Littlewood conjecture. Many of these proofs use a particularly geometric viewpoint.

The general principle of applying ergodic theory to geometry is now both well established and fundamental. This is bourne out by the examples of the fundamental and classical Mostow rigidity theorem (which, of course, show that in higher dimensions the Moduli space is trivial and emphasizes the interest in surfaces) and the seminal work of Margulis on lattice point and closed orbit counting for negatively curved manifolds, and super-rigidity for Lie groups.

Historically, ergodic theory has its roots in theoretical physics and, in particular, statistical mechanics, and is generally concerned with the long term stochastic behaviour of deterministic dynamical systems. Moreover, one of the key methods of our analysis, thermodynamic formalism, is a particularly fruitful branch of ergodic theory, with strong connections to statistical mechanics.

The underlying theme in the proposed programme of research is to study the application of ergodic theory and thermodynamic formalism in order to gain a better insight into metrics on Riemann surfaces and their geometry. The connection between ergodic theory and geometry in our proposal comes from the classical viewpoint of studying the dynamics of the geodesic flow. However, considering the flow on moduli spaces, instead of classical Riemannian manifolds, leads to more challenging technical problems.

The programme of proposed research is divided into four key areas. Firstly, studying the dynamics of the Weil-Petersson geodesic flow. This is an area in which there has been considerable progress in the past couple of years, and we have made particular contributions to this. In particular, the Weil-Petersson metric is one which has negative curvature(s) and thus is amenable to many classical techniques in ergodic theory, by analogy with the theory of scattering billiards (notwithstanding some considerable technical problems). Moreover, the subtle interplay between the dynamics and the geometry gives a greater insight into both aspects.

A second area is the study of the Teichmuller geodesic flow. This is a topic which has received considerable attention from leading experts in mathematics (e.g., Fields' medallists McMullen and Kontsevich). However, statistical properties of such flows can be studied using techniques from thermodynamic formalism since the flows can be conveniently realised as suspension flows over countable branch expanding maps.

A third area of investigation relates to the determinant of the laplacian, whose origins are related to mathematical physics. This is a function defined on the space of function whose behaviour is particularly mysterious. Using techniques we have developed over several years we will determine interesting values and points associated to the function. In particular, we expect to resolve a long standing problem of Sarnak in this area.

The final area of study is at the level of the surfaces themselves. We want to give a new interpretation for the canonical invariants discovered by Forni-Flaminio in the special case of surfaces of constant curvature and to extend the theory to more general surfaces. The basic approach uses recent work of ours on the dynamical zeta function. This offers the possibility of opening up a whole new field of research.

The general principle of applying ergodic theory to geometry is now both well established and fundamental. This is bourne out by the examples of the fundamental and classical Mostow rigidity theorem (which, of course, show that in higher dimensions the Moduli space is trivial and emphasizes the interest in surfaces) and the seminal work of Margulis on lattice point and closed orbit counting for negatively curved manifolds, and super-rigidity for Lie groups.

Historically, ergodic theory has its roots in theoretical physics and, in particular, statistical mechanics, and is generally concerned with the long term stochastic behaviour of deterministic dynamical systems. Moreover, one of the key methods of our analysis, thermodynamic formalism, is a particularly fruitful branch of ergodic theory, with strong connections to statistical mechanics.

The underlying theme in the proposed programme of research is to study the application of ergodic theory and thermodynamic formalism in order to gain a better insight into metrics on Riemann surfaces and their geometry. The connection between ergodic theory and geometry in our proposal comes from the classical viewpoint of studying the dynamics of the geodesic flow. However, considering the flow on moduli spaces, instead of classical Riemannian manifolds, leads to more challenging technical problems.

The programme of proposed research is divided into four key areas. Firstly, studying the dynamics of the Weil-Petersson geodesic flow. This is an area in which there has been considerable progress in the past couple of years, and we have made particular contributions to this. In particular, the Weil-Petersson metric is one which has negative curvature(s) and thus is amenable to many classical techniques in ergodic theory, by analogy with the theory of scattering billiards (notwithstanding some considerable technical problems). Moreover, the subtle interplay between the dynamics and the geometry gives a greater insight into both aspects.

A second area is the study of the Teichmuller geodesic flow. This is a topic which has received considerable attention from leading experts in mathematics (e.g., Fields' medallists McMullen and Kontsevich). However, statistical properties of such flows can be studied using techniques from thermodynamic formalism since the flows can be conveniently realised as suspension flows over countable branch expanding maps.

A third area of investigation relates to the determinant of the laplacian, whose origins are related to mathematical physics. This is a function defined on the space of function whose behaviour is particularly mysterious. Using techniques we have developed over several years we will determine interesting values and points associated to the function. In particular, we expect to resolve a long standing problem of Sarnak in this area.

The final area of study is at the level of the surfaces themselves. We want to give a new interpretation for the canonical invariants discovered by Forni-Flaminio in the special case of surfaces of constant curvature and to extend the theory to more general surfaces. The basic approach uses recent work of ours on the dynamical zeta function. This offers the possibility of opening up a whole new field of research.

### Planned Impact

The nature of pure mathematical research is that it is traditionally somewhat removed from immediate commercial and government usage. However, we might expect that the quality and cutting edge nature of this proposal will ultimately have far wider implications which are harder to predict at the moment. Moreover, we would expect there to be a significant impact, but only over the longer term.

Although the primary beneficiaries of this project will be researchers working in mathematics, in general,

and pure dynamical systems, in particular, we envisage that this project will eventually have impact in a broader setting.

Potential applications of this research could be in engineering (Hidden Markov Models, resonance phenomena, growth models, data compression), economics (modelling of stock market, overlapping generation models, etc.), biology (tumour growth) and all areas where thermodynamic formalism and ergodic theory has made important contributions in recent years. However, this project is not aiming to explore directly these applications, but as a more immediate consequence expect that it will have less quantifiable, but equally important, consequences. These would include:

(a) To strengthen the knowledge base within the field of dynamical systems, ergodic theory and geometry (and more generally mathematics) in the UK, maintaining our leadership in this field;

(b) To promote a positive and strong image of British mathematics abroad;

(c) To place the UK at the cutting edge of research in a particularly active and vibrant area of mathematics; and

(d) To help encourage and train a new generation of young mathematicians.

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

Within ergodic theory, majorization and optimization has some of its roots in practical problems in transportation theory. For example, the classical motivating example relates to the location of factories and mines and the most efficient transportation of ore. In this context there is a cost function corresponding to transportation costs, which one wants to minimize. The theoretical problems in this proposal are part of an idealized version of these real world problems (where given functions have their integrals minimized with respect to suitable generalized measures, i.e. distributions). There are also interesting connections between the mathematical problems considered and Economics. Since a great deal of economics is concerned with the maximisation of the expectation of convex or concave utility functions, this makes majorization a topic of considerable interest to economists.

One way in which we hope to ensure the widest impact beyond the immediate circle of experts, is by participating in a broad range of workshops and attending talks in different areas. Warwick is a particularly appropriate institution for an RA to be based, giving the opportunity to interact with visiting researchers and experts in very diverse fields, and thus provided the opportunity to meet and discuss scientific research with the broadest possible spectrum of scientists. We have every expectation that such discussions would be the basis for new research directions and interactions.

Finally, dynamical systems and geometry, the two main themes of this proposal, are both subject areas within mathematics which are probably easier to explain to a wider audience, and thus have a greater impact, than many other areas.

Although the primary beneficiaries of this project will be researchers working in mathematics, in general,

and pure dynamical systems, in particular, we envisage that this project will eventually have impact in a broader setting.

Potential applications of this research could be in engineering (Hidden Markov Models, resonance phenomena, growth models, data compression), economics (modelling of stock market, overlapping generation models, etc.), biology (tumour growth) and all areas where thermodynamic formalism and ergodic theory has made important contributions in recent years. However, this project is not aiming to explore directly these applications, but as a more immediate consequence expect that it will have less quantifiable, but equally important, consequences. These would include:

(a) To strengthen the knowledge base within the field of dynamical systems, ergodic theory and geometry (and more generally mathematics) in the UK, maintaining our leadership in this field;

(b) To promote a positive and strong image of British mathematics abroad;

(c) To place the UK at the cutting edge of research in a particularly active and vibrant area of mathematics; and

(d) To help encourage and train a new generation of young mathematicians.

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

Within ergodic theory, majorization and optimization has some of its roots in practical problems in transportation theory. For example, the classical motivating example relates to the location of factories and mines and the most efficient transportation of ore. In this context there is a cost function corresponding to transportation costs, which one wants to minimize. The theoretical problems in this proposal are part of an idealized version of these real world problems (where given functions have their integrals minimized with respect to suitable generalized measures, i.e. distributions). There are also interesting connections between the mathematical problems considered and Economics. Since a great deal of economics is concerned with the maximisation of the expectation of convex or concave utility functions, this makes majorization a topic of considerable interest to economists.

One way in which we hope to ensure the widest impact beyond the immediate circle of experts, is by participating in a broad range of workshops and attending talks in different areas. Warwick is a particularly appropriate institution for an RA to be based, giving the opportunity to interact with visiting researchers and experts in very diverse fields, and thus provided the opportunity to meet and discuss scientific research with the broadest possible spectrum of scientists. We have every expectation that such discussions would be the basis for new research directions and interactions.

Finally, dynamical systems and geometry, the two main themes of this proposal, are both subject areas within mathematics which are probably easier to explain to a wider audience, and thus have a greater impact, than many other areas.

## People |
## ORCID iD |

Mark Pollicott (Principal Investigator) |

### Publications

Aimino R
(2021)

*Thermodynamic Formalism - CIRM Jean-Morlet Chair, Fall 2019*
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
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)

*Equidistribution results for sequences of polynomials*in Journal of Number Theory
Bárány B
(2017)

*Ledrappier-Young formula and exact dimensionality of self-affine measures*in Advances in Mathematics
BÁRÁNY B
(2015)

*On the Ledrappier-Young formula for self-affine measures*in Mathematical Proceedings of the Cambridge Philosophical Society
Bárány B
(2018)

*Pointwise regularity of parameterized affine zipper fractal curves*in Nonlinearity
Bárány B
(2016)

*On the dimension of self-affine sets and measures with overlaps*in Proceedings of the American Mathematical Society
Colognese P
(2020)

*Dynamics: Topology and Numbers*Description | There were three different RAs on this grant, all of whom went to permanent positions afterwards. Thus from the point of training it was very successful. On the scientific level, considerable strides were taken forward in dimension theory. |

Exploitation Route | The impact on the study of dimension theory seems to be quite wide. Many of the ideas and techniques pioneered during this grant have been carried forward by others. |

Sectors | Digital/Communication/Information Technologies (including Software),Electronics,Financial Services, and Management Consultancy,Security and Diplomacy,Transport |

URL | http://personalpages.manchester.ac.uk/staff/jonathan.fraser/papers.html |