Some Generic Properties of Partially Hyperbolic Endomorphisms

TL;DR

This paper investigates the topological properties of partially hyperbolic endomorphisms, identifying obstructions to conjugacy with linear models and demonstrating generic properties in higher-dimensional manifolds.

Contribution

It introduces new topological obstructions to center leaf conjugacy and characterizes generic properties of partially hyperbolic endomorphisms in higher dimensions.

Findings

Obstructions related to the number of center directions.

Existence of a dense subset of non-special endomorphisms.

Examples illustrating the obstructions.

Abstract

In this work, we deal with a notion of partially hyperbolic endomorphism. We explore topological properties of this definition and we obtain, among other results, obstructions to get center leaf conjugacy with the linear part, for a class of partially hyperbolic endomorphism $C^1-$sufficiently close to a hyperbolic linear endomorphism. Indeed such obstructions are related to the number of center directions of a point. We provide examples illustrating these obstructions. We show that for a manifold $M$ with dimension $n \geq 3,$ admitting a non-invertible partially hyperbolic endomorphisms, there is a $C^1$ open and dense subset $\mathcal{U}$ of all partially hyperbolic endomorphisms with degree $d \geq n,$ such that any $f \in \mathcal{U}$ is neither $c$ nor $u$ special.