# Dynamical zeta functions and resonances for infinite area surfaces

Lead Research Organisation:
University of Warwick

Department Name: Mathematics

### Abstract

This proposal deals with complex functions first introduced by the famous norwegian mathematician and Fields medalist Atle Selberg in 1956, and subsequently called Selberg zeta functions. These were originally associated to compact surfaces of constant negative curvature.

Their definition was by analogy with the famous Riemann zeta function, except that the role of the prime numbers is replaced by the lengths of closed geodesics on the surface. The striking fact is that in this setting the zeros lie on specific lines, which is very similar to the famous Riemann Hypothesis, both one of the problems from Hilbert's famous list of 23 problems and the Clay Institute's Millennium Problems.

However, by contrast, in the case of many examples of open surfaces, or infinite area surfaces, the zeros of the associated Selberg zeta functions are much more complicated. These individual zeros are often called "resonances" and play a role similar to that of the eigenvalues of the laplacian for the compact case, and are important geometric and dynamical invariants for the surfaces

With the development of better computational methods and computer hardware over recent years a much clearer picture of the patterns of these zeros has emerged in some interesting cases. Somewhat surprisingly, the plots of these zeros had strikingly beautiful patterns. They appear to lie on very delicately defined curves in shapes reminiscent of lace embroidery. These plots of the zeros have their simplest structures when the underlying surface has more symmetries.

This work will help to understand these patterns of zeta function zeros and the information that it gives on both the zeta function and the associated surface.

Their definition was by analogy with the famous Riemann zeta function, except that the role of the prime numbers is replaced by the lengths of closed geodesics on the surface. The striking fact is that in this setting the zeros lie on specific lines, which is very similar to the famous Riemann Hypothesis, both one of the problems from Hilbert's famous list of 23 problems and the Clay Institute's Millennium Problems.

However, by contrast, in the case of many examples of open surfaces, or infinite area surfaces, the zeros of the associated Selberg zeta functions are much more complicated. These individual zeros are often called "resonances" and play a role similar to that of the eigenvalues of the laplacian for the compact case, and are important geometric and dynamical invariants for the surfaces

With the development of better computational methods and computer hardware over recent years a much clearer picture of the patterns of these zeros has emerged in some interesting cases. Somewhat surprisingly, the plots of these zeros had strikingly beautiful patterns. They appear to lie on very delicately defined curves in shapes reminiscent of lace embroidery. These plots of the zeros have their simplest structures when the underlying surface has more symmetries.

This work will help to understand these patterns of zeta function zeros and the information that it gives on both the zeta function and the associated surface.

### Planned Impact

In the medium to long term, the work also has the potential to impact on other areas of the sciences central to the future of the UK.This is via the alternative dynamical interpretation of the resonances in terms of the rate of decay of the correlation function. In the present context the dynamics is for the geodesic flow on a hyperbolic surfaces, but this is a simplified for more complicated chaotic systems. This is usually called "sensitive dependence on initial conditions" and is important, for example, in weather modelling and geophysical studies, based as they are on ``dynamical systems out of equilibrium''.

Aside from the purely scientific aspects of this research, the various plots of the zeros form very beautiful images. This suggests to the trained mathematician there is a specific technical underlying structure. But to others this might be viewed as an illustration of how to visualize complex ideas from mathematics in a way which is aesthetically pleasing.

More specifically, these images could represent the basis of an outreach project to the wider population. This would give a way to popularize some analytical aspects of mathematics (ergodic theory, dynamical zeta functions, resonances, etc.) in the same that geometry has succeeded in reaching the popular imagination (e.g, the exhibition ``Brilliant Geometry'' held from 13 May - 4 2017 at the Summerhall Gallery, Edinburgh; the exhibition ``Brilliant Geometry'' held from 13 May - 4 2017 at the Summerhall Gallery, Edinburgh; the European Society for Mathematics and the Arts exhibition from 12-23 March 2018 in Paris V city hall; and the design of the Winton Gallery in the Science Museum in London).

It would be a simple matter to design a number of eye catching posters to promote mathematics that would appeal, in particular, to school children.

The training of a PDRA in this area, which is highly relevant to many other disciplines,

would help contribute to the UK science base, essential for the economic development of the UK.

Furthermore, the scientific activity associated with this proposal would have a highly beneficial effect on the education

of the PhD students associated to

this research area. This would further lead to additional

people trained

with important skills.

Aside from the purely scientific aspects of this research, the various plots of the zeros form very beautiful images. This suggests to the trained mathematician there is a specific technical underlying structure. But to others this might be viewed as an illustration of how to visualize complex ideas from mathematics in a way which is aesthetically pleasing.

More specifically, these images could represent the basis of an outreach project to the wider population. This would give a way to popularize some analytical aspects of mathematics (ergodic theory, dynamical zeta functions, resonances, etc.) in the same that geometry has succeeded in reaching the popular imagination (e.g, the exhibition ``Brilliant Geometry'' held from 13 May - 4 2017 at the Summerhall Gallery, Edinburgh; the exhibition ``Brilliant Geometry'' held from 13 May - 4 2017 at the Summerhall Gallery, Edinburgh; the European Society for Mathematics and the Arts exhibition from 12-23 March 2018 in Paris V city hall; and the design of the Winton Gallery in the Science Museum in London).

It would be a simple matter to design a number of eye catching posters to promote mathematics that would appeal, in particular, to school children.

The training of a PDRA in this area, which is highly relevant to many other disciplines,

would help contribute to the UK science base, essential for the economic development of the UK.

Furthermore, the scientific activity associated with this proposal would have a highly beneficial effect on the education

of the PhD students associated to

this research area. This would further lead to additional

people trained

with important skills.

## People |
## ORCID iD |

Mark Pollicott (Principal Investigator) |

Description | We have developed two new algorithms which are very useful in analyzing classical problems: (a) We have found a method to get lower bounds on the dimension of Bernoulli convolutions, a problem dating back over a century. These are a parameterized family of measures and we show that for all parameter values the size of the measure (the dimension) is at least 0.96 (ii) We have a technique to show that certain harmonic measures corresponding to random products of measures is singular. This supports a classical conjecture of Kaimanovich-Le Prince. |

Exploitation Route | Not realistically |

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