Recent advances in ergodic theory and dynamics

This event will be held in Loughborough University 13-14 July 2023.

The first day of the meeting is a one-day ergodic theory meeting, which is part of the LMS Scheme 3 grant between Birmingham, Bristol, Durham, Exeter, Loughborough, Manchester, Open, Queen Mary, St. Andrews, and Warwick.

Schedule, titles and abstracts are below. 

Programme

Thursday 13 July

Dougall 10:00->11:00
Break
Melbourne 11:30->12:30
Lunch
Dougall 14:30-> 15:30
Break
Melbourne 16:00-> 17:00

Friday 14 July

Alves 10:00->11:00
Break
Alves 11:30->12:30
Lunch
Slipantschuk 14:30-> 15:30
Break
Slipantschuk 16:00-> 17:00

Titles and Abstracts

Speaker: José F. Alves (Porto)

Title: Entropy formula for systems with inducing schemes

Abstract: We show that Pesin's entropy formula holds for SRB measures with finite entropy given by inducing schemes, and we give applications to some classes of dynamical systems with singular sets where the classical results of Ruelle and Pesin cannot be applied. We also present examples of systems with SRB measures given by inducing schemes for which Ruelle's inequality is not valid.

Speaker: Rhiannon Dougall (Durham)

Title: Proving a ratio limit theorem for random walks on groups using the variational principle.

Abstract: (This talk will be close the topic of group extensions.) There are strong results for certain random walks on groups when we have enough structure, such as the random walk being symmetric and the group having some good structure. We can manage without symmetry if the group is small enough e.g. abelian. We will be interested in the case of a non-degenerate (non-symmetric in general) random walk on an amenable group. We are able to obtain a ratio limit theorem, which says that the probability to return to g in n steps is asymptotically proportional to that of returning to some fixed origin; and we have an explicit description of the constant in terms of g.

We'll describe how this result follows from large deviations and the variational principle.

This is joint work with Richard Sharp.

Speaker: Ian Melbourne (Warwick)

Title: Decay of correlations and statistical limit laws for geodesic flows on nonpositively curved surfaces Abstract: We discuss decay of correlations and statistical limit laws for certain classes of nonpositively curved surfaces. For surfaces where negative curvature breaks down at one periodic orbit, we obtain sharp polynomial rates of mixing and the standard CLT, etc. When there is a degenerate cylinder, we obtain decay rate 1/t and the CLT with nonstandard normalisation \sqrt{t log t}.

The arguments involve building a Young tower and are somewhat technical, but we will focus on the more palatable aspects. In particular, relatively simple calculations using Clairaut coordinates readily yield the appropriate decay rates. This is joint work with Yuri Lima and Carlos Matheus.

Speaker: Julia Slipantschuk

Title: Distribution of Pollicott-Ruelle resonances for Anosov maps on the torus.

Abstract: The purpose of this lecture is to present a complete description of Pollicott-Ruelle resonances for a class of rational Anosov diffeomorphisms of the two-torus. This allows us to show that every homotopy class of two-dimensional Anosov diffeomorphisms contains maps with the sequence of resonances decaying stretched-exponentially. These results are obtained by analysing the corresponding Koopman operator on an anisotropic Hilbert space, seen as direct sum of Hardy-Hilbert spaces on log-conical Reinhardt domains.