Title: The Forni Subspace: A Survey of Recent Results

Abstract: The Forni subspace, defined by Avila-Eskin-Moeller (in the context of translation surfaces), is a mechanism for producing Lyapunov exponents equal to zero in the Kontsevich-Zorich cocycle.  In this talk we will introduce the basic objects and survey various classification results.  We will explain some of the techniques and highlight possible future directions.  This will include joint work with Frederik Benirschke and Chaya Norton.

 

Title: TBA

Abstract: TBA

 

Title: Exotic Lyapunov Exponents.

Abstract: We construct two examples of continuous SL(2,R)-cocycles over hyperbolic dynamics whose Lyapunov spectra are "exotic". In the first example, the (top) Lyapunov exponent is the same positive number for all invariant (probability) measures, but nevertheless the cocycle is not uniformly hyperbolic. In the second example, there exists a unique invariant measure whose Lyapunov exponent is far from zero. Each construction starts with the selection of an appropriate non-uniformly hyperbolic cocycle over a uniquely ergodic base, and then proceeds with a careful extension of this cocycle. The resulting moduli of continuity are "bad"; in fact, such examples can never be Holder-continuous.

 

Title: Accumulation points of normalized approximations.

Abstract: Building on classical aspects of the theory of Diophantine approximation, we consider the collection of all accumulation points of normalized integer vector translates of points qα,  with α in R^d and q an integer. First we derive measure theoretic and Hausdorff dimension results about the set of α whose accumulation points are all of R^d. Then we focus primarily on the case when the coordinates of α together with 1 form a basis for an algebraic number field K. Here we show that, under the correct normalization, the set of accumulation points displays an ordered geometric structure which reflects algebraic properties of the underlying number field. For example, when d = 2, this collection of accumulation points can be described as a countable union of dilates of a single ellipse, or of a pair of hyperbolas, depending on whether or not K has a non-trivial embedding into the complex numbers.

 

Title: Thermodynamic formalism for non-compact locally CAT(-1) geodesic flows via Patterson-Sullivan measures.

Abstract:  On the geodesic flow of a manifold of pinched negative curvature, there is a geometric construction of equilibrium states for Holder potentials which has been thoroughly developed by the well-known work of Paulin, Pollicott, and Schapira. For the geodesic flow of a proper locally CAT(-1) space, the construction was not as clear due to discrepancies arising from non-uniqueness of extensions of geodesic segments. I will talk about joint work with Daniel Thompson showing how to use ideas from coarse hyperbolicity to handle these discrepancies in the construction, and how to still prove that the measure is the unique equilibrium state if it is finite. Our technique works for any bounded potential satisfying the Bowen property.

 

Title: Non tame cocycle rigidity above affine unipotent abelian actions on the torus.

Abstract: Cocycle rigidity with tame solutions is a crucial ingredient in KAM theory.  We are interested in cocycle rigidity above affine unipotent abelian actions on the torus with Diophantine translation data. We consider  unlocked actions whose rank one factors are non vanishing translations (the locked actions do not have any kind of stability). 

It follows from Katok and Robinson's observations that when one generator of the action is of step less or equal to 2 then cocycle rigidity with tame solutions holds. Moreover,  Damjanovic, Fayad and Saprykina

 proved that in this case almost cocycles also have almost solutions (with a low regularity control on the error), and from there concluded  KAM-rigidity of these actions. 

  In a joint work with S. Durham, we find examples of affine Z^2-actions on the torus above which smooth cocycle rigidity holds but is not tame. 

The linear part of the action is generated by unipotent matrices of step 3. Our examples show that KAM-rigidity for higher rank actions by affine unipotent toral actions does not hold in general when no element of the actions is of step less or equal to 2.

 

Title: Finiteness of totally geodesic hypersurfaces in variable negative curvature.

Abstract:  I will 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 an arithmetic hyperbolic manifold. I will provide some context, via analogous rigidity results in Teichmuller dynamics, as well as the case of homogeneous spaces. Joint work with David Fisher and Ben Lowe.

 

Title: Dynamics on character varieties.

Abstract: The classification of geometric structures on manifolds naturally leads to actions of mapping class groups on deformation spaces of flat bundles over surfaces. In joint work with Forni, these actions are suspended to give extensions of the Teichmuller flow.  This talk will survey the evolution of these ideas and report on some recent joint with Forni, Lawton and Matheus when the structure group is compact. If time permits I will speak on a program to analyze an action of the modular group on the variant of the Markoff surface x^2 + y^2 + z^2 - x y z = 20 where the dynamics bifurcates between ergodic (level < 20) and not ergodic (level > 20). 

 

Title: Reduction theory for Fuchsian groups with cusps.

Abstract: For any Fuchsian group of the first kind with at least one cusp and possibly with elliptic points, we construct a fundamental polygon such that the two-dimensional natural extension, a.k.a. reduction map, of the Bowen–Series-like map acting on the boundary of the upper half-plane in a piecewise manner by generators of the group, has a domain of bijectivity (and global attractor) with simple and finite rectangular structure. The fundamental polygon is related to the free product structure of the group, and its deformation in the Teichmüller space preserves the combinatorial structure and the marking. This answers a question posed by Don Zagier. (Joint project with I.Ugarcovici and A.Abrams).

 

Title: Patterson-Sullivan measures of Anosov groups are Hausdorff measures

Abstract: In the theory of Kleinian groups, Sullivan's classical theorem establishes the correspondence between Patterson-Sullivan measures and Hausdorff measures on the limit sets of convex cocompact Kleinian groups. This connection provides a geometric understanding of Patterson-Sullivan measures in terms of the internal metric on limit sets. Recent advancements in the theory of discrete subgroups of higher rank Lie groups have brought Anosov subgroups into focus as a natural extension of convex cocompact Kleinian groups. This raises an intriguing question: under what conditions do Patterson-Sullivan measures of Anosov subgroups emerge as Hausdorff measures on limit sets with appropriate metrics?

For all Zarski dense Anosov subgroups, we give a definitive answer to this question by showing that their limit sets are Ahlfors regular for intrinsic conformal premetrics, and  that a Patterson-Sullivan measure is equal to the Hausdorff measure if and only if the associated linear form is symmetric. This is the joint work with Subhadip Dey and Hee Oh.

 

Title: An average intersection estimate for families of diffeomorphisms

Abstract: Poincaré’s formula in integral geometry states that for any submanifolds of V, W of a compact homogeneous space M = G/H with G acting transitively on tangent subspaces, the average volume of the intersection between g(V) and W is equal to Cvol(V)vol(W) for C a constant depending only on the dimensions of V and W. We discuss an adaptation of this result to general (non-homogeneous) closed manifolds, where the transformation group G is replaced by a compact family of diffeomorphisms, and the formula now holds up to uniform multiplicative error. We then sketch some applications of this result to the dynamics of C^1 diffeomorphisms with exponential volume growth. This is joint work with Mike Shub. 

 

Title: Critical quasicircle maps

Abstract: There is an essentially complete renormalization theory for analytic critical circle maps which serves to justify the golden mean universality and other conjectures by physicists. In this talk, we will introduce a generalization called "critical quasicircle maps", i.e. analytic self homeomorphisms of a quasicircle with a single critical point. We will sketch the realization of such maps with prescribed combinatorics, and then discuss the state of the art renormalization picture together with its implications on rigidity, universality, and conjugacy classes.

 

Title: Ergodic averages along \Omega(n).

Abstract: Following Birkhoff's proof of the Pointwise Ergodic Theorem, it is natural to consider whether convergence still holds along various subsequences. In 2020, Bergelson and Richter showed that in uniquely ergodic systems, pointwise convergence holds along the number theoretic sequence \Omega(n), where \Omega(n) counts the number of prime factors of n with multiplicities. In this talk, we will see that by removing this assumption, a pointwise ergodic theorem does not hold along \Omega(n). In fact, \Omega(n) satisfies a rather strong non-convergence property. We will further classify the strength of this non-convergence behavior by considering weaker notions of averaging (based on current joint work with S. Mondal).

Daniel Mitsutani 

Title: Hidden Symmetry Rigidity of Geodesic Flows
Abstract: An old result of Bochner proves that closed Riemannian manifolds of negative Ricci curvature admit only finitely many isometries. On the other hand, work beginning with Eberlein, and later extended by Farb and Weinberger, shows that rigidity in the presence of too many isometries still occurs provided one looks at covers of a closed manifold of negative sectional curvature to find “hidden symmetries”: Eberlein proves that a closed Riemannian manifold of negative sectional curvatures admitting infinitely many isometries of its universal cover must be locally symmetric.
From the dynamical perspective, hyperbolic dynamical systems also display such a phenomenon: if the centralizer group of a hyperbolic dynamical system is too large often it is conjugate to an algebraic one. In this talk we will consider hidden symmetries of the hyperbolic dynamical system given by the geodesic flow of a manifold of negative sectional curvatures. We will introduce an appropriate notion of a centralizer for the geodesic flow on the universal cover, and prove that when it is not discrete the metric must be locally symmetric.

 

 

Title: Non-stationary Ito-Kawada and Ergodic Theorems for random isometries.

Abstract: I will consider a nonstationary sequence of independent random isometries of a compact metrizable space. Assuming that there are no proper closed subsets with deterministic images I establish a weak-* convergence to the unique invariant under isometries measure, Ergodic Theorem and Large Deviation Type Estimate. I also show that all the results can be carried over to the case of a random walk on a compact metrizable group. In particular, I prove a nonstationary analog of classical Ito-Kawada theorem and give a new alternative proof for the stationary case.

 

Title: Automorphic measures for critical circle maps.

Abstract: Let f be a C^2 circle diffeomorphism with irrational rotation number. As established by Douady and Yoccoz in the eighties, for any given s > 0 there exists a unique  automorphic measure of exponent s for f. In this talk, we shall discuss the proof that the same holds when f is a minimal critical circle map. We shall also briefly discuss two applications of this result. The first one is that these maps admit no invariant distributions of order 1 independent of the unique invariant measure. The second one is an improvement of the Denjoy-Koksma inequality for absolutely continuous observables.

 

Title:  Rigidity of Kleinian groups.

 Abstract: Discrete subgroups of PSL(2,C) are called Kleinian groups. After briefly reviewing the Mostow-Sullivan rigidity theorem, we will discuss new rigidity results for  Kleinian groups which we obtain using the dynamics of one parameter diagonal flows on higher rank homogeneous spaces of infinite volume. (This talk is based on joint work with Dongryul Kim).

 

Title: Renormalization of dissipative  diffeos of the disk

Abstract: It has been proven that any Henon map (with Jacobian smaller than 1/4) lying on the boundary of diffeomorphisms with zero entropy is infinitely renormalizable. Aiming to tackle the converse problem, the talk will discuss how to develop a non-perturbative renormalization theory for two-dimensional diffeomorphisms.

 

Title: A counterexample to a marked length spectrum semi-rigidity problem.

Abstract: Given a negatively curved surface (M,g), I'll discuss how to construct a perturbation (M,g') such that each closed geodesic becomes longer, and yet there is no diffeomorphism which contracts every tangent vector. This is joint work with Andrey Gogolev.

 

Title: Non-conical strictly convex divisible sets are maximally anisotropic\\

Abstract: Let U be a non-conical strictly convex divisible set. Even though the boundary S of U is not C^2, Benoist showed that S is C^{1+} and Crampon established that S has a sort of anisotropic Holder regularity -- described by a list L of real numbers - at almost all of its points. In this talk, we discuss our joint work with  P. Foulon and P. Hubert showing that S is maximally anisotropic (in the sense that the list L contains no repetitions) thanks to the features of the Hilbert flow. 

 

Title: Robustness of the measure of maximal entropy for smooth surface diffeomorphisms.

Abstract: Let f be a topologically transitive C-infinity surface diffeomorphism with positive topological entropy. Let m be the (unique) measure of maximal entropy. We show that for any invariant measure m' whose entropy is epsilon away from the maximal entropy,

(1) |m(h)-m'(h)|=O(\sqrt{\epsilon}) for all Holder functions h with unit Holder norm

(2) the Lyapunov exponents of m’ are O(\sqrt{\epsilon}) away from those of m

(3) the Oseledets splitting of m’ is O(\sqrt{\epsilon}) away from that of m’ (in a sense that will be made precise in the talk).

The case of Anosov diffeomorphisms is due to Kadyrov; the novelty of our work is that we do not need the Anosov assumption. (Joint with Jerome Buzzi and Sylvain Crovisier)

 

Title: How to do it on Cantor sets.

Abstract: I will talk about some recent results about minimal Cantor systems.

 

Title: Global smooth rigidity for toral automorphisms 

Abstract: Suppose  f is a diffeomorphism  on torus  whose linearization A  is weakly irreducible.  Let 

H be a conjugacy between f and A. We prove the following: 1. if A is hyperbolic and H is weakly differentiable 

2. if A is partially hyperbolic and H is C^{1+holder}.  Then H is C-infty.  Our result shows that 

the conjugacy in all local and global rigidity results for irreducible A  is C-\infty. This is a joint work with 

B.  Kalinin and V Sadovskaya.