A persistently singular map of $\mathbb{T}^n$ that is $C^1$ robustly transitive

TL;DR

This paper constructs an example of an endomorphism on the n-torus that remains singular and transitive under small $C^1$ perturbations, demonstrating robust transitivity in a singular setting.

Contribution

It provides the first example of a $C^1$ robustly transitive endomorphism of the n-torus that is persistently singular, expanding understanding of dynamical robustness.

Findings

Existence of a $C^1$ robustly transitive singular endomorphism on $ T^n$.

Persistence of singularity under $C^1$ perturbations.

Demonstration of robust transitivity in a singular context.

Abstract

Consider the set $E$ of endomorphisms of the n-torus endowed with the $C^1$ topology. A point in $E$ that is persistently singular and robustly transitive is exhibited.