Many important questions in dynamics can be reduced to the study of positivity or simplicity of Lyapunov exponents of a suitable cocycle over a suitably chaotic base. The most common approach in the literature is via coding, i.e., reduction to random matrix products induced by countable state Markov chains. In this talk we discuss a new approach to this problem that entirely bypasses coding. As a particular case, we apply our results to arbitrary SL(2,R) orbit closures of Abelian differentials, for which coding is not known to hold. This is joint work with DeWitt, Eskin, Gadre, Gutierrez-Romo, Lima, Matheus, Rafi, and Schleimer.
Motivated by the study of billiards in polyhedra, we study linear flows in a family of singular flat 3-manifolds which we call translation prisms. Using ideas of Furstenberg and Veech, we connect results about weak mixing properties of flows on translation surfaces to ergodic properties of linear flows on translation prisms, and use this to obtain several results about unique ergodicity of these prism flows and related billiard flows. Furthermore, we construct explicit eigenfunctions for translation flows in pseudo-Anosov directions with Pisot expansion factors, and use this construction to build explicit examples of non-ergodic prism flows, and non-ergodic billiard flows in a right prism over a regular n-gons for n=7,9,14,16,18,20,24,30. This is joint work with Nicolas Bedaride, Pat Hooper, and Pascal Hubert.
Peter Walters asked in 1986 whether any uniquely ergodic homeomorphism with a non-atomic invariant probability measure admits a non-uniform GL(2,R) cocycle. We discuss the history and context of this question and then present joint work with Artur Avila that gives an affirmative answer to it.
For systems with an invariant probability measure we take zero measure sets and look at the limiting return times statistics for shrinking neighbourhoods. If those return statistics are suitably rescaled then one obtains a limiting statistics which will be compound Poisson. The associated parameters are found through the approximating sets. The methods we use allow for situations when the rescaling is not according to Kac's theorem.
Multifractal analysis studies how local scaling behavior varies across points in a dynamical system. In conformal dynamical systems, it describes the distribution of quantities such as Lyapunov exponents, pointwise dimensions, and Birkhoff averages on invariant sets. While this theory has been extensively developed for conformal systems, much less is known in the non-conformal setting. In this talk, we discuss the multifractal analysis of equilibrium states associated with Hölder continuous potentials for hyperbolic holomorphic endomorphisms of CP^k. In this setting, we obtain results that parallel those known in the conformal case. The key ingredient is a new dimension theory. This is based on joint works with F. Bianchi and with N. Dalaklis.
The stabilized automorphism group of a dynamical system (X,T) is the group of all self-homeomorphisms of X that commute with some power of T. While this is an algebraic object, we show that it captures rich dynamical information. We begin by characterizing the stabilized automorphism groups of odometers and Toeplitz subshifts, establishing an invariance property in these settings. We then extend our results to a broader class of minimal systems, proving that if two such systems have isomorphic stabilized automorphism groups and each has a non-trivial rational eigenvalue, then they must share the same set of rational eigenvalues. We further identify a class of systems for which the assumption of having a non-trivial rational eigenvalue can be removed. Finally, we generalize a known result for mixing shifts of finite type to include all irreducible shifts of finite type.
In this talk, we will consider random dynamical systems and group actions on surfaces that are given by diffeomorphisms. We will discuss about the absolutely continuity of stationary measures, the classification of orbit closures, and the exact dimensionality of stationary measures. This talk is mostly based on joint work with Aaron Brown, Davi Obata, and Yuping Ruan.
While many geodesic flows in negative curvature exhibit exponential decay of cor- relations for the Liouville measure and satisfy classical limit theorems, the study of similar properties for geodesic flows in nonpositive curvature is far from a global understanding. In this talk, I will discuss results regarding these topics for certain classes of geodesic flows in surfaces of nonpositive curvature, which include polyno- mial decay of correlations and standard/nonstandard central limit theorems. Joint work with Carlos Matheus and Ian Melbourne.
In this talk we discuss rigidity of the smooth centralizer of diagonal actions on (twisted) homogeneous spaces of semisimple Lie group. We will review some recent developments and focus on cases when the diagonal elements is very non-generic. More concretely, we shall show that for any non-trivial diagonal matrix acting on SLnR/\Gamma, n\ge 5, a perturbation is smoothly conjugate to a diagonal matrix if the centralizer stays the same. This represents joint work with Amie Wilkinson.