An example of a map which is $C^2$ robustly transitive but not $C^1$ (robustly transitive)

TL;DR

This paper constructs a specific example of a map on the 2-torus that remains transitive under $C^2$ perturbations but not under $C^1$ perturbations, highlighting differences in robustness at different smoothness levels.

Contribution

It provides the first known example of an endomorphism that is $C^2$-robustly transitive but not $C^1$-robustly transitive, illustrating a nuanced distinction in dynamical robustness.

Findings

Demonstrates existence of a $C^2$-robustly transitive endomorphism

Shows this map is not $C^1$-robustly transitive

Highlights the difference in robustness between $C^2$ and $C^1$ perturbations

Abstract

The aim of this work is to exhibit an example of an endomorphism of $\T^{2}$ which is $C^2$-robustly transitive but not $C^1$-robustly transitive.