Alejandro Mesón

Invariant of dynamical systems: A generalized entropy

In this work the concept of entropy of a dynamical system, as given by Kolmogorov, is generalized in the sense of Tsallis. It is shown that this entropy is an isomorphism invariant, being complete for Bernoulli schemes.

Geometric constructions and multifractal analysis for boundary hyperbolic maps

In this paper we present geometric constructions based on the method by Series for coding geodesics in the hyperbolic disc. For this are used certain special types of one-dimensional maps defined on the boundary of the hyperbolic plane. These types of maps do not require additional assumptions to ensure the existence of Gibbs states. We connect geometrical constructions from limit sets of Fuchsian groups with multifractal analysis by modifying the definition of a local entropy by Takens and Verbitski.

An approach to the problem of phase transitions in the continuum

Based on the spectral properties of Ruelle transfer operators, we establish conditions for the absence of phase transitions in the continuum. Our approach differs in several aspects from those which use Kirkwood–Salsburg and Kirkwood–Ruelle operators. In particular the results follow in a concise and relatively direct way.

On the topological entropy of the irregular part of v-statistics multifractal spectra

Abstract Let (X; d) be a compact metric space and f : X → X, if Xr is the product of r—copies of X,r ≥ 1, and Φ : Xr → R then the multifractal decomposition for V –statistics is given by The irregular part, or historic set, of the spectrum is the set points x ∊ X; for which the limit does not exist. In this article we prove that for dynamical systems with specification, the irregular part of the V ≥statistics spectrum has topological entropy equal to that of the whole space X.

On the Irregular Part of V -Statistics Multifractal Spectra for Systems with Non-Uniform Specification

Abstract Let (X, f) be a dynamical system with X a compact metric space. Let Xr be the product of r-copies of X, r≥ 1, and Φ: Xr → R. The multifractal decomposition for V –statistics for Φ, f is defined as .The set of points x ∊ X, for which the limit does not exist is called the irregular part, or historic set, of the spectrum. In this article we analyze the irregular part of the V -statistics for systems satisfying a weak form of the known Bowen specification property, called the non-uniform specification property. This concept was introduced by P. Varandas and allows to work in a nonuniformly hyperbolic context.

Finer Structure of the Multifractal Spectrum of Local Entropies

Abstract We consider a finer decomposition for the multifractal spectrum of local entropies ε (α) =: htop (f, Kα := {x : hμ (f, x) = α}). This consists in taking a partition for each multifractal set of the form If ε (α, V) denotes htop (f, Kα V), then is established that ε (α, V) = ε (α) for a set of values of V.

Higher Multifractal Analysis for Simultaneous Actions of Fuchsian Groups

Abstract We consider groups Γ1, Γ2, …, Γ d acting on the hyperbolic plane H 2 and by the method of Series for coding hyperbolic geodesics we can do simultaneous multifractal decompositions from the action of these groups. We obtain a variational formula for a dimension of the multifractal sets which generalizes that presented in our previous article for d = 1.

Geodesic flow on an hyperboloid model and the two-body motion under repulsive Coulomb forces

We show that (in n dimension) the flow defined by the Hamiltonian system for two charged particles of the same sign, is mapped into the geodesic flow over the non-Euclidean space: Hn+⊔∂Hn−−{P}, say the hyperboloid with the axis along e0 which besides having its “boundaries at infinite” identified, it is also punctured at the point P≡(ξ0,ξ1,…,ξn)=(−1,0,…,0) (corresponding to the collision states) and whose metric is of negative curvature.

Large deviations for multiple ergodic averages

Abstract The main purpose of this work is to estimate how multiple ergodic averages appart from a given quantity. This problem can be studied by describing a large deviation process for empirical measures as obtained by using the contraction principle. The case of single ergodic averages for empirical measures was already studied by Pfister and Sullivan [Nonlinarity, 10 (2005) 237-261]. To have a more complete picture on empirical measures and V– statistics, we estimate the size of the sets GK = {x : Lr (x) ⊂ K }, where Lr(x) is the limit-point set of the sequence of empirical measures and K is a compact subset of ℳ(Xr) with ℳ(X) the set of measures on X. In pasrticular, we obtain a variational formula for the topological entropy of Gk. The result of this work about the dimension of the sets Gk can be compared with the one recently circulated by Fan, Schemeling and Wu [arXiv:1206.3214v1 (2012)].

Ergodic theorem for amenable groups and weakly integrable functions

Fil: Meson, Alejandro Mario. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Centro Cientifico Tecnologico Conicet - La Plata. Instituto de Fisica de Liquidos y Sistemas Biologicos. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Instituto de Fisica de Liquidos y Sistemas Biologicos; Argentina