Robust transitivity and domination for endomorphisms displaying critical points

TL;DR

This paper investigates the structure of robustly transitive endomorphisms on closed manifolds, revealing conditions under which they must have a dominated splitting or be local diffeomorphisms, thus providing topological obstructions.

Contribution

It establishes a link between robust transitivity and dominated splittings or local diffeomorphisms, offering new insights into the structure of endomorphisms with critical points.

Findings

Robustly transitive endomorphisms either have a non-trivial dominated splitting or are local diffeomorphisms.

Topological obstructions are identified for the existence of such endomorphisms.

Analysis of the kernel of the differential and recurrence to critical points is crucial for the results.

Abstract

We show that robustly transitive endomorphisms of a closed manifolds must have a non-trivial dominated splitting or be a local diffeomorphism. This allows to get some topological obstructions for the existence of robustly transitive endomorphisms. To obtain the result we must understand the structure of the kernel of the differential and the recurrence to the critical set of the endomorphism after perturbation.