Keeping things bounded without compactness and continuity - Lecture 2 | Vidéo

Collection New directions in thermodynamic formalism / Nouvelles orientations pour le formalisme thermodynamique

Organisateur(s) Buzzi, Jérôme ; Hasselblatt, Boris ; Kucherenko, Tamara ; Thompson, Daniel J.

Date(s) 22/06/2026 - 26/06/2026

URL associée https://conferences.cirm-math.fr/3493.html

Keeping things bounded without compactness and continuity - Lecture 2

Thermodynamic formalism involves many quantities that grow or decay exponentially fast. Comparing quantities with the same exponential rate results in ratios that are permitted to grow or decay sub-exponentially. A recurring theme is that when the underlying system is "sufficiently hyperbolic", many of these ratios turn out to be bounded away from 0 and infinity. The resulting uniform ratio bounds are central to the study of the measure of maximal entropy and other equilibrium measures. In these lectures, I will survey some of these bounds, their proofs, and their consequences for existence, uniqueness, and other results in thermodynamic formalism. In particular, I will describe recent work that extends this story beyond the classical setting of continuous systems on compact spaces. This includes geodesic flows on non-compact negatively curved manifolds under a strong positive recurrence condition, using a version of Bowen's specification property, and also includes Sinai billiard maps using growth-fragmentation lemmas and a Hausdorff measure construction.