Karma Dajani
Title: Dynamical behaviour of alternate base expansions
Abstract: Let \(\beta=(\beta_0,\ldots,\beta_{p-1})\) be a \(p\)-tuple of real numbers with \(\beta_i>1\). We consider a generalization of the \(\beta\) -expansion by applying cyclically the bases \(\beta_0,\ldots,\beta_{p-1}\). We refer to the resulting expansion as an (alternate base) \(\beta\) -expansion. Just as in the case of the classical \(\beta\) -expansion one has typically uncountably many expansions. In this talk, we will concentrate on the greedy expansion, we introduce a dynamical system generating them and study its ergodic properties. If time permits, we then de ne the alternate lazy expansion and show that the dynamical system underlying such expansions is isomorphic to the greedy counterpart. We end (conditioned on time) by comparing the alternate base \(\beta\) -expansion with the greedy \(\beta_0,\ldots,\beta_{p-1}\)-expansion with digits in some special set, and characterize when these expansions are the same. This is a joint work with Emilie Charlier and Celia Cisternino.
[button url=”http://eventos.cmm.uchile.cl/edynamicsxi/wp-content/uploads/sites/115/2021/04/DajaniExpandingShort.pdf” target=”self” style=”flat” background=”#C1272D” color=”#FFFFFF” size=”3″ wide=”no” center=”no” radius=”auto” icon=”icon: file-pdf-o” icon_color=”#FFFFFF” text_shadow=”none”]Slides (PDF)[/button] [button url=”http://eventos.cmm.uchile.cl/edynamicsxi/wp-content/uploads/sites/115/2021/04/DajaniExpandingLong.pdf” target=”self” style=”flat” background=”#C1272D” color=”#FFFFFF” size=”3″ wide=”no” center=”no” radius=”auto” icon=”icon: file-pdf-o” icon_color=”#FFFFFF” text_shadow=”none”]Slides for Break Out Room (PDF)[/button]Pascal Hubert
Title: Minimality, unique ergodicity and rigidity of families of interval exchange transformations
Abstract: In this talk, I will discuss joint results with Sébastien Ferenczi. In a few papers, we consider families of interval exchange transformations (iet) that contain square-tiled iet and Veech’s original example: a minimal and non uniquely ergodic iet. We have new results about rigidity. Working on these questions has led us to study old problems questions concerning minimality and unique ergodicity of skew products over rotations and iet. The proofs mix ideas from combinatorics, geometry and number theory.
Tianyu Wang
Title: Multifractal analysis in surface with no focal points
Abstract: In this presentation, we will work with the geodesic flow on a compact rank-one surface without focal points, which is non-uniformly hyperbolic. The main goal is to study the topological size of Lyapunov level sets with different exponents. We will show that, even with the existence of phase transition, we are able to obtain a complete description of the topological entropy of all non-empty Lyapunov level sets, and a lower bound for the Hausdorff dimension of such sets. The main idea is to make use of thermodynamic formalism of such flow with respect to the geometric-t potential, provided by Chen, Kao and Park, before phase transition, and a symbolic exhaustion of hyperbolicity, which is motivated by the work of Burns and Gelfert, at phase transition. The content is based on a joint work with Kiho Park.
David Kerr
Title: Entropy, orbit equivalence, and sparse connectivity
Abstract: It was shown by Tim Austin that if an orbit equivalence between probability-measure-preserving actions of finitely generated amenable groups is integrable then it preserves entropy. I will discuss some joint work with Hanfeng Li in which we show that the same conclusion holds for the maximal sofic entropy when the acting groups are countable and sofic and contain an amenable \(w\)-normal subgroup which is not locally virtually cyclic, and that it is moreover enough to assume that the Shannon entropy of the cocycle partitions is finite (what we call Shannon orbit equivalence). It follows that two Bernoulli actions of a group in the above class are Shannon orbit equivalent if and only if they are conjugate.
Asaf Katz
Title: Measure rigidity of Anosov flows via the factorisation method
Abstract: Anosov flows are central objects in dynamics, generalizing the basic example of a geodesic flow over a negatively curved surface. In the talk we will introduce those flows and their dynamical behavior. Moreover, we show how the factorization method, pioneered by Eskin and Mirzakhani in their groundbreaking work about measure rigidity for the moduli space of translation surfaces, can be adapted to smooth ergodic theory and in particular towards the study of Anosov flows. Using this adaption, we show that for a quantitatively non-integrable Anosov flow, every generalized u-Gibbs measure is absolutely continuous with respect to the whole unstable manifold. In the talk I will discuss the factorization method, the relations to previous works (Eskin-Mirzakhani, Eskin-Lindenstrauss) and the result together with some examples and applications.
Jayadev Athreya
Title: Local geometry of random geodesics on negatively curved surfaces
Abstract: In joint work with S. Lalley, J. Sapir, and M. Wroten, we show that the tessellation of a compact, negatively curved surface induced by a typical long geodesic segment, when properly scaled, looks locally like a Poisson line process. This implies that the global statistics of the tessellation — for instance, the fraction of triangles — approach those of the limiting Poisson line process.