Noticias en español
Analogies between the geodesic flow on a negatively curved manifold and countable Markov shifts
ABSTRACT: By the work of Bowen and Ratner it is known that the geodesic flow on a compact negatively curved manifold can be modeled as a suspension flow over a subshift of finite type. Unfortunately, a symbolic representation is not available if the manifold is non-compact. In this talk I will briefly explain some recent developments on the study of the thermodynamic formalism of the geodesic flow on non-compact negatively curved manifolds. Surprisingly some of the methods used to understand the geodesic flow have consequences to the theory of countable Markov shifts. I will explain such...
Read MoreNorm and pointwise averages of multiple ergodic averages and applications
ABSTRACT: Via the study of multiple ergodic averages for a single transformation, Furstenberg, in 1977, was able to provide an ergodic theoretical proof of Szemerédi’s theorem, i.e., every subset of natural numbers of positive upper density contains arbitrarily long arithmetic progressions. We will present some recent developments in the area for more general averages, e.g., for multiple commuting transformations with iterates along specific classes of integer valued sequences. We will also get numerous applications of the aforementioned study to number theory, as we will present the...
Read MoreMultidimensional continued fractions and symbolic dynamics for toral translations
ABSTRACT: We give a dynamical, symbolic and geometric interpretation to multi-dimensional continued fractions algorithms. For some strongly convergent algorithms, the construction gives symbolic dynamics of sublinear complexity for almost all toral translations; it can be used to obtain a symbolic model of the diagonal flow on lattices in $\mathbb R^3$.
Read MoreEntropías intermedias y temperatura nula en curvatura negativa
ABSTRACT: Un problema bastante general en teoría ergódica consiste en estudiar al conjunto de entropías de un sistema dinámico respecto a sus medidas ergódicas. Katok conjeturó que dicho conjunto contiene al intervalo $[0,h_{top}(f))$ en el caso de difeomorfismos suaves en variedades compactas. Si bien la conjetura permanece abierta, muchos avances se han logrado a la fecha. Se conoce, por ejemplo, que el flujo geodésico en variedades compactas a curvatura negativa verifica esta propiedad. La demostración de esto último recae en la realización del flujo geodésico como un flujo de...
Read MoreInvariant Random Subgroups of Full Groups of Bratteli diagrams
ABSTRACT: In the talk, we will classify the ergodic invariant random subgroups (IRS) of simple AF full groups. AF full groups arise as the transformation groups of Bratteli diagrams that preserve the cofinality of infinite paths in the diagram. AF full groups are complete (algebraic) invariants for the isomorphism of Bratteli diagrams. Given a simple AF full group G, we will prove that every ergodic IRS of G arises as the stabilizer distribution of a diagonal action on X^n for some n, where X is the path-space of the Bratteli diagram associated to G. This is joint work with Artem...
Read MoreLocal rules for planar tilings
ABSTRACT: The cut and project method is one of the prominent method to define quasiperiodic tilings. In order to model quasicrystals, where energetic interactions are only short range, it is important to know which of these tilings can be characterized by local configurations (in dynamical terms: which of these tiling spaces are of finite type or sofic). In this talk we shall review known results, in particular those obtained these last years with Nicolas Bedaride and Mathieu Sablik.
Read More