Category: Uncategorized

Rigidity of foliations on surfaces and renormalization

Abstract: A class of dynamical systems is called (geometrically) rigid if the existence of a topological conjugacy implies automatically that the conjugacy is differentiable. Several classes of geometrically rigid system have been discovered in one-dimensional dynamics. 

In particular, it follows from a celebrated result by Michel Herman on circle diffeomorphisms (later improved by Yoccoz) that minimal smooth orientable foliations on surfaces of genus one, under a full measure arithmetic condition, are geometrically rigid.

In very recent joint work with Selim Ghazouani, we prove a generalization of this result to genus two, in particular by showing that smooth, orientable foliations with non-degenerate (Morse) singularities on surfaces of genus two, under a full measure arithmetic condition, are geometrically rigid. This in particular proves the genus two case of a conjecture by Marmi, Moussa and Yoccoz (formulated in the language of the Poincare maps, namely generalized interval exchange transformations.

During the talk, after motivating and explaining the result, we will give a brief survey of some of the key results in the theory of circle diffeos and in the study of generalized interval exchange maps and then an brief overview the strategy of the proof, which is based on renormalization.

Slides

Totally geodesic submanifold, superrigidity and arithmeticity

Abstract: I will discuss some recent work showing that finite volume real and complex hyperbolic manifolds having infinitely many maximal totally geodesic submanifolds are arithmetic.  I will put this in the context of what we do and (mostly) don’t know about real  and complex hyperbolic manifolds, particularly in dimension at least 4.  Key ingredients in the proofs depend on homogeneous dynamics and algebraic dynamics and it is tempting to believe these might be relevant to some remaining mysteries.  This is based on joint work with Bader, Miller and Stover and also some earlier joint work with Lafont, Miller and Stover.

Slides

Pointwise normality and Fourier decay for self-conformal measures

Projective cones and Billiards

In the moduli space of translation surfaces, large orbit closures are strata or loci of double covers

The strong unstable foliation of an Anosov diffeomorphism

Abstract: I will discuss recent work with Avila and Crovisier (and related work with Eskin, Potrie and Zhang as well) on the following problem and some higher dimensional analogues: Let f be an Anosov diffeomorphism in dimension 3.  Assume the unstable bundle is 2 dimensional and admits a dominated splitting into weak and strong unstable bundles.  Under what hypotheses is the strong unstable foliation minimal

Notes

On ergodic and statistical properties of smooth systems

Abstract: We will recall some basic ergodic and statistical properties  such as: ergodicity, (quantitative) mixing, K property, Bernoullicity, central limit theorem. We will be interested in rigidity and flexibility of these properties for smooth diffeomorphisms preserving a smooth measure. Our main rigidity result is that smooth diffeomorphisms which are exponentially mixing are Bernoulli (joint with D. Dolgopyat and F.Rodriguez-Hertz). For flexibility results we show existence of C^r smooth systems which satisfy the (non-trivial) central limit theorem and are of zero entropy. Moreover we show that there are smooth K, non-Bernoulli systems which satisfy (non-trivial) central limit theorem (joint with D. Dolgopyat, C. Dong, P.Nandori).

Critical exponents and amenability of covers

Abstract: In this joint work with R Coulon, R Dougall and  S Tapie, we prove the following result. Let be two discrete groups acting properly isometrically on a hyperbolic space . Then their critical exponents coincide if and only if  the small group is coamenable in .  In this talk, I will explain the statement and sketch the main steps of the proof, which involves a strange construction of so-called twisted Patterson-Sullivan measures.

Slides

Injectivity radius of discrete subgroups of higher rank Lie groups.

Abstract: Let be a simple higher rank Lie group and let be the associated symmetric space. Margulis conjectured that any discrete subgroup of such that has uniformly bounded injectivity radius must be a lattice. I will present the proof of this conjecture and explain how stationary random subgroups play the central role in the argument. The talk is be based on a recent joint work with Tsachik Gelander.