Organisers

Wael Bahsoun, Loughborough University

Viviane Baladi, CNRS and Sorbonne University

This school will provide courses from experts on several aspects of the spectrum of transfer operators. Special attention will be provided to the theory of anisotropic spaces, and to applications in geometrical or physical settings.

The target audience includes PhD students and young postdocs.

► Lecturer: Mark Demers
Course structure: 6 lectures, 50 minutes
Title: Anisotropic Banach Spaces and Thermodynamic Formalism for Dispersing Billiard Maps.

Abstract: Mathematical billiards with dispersing boundaries comprise a physically interesting class of hyperbolic systems with singularities. Starting from some toy models of hyperbolic maps, we will build up a set of tools (Banach spaces and transfer operators) that we shall use to analyze the statistical properties of these systems. This framework will allow us to present recent progress in the theory of equilibrium states and topological pressure in the context of dispersing billiards, including the existence and uniqueness of a measure of maximal entropy.

► Lecturer: Semyon Dyatlov
Tutor: Malo Jézéquel
Course structure: 4 lectures, 55 minutes + 2 tutorials, 40 minutes
Title: Fractal uncertainty principle and spectral gaps.

Abstract: This minicourse describes the spectral gap problem for convex co-compact hyperbolic surfaces and the approach to it which uses the fractal uncertainty principle, a recently developed tool in harmonic analysis. Roughly speaking, the combinatorial and harmonic-analytic properties of the limit set of the surface make it possible to prove statements such as exponential remainders in the Prime Orbit Theorem or exponential local energy decay for the wave equation. Here are some of the topics that will be covered:

• Convex co-compact hyperbolic surfaces and their Schottky representations. Schottky limit sets and Patterson-Sullivan measure. Transfer operators.
• The spectral gap problem, a brief overview of its applications, and known results.
• Fractal uncertainty principles, a brief overview of the three known approaches.
• A proof that fractal uncertainty principle gives a spectral gap using transfer operators (following Dyatlov-Zworski 2017).
• A proof of the fractal uncertainty principle for the special case of arithmetic Cantor sets, done using each of the three approaches.

► Lecturer: Sébastien Gouézel
Course structure: 6 lectures, 50 minutes
Title: Ruelle resonances for geodesic flow on noncompact manifolds.

Abstract: Ruelle resonances are complex numbers describing the fine asymptotic properties of the correlations of smooth functions under a given flow. We will survey different situations in which one can make sense of this notion, starting with the simplest situation of expanding semi flows, and adding progressively technical tools, to be able in the end to cover more complicated flows of geometric origin, such as the geodesic flow on compact or noncompact negatively curved manifolds.

► Lecturer: Masato Tsujii
Course structure: 6 lectures, 50 minutes
Title: 3-dimensional Anosov flows.

Abstract: In this mini-course, I am going to discuss mainly about local geometric structure of the(strong) stable and unstable lamination of smooth Anosov flows, particularly in the lowest dimensional case, that is, in dimension 3. We will also explain how that structure leads to (exponential) mixing property of the flow, using a simplified model of the U(1)-extensions of Anosov diffeomorphisms. The content will be based on my joint work with Zhiyuan Zhang.

This project has also received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 787304).