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Fractal geometry of non-uniformly hyperbolic horseshoes

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 Poincaré 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.