Title: From Selberg to Ihara via Berkovich
Abstract:
In 1998, McMullen studied the vanishing rate of the Hausdorff dimension of the limit sets of certain degenerating families of Schottky groups. In 2024, Dang and Mehmeti vastly extended McMullen's result by using the philosophy that degenerating complex objects are naturally described by the limiting non-Archimedean objects.
In this talk, we pursue this philosophy by showing that the Selberg zeta functions attached to degenerating families of Schottky groups converge (after suitable rescalings) to Ihara zeta functions of non-Archimedean Schottky groups acting on Berkovich projective lines. Here, one key idea is the introduction of a middle zeta function interpolating between Selberg and Ihara zeta functions. As a by-product, we improve upon theoretical and numerical results of Weich, Borthwich and Pollicott-Vytnova on the convergence of Selberg zeta functions, and we obtain exponential error terms in the asymptotics formulas by McMullen and Dang-Mehmeti. These results were obtained jointly with Jialun Li, Wenyu Pan and Zhongkai Tao.
Title: Lyapunov–Oseledets spectrum for transfer operator cocycles under perturbations: two examples
Abstract:
In recent years, the study of transfer operators has been combined with multiplicative ergodic theory to shed light on ergodic-theoretic properties of random dynamical systems. The so-called Lyapunov– Oseledets spectrum associated to the transfer operator cocycle contains fundamental information about invariant measures, exponential decay rates and coherent structures which characterize dominant global transport features of the system. While the scope of this framework is broad, it is challenging to identify and approximate this spectrum. In this talk, we present two examples where the Lyapunov– Oseledets spectrum can be understood and analyzed under perturbations: Blaschke product cocycles and random metastable systems. This talk is based on joint works with Anthony Quas and Joshua Peters.
Title: Transfer operators, atomic decomposition and Besov spaces.
Abstract:
Since Ruelle's groundbreaking contributions, the study of transfer operators has become a main tool for understanding the ergodic theory of expanding maps, which are discrete dynamical systems that locally expand distances. Questions regarding the existence of interesting invariant measures and statistical properties of such dynamical systems, such as exponential decay of correlations and the Central Limit Theorem, can be answered by studying the spectral properties of the operators' action on suitable spaces of functions. By the use of the atomic decomposition method, we consider new Banach spaces of functions, which in some cases coincide with Besov spaces. These spaces have a remarkably simple definition and allow us to obtain very general results on the quasi-compactness of the transfer operator acting in these spaces, even when the underlying phase space and expanding map are highly irregular. This work was conducted in collaboration with Alexander Arbieto (UFRJ-Brazil).
Title: Entropy and measures at the boundary
Abstract:
We describe equivariant families of measures on the boundary of the universal cover of a closed Riemannian manifold with negative curvature.
We discuss the associated entropy and its rigidity properties. The same formalism can describe:
1.the Patterson-Sullivan family and the associated Burger-Roblin measure,
2. the Lebesgue family and the Liouville measure,
3. the harmonic measures and the drifted harmonic measures,
4. the Mohsen family giving the Rayleigh quotient, and
5. the Gibbs-Patterson families.
Title: Stochastic behaviour of chaotic billiards
Abstract:
We consider a point particle (with random initial position and velocity) evolving in a deterministic chaotic billiard (such as the Sinai billiard in the torus, the periodic Lorentz gas with finite or infinite horizon, etc). The initial randomness combined with the hyperbolicity of the dynamics gives rise to interesting stochastic properties. The aim of this talk is to present probabilistic limit theorems in this context.
Title: Critical circle maps: classical and new results
Abstract:
Critical circle maps are circle homeomorphisms with a finite number of non-flat critical points. These maps belong to the boundary of chaos, i.e. between circle diffeomorphisms and chaotic circle maps.
In this talk we will discuss some classical and new results, including some recent progress, concerning the smoothness of the conjugacy for maps with more than one critical point.
Title: Natural measures and statistical properties of maps with several neutral fixed points and no physical measure
Abstract:
We show that infinite measure-preserving intermittent maps that do not admit physical measures nevertheless exhibit strong statistical properties. In particular, we prove existence of a distinguished "natural" measure such that the pushforwards of any absolutely continuous probability measure converge to it. We also obtain results on almost sure and distributional convergence for empirical measures. (Joint with Douglas Coates and Amin Talebi.)
Title: Infinite sumsets inside sets of positive density
Abstract:
A central question in Ramsey theory asks about what patterns can be found in sets of integers with positive density. A method introduced by Furstenberg in the 1970's allows for techniques from ergodic theory to be used to study this problem. More recently, a new variant of this method was developed that allows one to study infinite patterns in sets with positive density. I will describe recent joint work with Kra, Richter and Robertson where we use ergodic theory to study infinite sumsets.
Title: Dispersion of orbits with sub-exponential rates.
Abstract:
In a joint work with Enrique Pujals, we developed the concept of generalized entropy that allows us to quantify the dispersion of a system's orbits in the space of growth orders. This object allows us to classify families of dynamical systems with classical entropy 0. The idea of this talk is to make an introduction, present the results obtained so far (with Enrique Pujals and Hellen de Paula) and discuss new projects in this regard.
Title: The amount of nonhyperbolicity of partially hyperbolic diffeomorphisms
Abstract:
We study a wide range of partially hyperbolic diffeomorphisms having a one-dimensional center bundle. These include well-known examples like robustly transitive perturbations of skew products, DA diffeomorphisms, and the time one-map of transitive Anosov flows. These systems exhibit a mix of hyperbolic behaviors alongside regions lacking any hyperbolicity. To determine ``which behavior dominates'', we conduct a multifractal analysis of the central Lyapunov exponent and detail the topological entropy of the related level sets. In particular, we investigate the subset of points with exponent zero. This is joint work with L.J. Díaz, B. Santiago, and J. Zhang.
Title: To be or not to be (hyperbolic): A conservative dilemma.
Abstract:
In dimension three, we present three conservative partially hyperbolic settings (derived from Anosov systems, time one-maps of geodesic flows on negatively curved surfaces, and skew-product with circle fibers) where the following dichotomy holds: the diffeomorphism is either Anosov or it supports and (ergodic) nonhyperbolic measure. Join work with J, Yang (UFF, Niterói, Brazil) J. Zhang (Beihaan, Beijing, China).
Title: The Mandelbrot set and its Satellite copies
Abstract:
For a polynomial on the Riemann sphere, infinity is a (super) attracting fixed point, and the filled Julia set is the set of points with bounded orbit. Consider the quadratic family P_c(z)=z^2+c. The Mandelbrot set M is the set of parameters c such that the filled Julia set of P_c is connected.
Computer experiments quickly reveal the existence of small homeomorphic copies of M inside itself; the existence of such copies was proved by Douady and Hubbard. Each little copy is either primitive (with a cusp on the boundary of its main cardioid region) or a satellite (without a cusp). Lyubich proved that the primitive copies of M satisfy a stronger regularity condition: they are quasiconformally homeomorphic to M. The satellite copies are not quasiconformally homeomorphic to M (as we cannot straighten a cusp quasiconformally), but are they mutually quasiconformally homeomorphic? In joint work with C. Petersen we prove that the answer is negative in general, but positive in the case the satellite copies have rotation numbers with the same denominator.
Title: Uniformly simplicity for subgroups of piecewise continuous bijections.
Abstract:
A group G is uniformly simple if there exists a positive integer N such that for any f,ϕ∈G∖{Id}, the element ϕ can be written as a product of at most N conjugates of f or f−1.
We provide conditions which guarantee that a subgroup G of the group of piecewise continuous bijectiones of the unit interval is uniformly simple. As corollaries we get that several important groups are uniformly simple.
Title: Regularity of the Lyapunov Exponents of 2 dimensional cocycles.
Abstract:
We will provide a comprehensive survey of recent developments in the study of the regularity of Lyapunov exponents associated with two dimensional cocycles.
Title: Shub's example revisited
Abstract:
Shub gave the first examples of robustly transitive diffeomorphisms on the 4-torus which are not uniformly hyperbolic (they are though partially hyperbolic). We will show that these examples have further interesting properties: under some bunching conditions, there exists an open dense subset of diffeomorphisms in this class exhibiting a unique homoclinic class of index two supported on the whole torus. This has important consequences in the study of equilibrium states. Join work with C. Liang, F. Yang and J. Yang.
Title: Rotation theory on Two Torus
Abstract:
In this talk, we will discuss some recent results on rotation theory on the two-torus. After a brief introduction to the definitions and some connections to the general theory of surface dynamics, we will focus more on specific shapes of rotation sets. In particular, when the rotation set is a line segment from the point (0,0) to a point with an irrational slope, we will discuss bounded and unbounded deviation properties along different directions. Specifically, we will apply a recent, powerful forcing theory to prove certain results and relate the bounded deviation phenomenon to the existence of non-contractible periodic orbits.
Some new results discussed here are joint works with Fabio Tal and Salvador Zanata.