# Partially hyperbolic endomorphisms with expanding linear part

## TL;DR

This paper investigates the transitivity of certain partially hyperbolic endomorphisms on the two torus, establishing conditions under which these maps are robustly transitive and characterizing their dynamical behavior.

## Contribution

It introduces a new criterion for robust transitivity based on the Jacobian and eigenvalues, and provides a complete dichotomy for special cases of these endomorphisms.

## Key findings

- Robust transitivity when Jacobian exceeds the largest eigenvalue modulus
- Complete classification of special partially hyperbolic endomorphisms
- Existence of wandering or periodic annuli in unstable foliations

## Abstract

In this paper we study transitivity of partially hyperbolic endomorphisms of the two torus whose action in the first homology has two integer eigenvalues of moduli greater than one. We prove that if the Jacobian is everywhere greater than the modulus of the largest eigenvalue, then the map is robustly transitive. For this we introduce Blichfedt's theorem as a tool for extracting dynamical information from the action of a map in homology. We also treat the case of specially partially hyperbolic endomorphisms, for which we obtain a complete dichotomy: either the map is transitive and conjugated to its linear part, or its unstable foliation must contain an annulus which may either be wandering or periodic.