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...
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...
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...
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...
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...
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...
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.
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...
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...
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)...
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.
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...
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...
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...
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...
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...
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...
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.
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...
