# Ergodic Theory Network

This is the webpage for a network of collaborative meetings in ergodic theory, supported by a London Mathematical Society Scheme 3 grant. The universities in the network are Birmingham, Bristol, Exeter, Loughborough, Manchester, Queen Mary, St Andrews, Surrey and Warwick.

Next meeting: **Wednesday 17th April 2019** at the Loughborough University, Brockington Building, Room U020

2:00pm: **Henna Koivusalo** (Vienna)

Title: *Mass transference principle: From balls to arbitrary shapes*

Abstract: The mass transference principle, proved by Beresnevich and Velani in 2006, gives a lower bound for the Hausdorff dimension for a limsup sets of a sequence of balls. Heuristically speaking, it says the following: if a sequence of balls gives a limsup set of full measure, then the limsup set of balls with the same centres as the orginal balls, but radii shrunk in a controlled way, cannot have Hausdorff dimension too low. This powerful result has found applications in number theory and geometric measure theory. I present a version of the mass transference principle for the case where the shrunk balls are replaced by sets of arbitrary shape. The work is joint with Michal Rams.

3:00pm: **Anke Pohl** (Bremen)

Title: *Automorphic functions twisted with non-expanding cusp monodromies, and dynamics*

Abstract: Automorphic functions play an important role in several subareas of mathematics and mathematical physics. The correspondence principle between quantum und classical mechanics suggests that automorphic functions are closely related to geometric and dynamical entities of the underlying locally symmetric spaces. Despite intensive research, the extent of this relation is not yet fully understood. A seminal and very influencial result in this direction has been provided by Selberg, showing that for any hyperbolic surface X of finite area, the Selberg zeta function (a generating function for the geodesic length spectrum of X) encodes the spectral parameters of the untwisted automorphic forms for X among its zeros. Subsequently, this type of relation was generalized to automorphic forms twisted by unitary representations, to resonances and to spaces of infinite area by various researchers. Recently, a deeper relation could be established by means of transfer operator techniques. For the 3 so-called Maass cusp forms, these methods allow us to provide a purely dynamical characterization of these automorphic functions themselves, not only of their spectral parameters. The structure of this approach indicates that an extension to automorphic forms with well-behaved, also non-unitary twists should be expected. While such a generalization seems to be a long-term goal, we could already show first steps in this direction on the level of the Selberg zeta functions. After surveying the transfer operator techniques, we discuss the current state of art regarding dynamical approaches towards automorphic forms with non-unitary twists.

4:30pm **Mark Pollicott** (Warwick)

Title: *Volume entropy and equidistribution on translation surfaces*

Abstract: A translation surface can be viewed as a polygon in the plane with paired sides which are parallel and of equal length. This gives a flat metric, aside from the singularities coming from the corners of the polygon. To generalize A.K.Manningâ€™s beautiful definition of (volume) entropy we can consider a ball whose radii are locally distance minimizing geodesics. The presence of singularities leads to it having exponential volume growth rate (in a suitable sense) and so an interesting notion of positive entropy. We show that this is usually an asymptotic and consider related distributions. This is joint work with Paul Colognese.

This is part of a school on smooth ergodic theory and partially hyperbolic systems during 15-18 April 2019, organised by Wael Bahsoun. See here for details.

Future meetings:

**Tuesday 25th June 2019** at the University of Birmingham (School of Mathematics, Lecture Theatre C)

Speakers: Christian Bick (Exeter), Alex Clark (QMUL), Piotr Oprocha (AGH University of Science and Technology)