# Topology of pre-images under Anosov endomorphisms

## TL;DR

This paper investigates the relationship between the density of pre-images and transitivity in Anosov endomorphisms, establishing conditions under which the inverse property holds for these systems.

## Contribution

It proves that under certain conditions, the inverse of dense pre-image sets implies transitivity for Anosov endomorphisms on closed manifolds.

## Key findings

- Pre-images are dense for a residual set of points in certain conditions.
- Anosov endomorphisms are covering maps, which aids in the analysis.
- The inverse property of pre-image density and transitivity is established under specific conditions.

## Abstract

For an endomorphism it is known that if all the points in the manifold have dense sets of pre-images then the dynamical system is transitive. The inverse has been shown for a residual set of points but the the exact inverse has not yet been investigated before. Here we are going to show that under some conditions it is true for Anosov endomorphisms on closed manifolds, by using the fact that Anosov endomorphisms are covering maps.