Group Actions with Hyperbolicity and Measure Rigidity

27 mai 2024, 09:00 → 31 mai 2024, 18:15 Europe/Paris

Amphithéâtre Hermite / Darboux (Institut Henri Poincaré)

Group Actions and Rigidity: Around the Zimmer Program

Workshop: Group Actions with Hyperbolicity and Measure Rigidity 

May 27 to 31, 2024 - IHP, Paris

Actions with some degree of hyperbolicity arise naturally in many mathematical contexts. Many natural problems reduce ti the classification of invariant measures or closed invariant sets. For instance, given a manifold equipped with an action by a large (not virtually-cyclic, though not necessarily higher-rank) group, under certain dynamical, geometric, or algebraic criteria on the action, one might hope to classify (1) all stationary or invariant measures and (2) all orbit closures for the action. In a geometrically related setting, given a partially hyperbolic diffeomorphism, one might ask when a u-Gibbs measure is necessarily SRB or when an equivariant foliation is necessarily minimal.  

This workshop will bring together experts working in homogeneous dynamics, smooth hyperbolic dynamics, complex and arithmetic dynamics, and group actions to discuss new results and techniques related to the rigidity of invariant measures and orbit closures for hyperbolic group actions.

Confirmed speakers:

• Timothée Bénard (University of Warwick)
• Serge Cantat (Université de Rennes & CNRS)
• Sylvain Crovisier (Université Paris-Saclay & CNRS)
• Nicolas de Saxce (Université Sorbonne Paris-Nord & CNRS)
• David Fisher (Rice University)
• Seung uk Jang (Université de Rennes)
• Asaf Katz (University of Michigan)
• Victor Kleptsyn (CNRS & Université de Rennes)
• Manuel Luthi (EPFL)
• Hee Oh (Yale University)
• Rafael Potrie (Universidad de la República)
• Megan Roda (University of Chicago)
• Roland Roeder (Indiana University)
• Federico Rodriguez-Hertz (PennState University)
• Bruno Santiago (Universidade Federal Fulminense)
• Cagri Sert (University of Zürich)
• Omri Solan (Hebrew University)
• Rosemary Eliott Smith (University of Chicago)
• Zhiyuan Zhang (CNRS & Université Paris-Nord)

 

Organising Committe:

• Aaron Brown (Northwestern Universty)
• Romain Dujardin (Sorbonne Université)
• Simion Filip (University of Chicago)
• David Fisher (Rice University)
• Davi Obata (Brigham Young University)

 

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lundi 27 mai

09:00 Registration/ Welcome coffee

1 Zimmer's conjecture for non-split semisimple Lie groups

Zimmer conjectures that a higher rank lattice in a semisimple Lie group cannot have essentially non-trivial action on manifolds with dimension smaller than certain quantity related to the Lie group. This conjecture is proved in many cases by Aaron Brown, David Fisher and Sebastian Hurtado. Their result provides sharp dimension bounds for real split semisimple real Lie groups but not sharp otherwise. We discuss a recent generalisation of their result with sharp bound to some other (non-real split) semisimple Lie groups. This is a joint work with Jinpeng An and Aaron Brown.

Orateur: Zhiyuan Zhang (Imperial College)

10:30 Coffee Break

2 Fourier decay of multiplicative convolutions

Abstract: In the first part of the talk, we shall recall the main ideas involved in the proof of Fourier decay for multiplicative convolutions of Frostman measures on the real line. In particular, we will present a recent result obtained in collaboration with Tuomas Orponen and Pablo Shmerkin on the minimal number of convolution products necessary to obtain some Fourier decay. The second part of the talk will be devoted to the application of these ideas to the study of linear random walks on the torus, and will be based on joint work with Weikun He.

Orateur: Nicolas de Saxcé (Université Paris-Nord)

12:00 Lunch

3 On measure rigidity of u-Gibbs states

Abstract: Physical measures are an important tool in the study of hyperbolic dynamics, governing, for example, the statistical properties of the orbit of almost every point with respect to volume (in the dissipative setting). The well-studied uniformly hyperbolic (Anosov) diffeomorphisms and flows always have ergodic physical measures, whereas the more general class of partially hyperbolic systems lose this property. For these systems, we are instead guaranteed the existence of at least one, and possibly infinite, ergodic u-Gibbs measure(s). In the case of a unique u-Gibbs measure, that measure is automatically physical.

Thus, a natural question in the partially hyperbolic setting is the following: under what conditions is there a unique u-Gibbs measure? More generally, which u-Gibbs measures are physical? This question was partially answered in dimension three by Eskin, Potrie, and Zhang. Here we partially extend this result to arbitrary dimensions, and discuss the dichotomy that arises: roughly, a u-Gibbs measure is physical if and only if it is not jointly integrable of some order.

Orateur: Rosemary Eliott Smith (University of Chicago)

15:00 Coffee Break

4 On the rigidity of u-Gibbs measures for partially hyperbolic diffeomorphisms in dimension three.

We consider diffeomorphisms of three manifolds having a non-trivial $Df$-invariant splitting $E^s\oplus E^c\oplus E^u$. In this setting, u-Gibbs measures are invariant measures with smooth conditionals along unstable manifolds. The set of all u-Gibbs measures plays a prominent role in the ergodic theory of these systems, for it contains the sets of all physical measures and SRB measures. When the center exponent is non positive, the measure is automatically SRB and even physical if the exponent is negative. Thus, a natural question is: given an ergodic u-Gibbs measure with positive exponent along the center, can we deduce that the measure is SRB, that is, that it has smooth conditionals along center-unstable manifolds? When the answer is yes, one also obtains that the measure is a physical measure. This question can be seen as a measure rigidity question since one desires to get some additional invariance along center manifolds. In this talk I intend to survey some developments on this question, and pose some problems.

Orateur: Bruno Santiago (Universidade Federal Fluminense)

5 Questions about the holomorphic group action dynamics on a natural family of affine cubic surfaces

I will describe the dynamics by the group of holomorphic automorphisms of the affine cubic surfaces
$$ S_{A,B,C,D} = \{(x,y,z) \in \mathbb{C}^3 : \textrm{ } x^2 + y^2 + z^2 +xyz = Ax + By+Cz+D\}, $$ where $A,B,C,$ and $D$ are complex parameters. This group action describes the monodromy of the famous Painlevè 6 Equation as well as the natural dynamics of the mapping class group on the $\mathrm{SL}(2,\mathbb{C})$ character varieties associated to the once punctured torus and the four times punctured sphere. For these reasons it has been studied from many perspectives by many people including Bowditch, Goldman, Cantat-Loray, Cantat, Tan-Wong-Zhang, Maloni-Palesi-Tan, and many others.

In this talk I will describe my recent joint with Julio Rebelo and I will focus on several interesting open questions that arose while preparing our work ``Dynamics of groups of automorphisms of character varieties and Fatou/Julia decomposition for Painlevé 6" and during informal discussions with many people.

Orateur: Roland Roeder (Indiana University-Purdue University Indianapolis)

mardi 28 mai

6 Stationary probability measures on projective spaces

We give a description of stationary probability measures on projective spaces for an iid random walk on $\mathrm{PGL}_d(\mathbb{R})$ without any algebraic assumptions. This is done in two parts. In a first part, we study the case (non-critical or block-dominated case) where the random walk has distinct deterministic exponents in the sense of Furstenberg--Kifer--Hennion. In a second part (critical case), we show that if the random walk has only one deterministic exponent, then any stationary probability measure on the projective space lives on a subspace on which the ambient group of the random walk acts semisimply. This connects the critical setting with the work of Guivarc'h--Raugi and Benoist--Quint. Combination of all these works allow to get a complete description. Joint works with Richard Aoun.

Orateur: Çagri Sert (University of Zurich)

10:30 Coffee Break

7 Clay Lecture: Rigidity of Kleinian groups

Discrete subgroups of $\mathrm{PSL}(2,\mathbb{C})$ are called Kleinian groups. After a brief review of the Mostow-Sullivan rigidity theorem, we will discuss a new rigidity theorem for general Kleinian groups and its various implications including a cross-ratio rigidity theorem. For convex cocompact Kleinian groups, we also present a measure-theoretic rigidity theorem. We will discuss how we use the dynamics of one-parameter diagonal flows on higher rank homogeneous spaces of infinite volume to prove these results.
(This talk is based on joint work with Dongryul Kim).

Orateur: Hee Oh (Yale University)

12:00 Group photo

12:10 Lunch

8 The horocycle flow in the moduli space of translation surfaces

Abstract: I will report on recent progress about the horocycle flow in the moduli space of translation surfaces. This is a joint work with J.Chaika and B.Weiss.

Orateur: Florent Ygouf (Université de Rennes)

15:00 Coffee Break

9 Gap in critical exponents of SL2(R) orbits in nonarithmetic quotients of SL2(C)

We will discuss the following result. For every nonarithmetic lattice $\Gamma < \mathrm{SL}_2(\mathbb{C})$ there is $\varepsilon \Gamma$ such that for every $g \in \mathrm{SL}_2(\mathbb{C})$ the intersection $g\Gamma g^{-1} \cap \mathrm{SL}_2(\mathbb{R})$ is either a lattice or a has critical exponent $\delta(g\Gamma g^{-1} \cap \mathrm{SL}_2(\mathbb{R})) \leq 1-\varepsilon \Gamma$. This result extends Mohammadi-Margulis and Bader-Fisher-Milier-Strover. We will focus on an ergodic component of the proof, asserting certain preservation of entropy-contribution under limits of measures.

Orateur: Omri Solan (Hebrew University)

10 Surpise Talk

18:00 Cocktail dinner party

mercredi 29 mai

11 Pushes of fractal measures on periodic horocycles

We examine pushes of fractal measures on certain periodic horospherical orbits with the aim to study Diophantine approximation on fractals. I will present an ongoing collaboration with Osama Khalil and Barak Weiss, where we partially extend earlier results by Khalil and myself, more explicitly proving ineffective equidistribution of pushes of the fractal measure coming from a special class of rational self-similar rotation-free iterated function systems. Notably, the class contains the Hausdorff measure on the middle third Cantor set which isn't covered by earlier work on this subject. The proof uses a partial classification of stationary measures for a random walk corresponding to the iterated function system on an S-arithmetic homogenous space. I will explain the main theorems, some of the ideas of the proof, and some of the differences to existing results on stationary measures.

Orateur: Manuel Luethi (Ecole Polythechnique Fédérale de Lausanne)

10:30 Coffee break

12 Hölder regularity of stationary measures

One of the main tools of the theory of dynamical systems are the invariant measures; for random dynamical systems, their role is taken by stationary measures.

In a recent work with A. Gorodetski and G. Monakov, we show that stationary measures almost always (under extremely mild assumptions) satisfy the H\"older regularity property: the measure of any ball is bounded by (a constant times) some positive power of its radius.

Orateur: Victor Kleptsyn (Université de Rennes)

jeudi 30 mai

13 Stationary Measures on Markov surfaces

Markov surfaces are certain (non-compact) algebraic, cubic surfaces in the affine 3-space. There is a natural group acting on them by algebraic, polynomial transformations. I will study stationary probability measures for this group action. This is a joint work with Christophe Dupont and Florestan Martin-Baillon.

Orateur: Serge Cantat (Université de Rennes)

10:30 Coffee Break

14 Markoff Surfaces in the p-adic World

Markoff surfaces appear in studies of the character variety of the
$1$-punctured torus or the $4$-punctured sphere, which have many algebraic
automorphisms. When we sketch their real points, say, we often observe
'hyperbolic' and "spherical'' parts. The dynamical nature of the
algebraic automorphisms on these respective parts is well-known for real
(or complex) points.

In the talk, we will discuss what happens when we ask an analogous
question for $p$-adic numbers. It turns out that (a) the tropicalization
of the variety gives rise to a copy of the hyperbolic plane, and (b)
there is a finite list of bounded, automorphism-invariant closed subsets
over $p$-adic points. These correspond to the behaviors of "hyperbolic"
and "spherical'' parts in the $p$-adic case.

Orateur: Seung uk Jang (Université de Rennes)

12:00 Lunch

15 Classifying ergodic hyperbolic stationary measures on K3 surfaces with large automorphism groups

Let $\mathrm{X}$ be a K3 surface with a large automorphism group $\mathrm{Aut}(\mathrm{X})$ (we do not assume that it contains any parabolic elements). Consider a probability measure $\mu$ on $\mathrm{Aut}(\mathrm{X})$; using the work of Cantat and DuJardin (2020) we study hyperbolic, ergodic $\mu$-stationary probability measures, and the supports of their conditional measures on the stable and unstable manifolds (which are a.e. biholomorphic to $\mathbb{C}$) using the techniques of Benoist and Quint (2011), and Eskin and Mirzakhani (2018).

Orateur: Megan Roda (University of Chicago)

15:00 Coffee Break

16 SH property for unstable laminations and applications to uniqueness of u-Gibbs

Abstract: The SH-property for unstable laminations is a topological analog to positive center exponents for u-Gibbs measures. I plan to explain some abundance results for this property in the context of volume preserving partially hyperbolic diffeomorphisms with one-dimensional center and how this implies results on rigidity of u-Gibbs measures in dimension 3. This is part of a joint project in many combinations with A. Avila, S. Crovisier, A. Eskin, A. Wilkinson and Z. Zhang.

Orateur: Rafael Potrie (Universidad de la Republica)

17 TBA

Orateur: Artur Ávila (Universität Zürich & IMPA)

vendredi 31 mai

18 Finiteness of totally geodesic hypersurfaces in variable negative curvature

Abstract: When can a negatively curved manifold admit infinitely many totally geodesic submanifolds of dimension at least two? I will explain some motivations for this question coming from different parts of mathematics. I will also explain a proof of the fact that a compact manifold with a real-analytic negatively curved metric admits only finitely many totally geodesic hypersurfaces, unless it is a hyperbolic manifold. And also state a more general conjecture. This is joint work with Simion Filip and Ben Lowe.

Orateur: David Fisher (Rice University)

10:30 Coffee break

19 Effective equidistribution of random walks

I will explain why a random walk on $\text{SL}_{2}(\mathbb{R})/\text{SL}_{2}(\mathbb{Z})$ equidistributes with an explicit rate toward the Haar measure, provided the walk is not trapped in a finite orbit and the driving measure is supported by algebraic matrices generating a Zariski-dense subgroup. The argument is based on a multislicing theorem which extends Bourgain's projection theorem and presents independent interest. Joint work with Weikun He.

Orateur: Timothée Bénard (CNRS/Université Sorbonne Paris Nord)

12:00 Lunch

20 Polynomial Fourier decay and a cocycle version of Dolgopyat's method for self-conformal measures

We prove polynomial Fourier decay for self conformal measures w.r.t a real C^2 smooth IFS, under mild non-linearity assumptions. One of the key ingredients in our argument is a cocycle version of Dolgopyat's method that is used to study the transfer operator. In particular, our version of Dolgopyat's method does not require the cylinder covering of the underlying fractal to be a Markov partition.
Joint with Amir Algom and Zhiren Wang.

Orateur: Federico Rodriguez Hertz (Penn State University)

15:00 Coffee Break

21 Minimality of strong unstable foliations

Abstract: I will discuss the minimality of the unstable foliations of partially hyperbolic diffeomorphisms with one-dimensional center, and in particular the abundance and the robustness of this property. This is part of a joint project in many combinations with A. Avila, S. Eskin, R. Potrie, A. Wilkinson and Z. Zhang.

Orateur: Sylvain Crovisier (Université Paris-Saclay)

22 Rigidity of u-Gibbs states in partially hyperbolic dynamics

SRB measures, being physical measures, are of prime importance in partially hyperbolic systems. Their existence is an open problem - in general. Nevertheless, a related, more general class of measures - known as u-Gibbs states, were known to exist by a theorem of Pesin-Sinai. I will explain how one can adapt the factorization technique, pioneered by Eskin-Mirzakhani, to the setting of smooth dynamics and prove that for quantitatively non-integrable systems a (generalized) u-Gibbs state must be an SRB measure. If time permits, I will try to describe some of the key ideas and constructions of the Eskin-Mirzakhani technique.

Orateur: Asaf Katz (University of Michigan)