Fractal geometry of non-uniformly hyperbolic horseshoes
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Publication:5168139
zbMATH Open1351.37179arXiv1311.1452MaRDI QIDQ5168139
Publication date: 3 July 2014
Abstract: This article is an expanded version of some notes for my talk at the ``Ergodic Theory and Dynamical Systems Workshop (from March 22 to March 25, 2012) held at the Department of Mathematics of the University of North Carolina at Chapel Hill. In the aforementioned talk, it was discussed some recent results on the fractal geometry of certain objects -- emph{non-uniformly hyperbolic horseshoes} -- constructed by Jacob Palis and Jean-Christophe Yoccoz in their recent emph{tour-de-force} work around emph{heteroclinic bifurcations of surface diffeomorphisms}. The goal of the present article is two-fold. The first part will be a modest survey on the history of the study of homoclinic/heteroclinic bifurcations of surface diffeomorphisms: starting from the seminal works of Henri Poincar'e on Celestial Mechanics, we will recall some landmark results on bifurcations of homoclinic/heteroclinic tangencies associated to emph{uniformly hyperbolic horseshoes}. Our discussions in this (first) part will culminate with a brief presentation of the scheme of the proof of the main theorems of J. Palis and J.-C. Yoccoz on this subject. In particular, we will highlight the main features of the so-called non-uniformly hyperbolic horseshoes, an object at the heart of the work of J. Palis and J.-C. Yoccoz. Then, the second part will be a sort of research announcement where we will discuss some results obtained in collaboration with J. Palis and J.-C. Yoccoz on the (fractal) geometry of non-uniformly hyperbolic horseshoes.
Full work available at URL: https://arxiv.org/abs/1311.1452
Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces (37E30) Homoclinic and heteroclinic orbits for dynamical systems (37C29) Dimension theory of smooth dynamical systems (37C45) Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory (37-02)
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