C^k-Robust transitivity for surfaces with boundary

TL;DR

This paper investigates the robustness of transitivity in surface diffeomorphisms with boundary, showing non-existence in C^1 but providing examples in C^k for k≥2, challenging existing conjectures.

Contribution

It proves the non-existence of C^1-robustly transitive diffeomorphisms on surfaces with boundary and constructs C^k-robustly transitive examples for k≥2, revealing new dynamics.

Findings

C^1-robust transitivity does not occur on surfaces with boundary.

Constructed C^k-robustly transitive diffeomorphisms for k≥2.

Blow-up of pseudo-Anosov diffeomorphisms yields C^2-robust topological mixing.

Abstract

We prove that C^1-robustly transitive diffeomorphisms on surfaces with boundary do not exist, and we exhibit a class of diffeomorphisms of surfaces with boundary which are C^k-robustly transitive, with k greater or equal than 2. This class of diffeomorphisms are examples where a version of Palis' conjecture on surfaces with boundary, about homoclinic tangencies and uniform hyperbolicity, does not hold in the C^2-topology. This follows showing that blow-up of pseudo-Anosov diffeomorphisms on surfaces without boundary, become C^2-robustly topologically mixing diffeomorphisms on a surfaces with boundary.