Some hyperbolicity revisited and robust transitivity

TL;DR

This paper revisits the concept of Some Hyperbolicity, generalizes its definition to include symplectic systems, and constructs robustly transitive diffeomorphisms with mixed center behavior.

Contribution

It introduces a broader definition of Some Hyperbolicity applicable to symplectic systems and constructs new examples of robust transitivity derived from Anosov diffeomorphisms.

Findings

Generalized the notion of Some Hyperbolicity to symplectic contexts

Constructed $C^1$ robustly transitive diffeomorphisms with mixed behavior

Extended applicability of hyperbolic concepts to broader dynamical systems

Abstract

In this article we revisit the notion of \textit{Some Hyperbolicity} introduced by Pujals and Sambarino in \cite{PuSa}. We present a more general definition, that in particular can be applied to the symplectic context (something that the previous couldn't). As an application we construct $C^1$ robustly transitive derived from Anosov diffeomorphisms with mixed behaviour on center leaves.