Scientific Committee
Comité scientifique
Carlangelo Liverani (University of Rome)
Françoise Pène (Université de Brest)
Yakov Pesin (Pennsylvania State University)
Omri Sarig (Weizmann Institute of Science)
Barbara Schapira (Université de Montpellier)
Mariusz Urbanski (University of North Texas)
Organizing Commitee
Comité d’organisation
Jérôme Buzzi (CNRS, Université Paris-Saclay )
Boris Hasselblatt (Tufts University)
Tamara Kucherenko (The City University New York)
Daniel J. Thompson (The Ohio State University)
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The thermodynamic formalism was first developed in 1970s for uniformly hyperbolic dynamics. Its initial motivation was the building of Sinaï-Ruelle-Bowen measures which are invariant measures with natural geometric properties. It has been an extremely successful tool to study important dynamical systems such as geodesic flows on compact manifold with negative curvature. Though the generalization of thermodynamical formalism beyond the uniformly hyperbolic setting dates back to the deep work of Ledrappier and Young in the 1980s, the development of these ideas has now reached a critical stage. A number of groundbreaking results have been obtained for surface diffeomorphisms and geodesic flows in nonpositive curvature, as well as non-compact systems. The techniques are very diverse: symbolic dynamics, geometric measure theory, combinatorial estimates, spectral techniques, etc. This conference will focus on several rapidly emerging directions with the main objective of establishing connections between various approaches.
Le formalisme thermodynamique a été développé pour la première fois dans les années 1970 pour la dynamique hyperbolique uniforme. Sa motivation initiale était la construction des mesures de Sinaï-Ruelle-Bowen, qui sont des mesures invariantes avec des propriétés géométriques naturelles. Cela a été un outil extrêmement réussi pour étudier des systèmes dynamiques importants, tels que les flux géodésiques sur des variétés compactes à courbure négative. Bien que la généralisation du formalisme thermodynamique au-delà du cadre hyperbolique uniforme remonte aux travaux approfondis de Ledrappier et Young dans les années 1980, le développement de ces idées a maintenant atteint un stade critique. Un certain nombre de résultats révolutionnaires ont été obtenus pour les difféomorphismes de surfaces et les flux géodésiques dans des espaces `a courbure non positive, ainsi que pour des systèmes non compacts. Les techniques sont très diverses : dynamique symbolique, théorie géométrique de la mesure, estimations combinatoires, techniques spectrales, etc. Cette conférence se concentrera sur plusieurs directions émergentes rapidement, avec pour principal objectif d’établir des connexions entre les différentes approches.
MINCOURSES
Vaughn Climenhaga (University of Houston) - Keeping things bounded without compactness and continuity
Abstract. 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.
Omri Sarig (Weizmann Institute of Science) - Irregularities in Uniform Distribution
Abstract. Suppose a is an irrational number. Weyl proved that na mod 1 is uniformly distributed on the unit interval. i.e., the frequency of visits of na mod 1 to a subinterval of [0,1] tends to the length of the subinterval. It has long been known that the error term in this limit theorem can exhibit strong bias, reflecting an “irregularity” in uniform distribution in the higher-order term. This bias depends on the fine number theoretic properties of a. For example, the square roots of two and three do not behave the same way (!) I will describe some recent joint work with Dmitry Dolgopyat on the equidistribution of the error term. There are amusing connections to the geometry of translation surfaces with infinite genus, and to local limit theorems of inhomogeneous Markov chains.
(Joint with Dmitry Dolgopyat)
SPEAKERS
Martin Bridgeman (Boston College)
Sylvain Crovisier (CNRS, Université Paris-Saclay)
Mark Demers (Fairfield University)
Katrin Gelfert (Federal University of Rio de Janeiro)
Cecilia Gonzalez-Tokman (University of Queensland)
Nyima Kao (The George Washington University)
Renaud Leplaideur (Université de la Nouvelle-Calédonie)
Yuri Lima (University of São Paulo)
Carlangelo Liverani (University of Roma)
Juan-Carlos Mongez (University of São Paulo)
Davi Obata (Brigham Young University)
Françoise Pène (Université de Brest)
Yakov Pesin (Pennsylvania State University)
Anthony Quas (University of Victoria)
Jana Rodriguez-Hertz (University of the Republic)
Barbara Schapira (Université de Montpellier)
Richard Sharp (Universoity of Warwick)
Ali Tahzibi (University of São Paulo)
Xueting Tian (Fudan University)
Masato Tsujii (Kyushu University)
Mariusz Urbański (University of North Texas)
Anibal Velozo (Catholic University of Chile)
Mary He Yan (University of Oklahoma)
SPONSORS
