On the geometry of horseshoes in higher dimensions

Abstract

The criterion of the recurrent compact set was introduced by Moreira and Yoccoz to prove that stable intersections of regular Cantor sets on the real line are dense in the region where the sum of their Hausdorff dimensions is bigger than 1. We adapt this concept to the context of horseshoes in ambient dimension higher than 2 and prove that horseshoes with upper stable dimension bigger than 1 satisfy, typically and persistently, the adapted criterion of the recurrent compact set. As consequences we show some persistent geometric properties of these horseshoes. In particular, typically and persistently, horseshoes with upper stable dimension bigger than 1 present blenders.