13 research outputs found
The Martingale approach after Varadhan and Dolpogpyat
No full textWe present, in the simplest possible form, the so called {\em martingale problem} strategy to establish limit
theorems. The presentation is specially adapted to problems arising in partially hyperbolic dynamical systems. We will
discuss a simple partially hyperbolic example with fast-slow variables and use the martingale method to prove an
averaging theorem and study fluctuations from the average. The emphasis is on ideas rather than on results. Also, no
effort whatsoever is done to review the vast literature of the field
Dinamica del gruppo di rinormalizzazione dei modelli di Potts gerarchici
No full textOggetto di questa tesi è il comportamento analitico delle funzioni termodinamiche di sistemi particolari, denominati modelli gerarchici.
Si tratta di modelli di Potts su reticoli che sono invarianti per una opportuna procedura di coarse-graining. Questa proprietà permette il calcolo esatto del gruppo di rinormalizzazione, dal quale si può ottenere una descrizione dei domini di analiticità delle funzioni termodinamiche dei sistemi ad essi associati. Tale descrizione è legata alla dinamica delle iterazioni di una opportuna mappa razionale in più variabili complesse che esprime l'operatore di rinormalizzazione associato al modello.
Si tratta di un argomento di ricerca estremamente nuovo e molti dei risultati ottenuti sono originali. All'esposizione dei risultati generali rigorosi si abbina inoltre una parte esemplificativa anche numerica, con il doppio fine di chiarire la trattazione e di esplicitare la sua generalità
Dynamics of Some Fermi–Ulam Models
No full textFermi acceleration is a mechanism which allows an interacting particle to extract arbitrarily large amounts of energy from a stationary environment. In the early 1940s, Fermi and Ulam designed simple dynamical systems to model this phenomenon, conducted numerical experiments, and obtained results supporting the ideas used to explain high-energy cosmic rays (cf. Fermi, 1949). In this talk, I will describe the dynamics of a number of dynamical systems inspired by their work. I will report on recent and ongoing work based on a combination of ideas from classical KAM (Kolmogorov–Arnold–Moser) theory and modern techniques in hyperbolic dynamics. This is part of a joint project with D. Dolgopyat
Abundance of escaping orbitsin a family of anti-integrable limitsof the standard map
No full textWe give quantitative results about the abundance of escaping orbits in a family of exact twist maps preserving Lebesgue measure on the cylinder T × R; geometrical features of maps of this family are quite similar to those of the well-known Chirikov-Taylor standard map, and in fact we believe that the techniques presented in this work can be further improved and eventually applied to studying ergodic properties of the standard map itself.
We state conditions which assure that escaping orbits exist and form a full Hausdorff dimension set. Moreover, under stronger conditions we can prove that such orbits are not charged by the invariant measure. We also obtain prove that, generically, the system presents elliptic islands at arbitrarily high values of the action variable and provide estimates for their total measure
AN INTEGRABLE DEFORMATION OF AN ELLIPSE OF SMALL ECCENTRICITY IS AN ELLIPSE
No full textAbstract. The classical Birkhoff conjecture says that the only integrable convex domains are circles and ellipses. In the paper we show that a version of this conjecture is true for small perturbations of ellipses of small eccentricity. 1
Marked Length Spectral determination of analytic chaotic billiards with axial symmetries
No full textWe consider billiards obtained by removing from the plane finitely many
strictly convex analytic obstacles satisfying the non-eclipse condition. The
restriction of the dynamics to the set of non-escaping orbits is conjugated to
a subshift, which provides a natural labeling of periodic orbits. We show that
under suitable symmetry and genericity assumptions, the Marked Length Spectrum
determines the geometry of the billiard table.Comment: 65 pages, 8 figure
