Noticias en español
Boundedness of hyperbolic components in moduli space.
ABSTRACT: A complex rational map of degree at least 2 is hyperbolic if each of its critical points is attracted to an attracting cycle. For a fixed degree, the hyperbolic rational maps form an open set in the space of rational maps. This open set deduces an open set in the moduli space of rational maps, modulo the Möbius conjugacy. Each component of the deduced open set is a hyperbolic component. In this talk, I will present some precompactness results on hyperbolic components. In particular, I will focus on the space of quartic Newton maps. This is a joint work with Y. Gao.
Read MoreAutomorphisms and extended symmetries of number-theoretic positive entropy subshifts.
ABSTRACT: We will discuss one- and multidimensional subshifts constructed via number-theoretically defined subsets of the integers (e.g. the visible lattice points in the plane, k-free integers, etc.), focusing our interest on their groups of automorphisms and extended symmetries, which are naturally defined conjugacy invariants. This type of shift space exhibits symmetry rigidity (that is, its group of automorphisms is essentially trivial), but is compatible with positive entropy and shows interesting variations on their extended symmetry groups, which may be small (finite) or large...
Read MoreHeterodimensionality of skew-products with concave fiber maps.
ABSTRACT: I will discuss examples of skew-products with concave interval fiber maps over a certain subshift. Here the subshift occurs as the projection of those orbits that stay in a given neighborhood and gives rise to a new type of symbolic space which is (essentially) coded. The fiber maps have expanding and contracting regions. As a consequence, the skew-product dynamics has pairs of horseshoes of different types of hyperbolicity. In some cases, they dynamically interact due to the superimposed effects of the (fiber) contraction and expansion, leading to nonhyperbolic dynamics...
Read MoreVariational principles and equilibrium states for (semi)group actions.
ABSTRACT: The search for a thermodynamic formalism for dynamical systems aims to prove the existence of invariant probability measures which maximize the topological pressure, besides reporting on their statistical properties. A special feature of the classical thermodynamic formalism for Ruelle expanding maps, which becomes an important tool in many applications including applications to multifractal analysis or large deviations, is the upper-semicontinuity of the Kolmogorov-Sinai entropy map. Simple examples illustrate that this property breaks down in the absence of expansiveness. In this...
Read MorePhase Transitions in Statistical Mechanics, Countable Markov Shifts, and Operator Algebras.
ABSTRACT We discuss the concept of phase transition in different settings and explore the connections among them. We will give an overview of the results about phase transitions in Ising-like models showing concrete examples where the lack of the analyticity of the pressure it is not connected with the changes in the set of Gibbs (DLR) measures. After, we describe two new types of phase transitions on countable Markov shifts and on their generalized versions: the volume-type and length-type. We give examples of potentials on countable Markov shifts where the set of conformal measures (a...
Read MoreAcerca de la derivada de una conjugación entre dos transformaciones de tipo Gauss.
ABSTRACT: A principios del siglo XX, H. Minkowski definió una función (“Question-Mark function”, Q) que establece una correspondencia uno a uno entre racionales y soluciones reales en [0,1] de polinomios cuadráticos con coeficientes enteros. Desde un punto de vista dinámico, Q es una conjugación topológica entre la transformación de Farey y el Tent-map. Entre sus propiedades se encuentra también que Q es una función singular, es decir, Lebesgue-c.t.p su derivada existe y es igual a cero (Salem,1943). En trabajos más recientes, Kesseböhmer y Stratmann calcularon explícitamente las...
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