A persistently singular map of $\mathbb{T}^n$ that is $C^2$ robustly transitive but is not $C^1$ robustly transitive

TL;DR

The paper constructs a specific endomorphism of the n-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 an explicit example of a map that is $C^2$ robustly transitive but not $C^1$ robustly transitive, illustrating the nuanced behavior of dynamical robustness.

Findings

Existence of a $C^2$ robustly transitive map on $ T^n$

The map is not $C^1$ robustly transitive

Demonstrates the difference in robustness between $C^1$ and $C^2$ topologies

Abstract

Let $\mathcal{E}$ be the set of endomorphisms of the $n$-torus. We exhibit an example of a map such that is robustly transitive if $\mathcal{E}$ is endowed with the $C^2$ topology but is not robustly transitive if $\mathcal{E}$ is endowed with the $C^1$ topology.