Dynamics Beyond Uniform Hyperbolicity
Dynamique au-delà de l’hyperbolicité uniforme

13 – 24 May 2019

Scientific Committee
Comité scientifique

Keith Burns (Northwestern University)
Lorenzo Diaz (Pontifical Catholic University of Rio de Janeiro)
Marcelo Viana (IMPA)
Lan Wen (Peking University)
Amie Wilkinson (University of Chicago)

Organizing Committee
Comité d’organisation

Christian Bonatti (CNRS /Université de Bourgogne)
Jérôme Buzzi (CNRS / Université Paris-Sud)
Sylvain Crovisier (CNRS / Université Paris-Sud)
Shaobo Gan (Peking University)
Maria José Pacifico (
Federal University of Rio de Janeiro)


In the last 20 years, the body of research on smooth dynamical systems beyond uniform hyperbolicity has grown rapidly and substantially. Among recent achievements, important progresses have been obtained towards the conjectures of Pugh and Shub, and of Palis. We plan to hold a meeting (school and conference) at the CIRM (Marseilles) over two weeks in May or June 2019 focusing on the global qualitative study (topological and ergodic) of differentiable dynamical systems. Topics include : non-uniform, partial and singular hyperbolicity, Lyapunov exponents, statistical properties, stability and bifurcations, entropy and symbolic extensions, connections with foliations and group actions. . .

A main focus is on disseminating new results, attracting promising junior researchers and graduate students and training researchers. This meeting will also federate and stimulate the research of several groups of mathematicians working in several countries (France, Brazil, China, USA,. . . ) It belongs to a series of successful conferences organized since 2001 : they have been instrumental for collaborations among the participants, furthering the subject and leading to new connections between its different aspects.

We would like to organize this meeting over two weeks, as it was the case previously. This would allow a three-pronged program :
– a school : half of the talks for mini-courses, designed to expound on recent significant advances in smooth dynamics,
– a conference : half of the talks for research presentations,
​– informal time for discussions and improvised seminars.

The 2nd Zhang Zhifen Prize in Mathematics will be awarded on Monday, May 13th.

Les recherches de ces 20 dernières années ont permis un important développement dans le domaine de la dynamique différentiable, au-delà du cadre uniformément hyperbolique. Parmi les résultats récents, des progrès importants ont été réalisés dans la direction des conjectures de Pugh-Shub et de Palis. Nous avons le projet d’organiser une rencontre (école et conférence) au CIRM (Marseille) sur deux semaines en mai ou juin 2019, portant sur l’étude qualitative globale (topologique et ergodique) des systèmes dynamiques différentiables. Les thèmes abordés seront : l’hyperbolicité non-uniforme, partielle et singulière, les exposants de Lyapunov, les propriétés statistiques, la stabilité et les bifurcations, l’entropie et les extensions symboliques, les interactions avec les feuilletages et les actions de groupes,…

L’un des objectifs principaux sera de disséminer de nouveaux résultats, d’attirer des étudiants et des jeunes chercheurs, de former les collègues. Cette rencontre permettra également de fédérer et de stimuler les recherches de plusieurs groupes de mathématiciens répartis dans différents pays (France, Brésil, Chiné, États-Unis,. . . ) Elle s’inscrira dans la continuité de conférences organisées depuis 2001 : elles ont été essentielles pour la conduite de collaborations entre les participants, permettant l’avancement du sujet et l’émergence de nouvelles interactions entre différentes thématiques.

Nous souhaitons organiser cette rencontre sur deux semaines, comme lors des sessions précédentes, afin de proposer un programme en trois parties : – une école : la moitiés des exposés de la rencontre seront des mini-cours présentant des avancées notables en dynamique différentiable,
– une conférence composée d’exposés de recherche.
​– des moments informels pour les discussions et des séminaires improvisés.

Le second Zhang Zhifen Prize in Mathematics sera décerné lundi 13 mai.

Functional analysis of transfer operators by Viviane BALADI and Masato TSUJII

Part I: (Masato Tsujii – 3 x 50 min)
 “Transfer operators for Anosov flows”

We present a functional-analytic approach to the study of transfer operators for Anosov flows. To study transfer operators, a basic idea in semi-classical analysis suggests to look at the action of the flow on the cotangent bundle. Though this idea is simple and intuitive (as we will explain in the lectures), we need some framework to make it work. In the lectures, we present such a framework based on a wave-packet transform.

Part II:(Viviane Baladi – 3 x 50 min)
 « Transfer operators for Sinai billiards. »

We will discuss an approach to the statistical properties of two-dimensional dispersive billiards (mostly discrete-time) using transfer operators acting on anisotropic Banach spaces of distributions. The focus of this part will be our recent work with Mark Demers on the measure of maximal entropy but we will also survey previous results by Demers, Zhang, Liverani, etc on the SRB measure.

Beyond Bowen’s specification property (I) by Vaughn CLIMENHAGA

Rufus Bowen introduced the specification property for uniformly hyperbolic dynamical systems and used it to establish uniqueness of equilibrium states, including the measure of maximal entropy.  After reviewing Bowen’s argument, we will present our recent work on extending Bowen’s approach to non-uniformly hyperbolic systems.  We will describe the general result, which makes precise the notion of « entropy (or pressure) of obstructions to specification » using a decomposition of the space of finite-length orbit segments, and then survey various applications, including factors of beta-shifts, derived-from-Anosov diffeomorphisms, and geodesic flows in non-positive curvature and beyond.

Rigidity in rank one: dynamics and geometry by Andrey GOGOLEV

A dynamical system is called rigid if a weak form of equivalence with a nearby system, such as coincidence of some simple invariants, implies a strong form of equivalence. In this mini-course we     will discuss smooth rigidity of hyperbolic dynamical systems and related geometric questions such as marked length spectrum rigidity of negatively curved manifolds. We will consider the following moduli: lengths of periodic orbits, spectra of Poincar\’e return maps of the periodic orbits, volume Lyapunov exponents. After a brief overview of some classical results we will focus on recent developments in rigidity of Anosov and partially hyperbolic systems as well as connections to geometric rigidity. The latter is based on joint work with B. Kalinin and V. Sadovskaya and with F. Rodriguez Hertz.

Additive combinatorics methods in fractal geometry by Pablo SHMERKIN and Peter VARJU

In the last few years ideas from additive combinatorics were applied to problems in fractal geometry and led to progress on some classical problems, particularly on the smoothness of Bernoulli convolutions and other self-similar measures. We will introduce some of these tools from additive combinatorics and present some of the main applications, including the smoothness of Bernoulli convolutions outside of a small set of exceptions, and for explicit values of the parameter.

Big Mapping Class Groups by Juliette BAVARD and Danny CALEGARI

Part I – Theory :
       In the « theory » part of this mini-course, we will present recent objects and phenomena related to the study of big mapping class groups. In particular, we will define two faithful actions of some big mapping class groups. The first is an action by isometries on a Gromov-hyperbolic graph. The second is an action by homeomorphisms on a circle in which the vertices of the graph naturally embed. We   will describe some properties of the objects involved, and give some fruitful relations between the dynamics of the two actions. For example, we will see that loxodromic elements (for
the first action) necessarily have rational rotation number (for the second action). Using these relations, we will explain how to construct non trivial quasimorphisms on subgroups of big mapping class groups. This includes joint work with Alden Walker. 

Part II – Examples :
      In this part we will discuss a number of natural examples in which big mapping class groups and their subgroups arise. These include the
inverse limit constructions of de Carvalho-Hall, the theory of finite depth (taut) foliations of 3-manifolds, the theory of “Artinization” of Thompson-like groups, two dimensional smooth dynamics, one dimensional complex dynamics  (topology of the shift locus, Schottky spaces) and several other contexts. We will try to indicate how viewing these examples from the perspective of (big) mapping class groups is a worthwhile approach.

​J. Bavard had to cancel her participation. The whole course will be given by D. Calegari.

Some new dynamical applications of smooth parametrizations for C∞ systems by David BURGUET

Smooth parametrizations of semi-algebraic sets were introduced by Yomdin in order to bound the local volume growth in his proof of Shub’s entropy conjecture for Cmaps. In this minicourse we will present some refinement of Yomdin’s theory which allows us to also control the distortion. We will give two new applications: – for any Csurface diffeomorphism f with positive entropy the saddle periodic points with Lyapunov exponents δ-away from zero for δ ]0,htop(f)[ are equidistributed along measures of maximal entropy. – for Cmaps the entropy is physically greater than or equal to the top Lyapunov exponents of the exterior powers. 

Ergodic optimization by Gonzalo CONTRERAS

We will show the proof that for generic Lipschitz functions on an expanding map there is a unique maximizing measure, and it is supported on a periodic orbit.
Bibliography: G. Contreras. Ground states are generically a periodic orbit. Inventiones Mathematicae 205(2), 383-412 (2016).

Beyond Bowen’s specification property (II) by Dan THOMPSON

These lectures are a mostly self-contained sequel to Vaughn Climenhaga’s talks in week 1. The focus of the week 2 lectures will be on uniqueness of equilibrium states for rank 1 geodesic flows, and their mixing properties. Burns, Climenhaga, Fisher and myself showed recently that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. I will discuss the proof of this result. With this result in hand, the question of when the “pressure gap” hypothesis can be verified becomes crucial. I will sketch our proof of the “entropy gap”, which is a new direct constructive proof of a result by Knieper. I will also describe new joint work with Ben Call, which shows that all the unique equilibrium states provided above have the Kolmogorov property. When the manifold has dimension at least 3, this is a new result even for the Knieper-Bowen-Margulis measure of maximal entropy.  The common thread that links all of these arguments is that they rely on weak orbit specification properties in the spirit of Bowen.



Non Uniform Hyperbolicity in Global Dynamics