Robust Transitivity for Endomorphisms

TL;DR

This paper investigates conditions under which endomorphisms on the n-dimensional torus maintain dense orbits despite perturbations, focusing on volume expansion and orbit behavior within expanding regions.

Contribution

It provides sufficient conditions for robust transitivity of volume-expanding endomorphisms on the torus, covering a broad class of examples.

Findings

Endomorphisms with volume expansion are robustly transitive.

Large connected arcs contain points with orbits in expanding regions.

Conditions ensure persistence of dense orbits under perturbations.

Abstract

We address the problem about under what conditions an endomorphism having a dense orbit, verifies that a sufficiently close perturbed map also exhibits a dense orbit. In this direction, we give sufficient conditions, that cover a large class of examples, for endomorphisms on the n-dimensional torus to be robustly transitive: the endomorphism must be volume expanding and any large connected arc must contain a point such that its future orbit belong to an expanding region.