Advances on Stable Ergodicity of Toral Automorphisms

TL;DR

This paper proves that all ergodic automorphisms of the N-dimensional torus with a 2D center are stably ergodic, extending previous results and providing a new minimality criterion for the proof.

Contribution

It generalizes prior work by removing algebraic restrictions and establishes stable ergodicity for a broader class of toral automorphisms.

Findings

All ergodic automorphisms with 2D center are stably ergodic.

Includes all automorphisms in dimensions N≤5 or N=7.

Introduces a minimality criterion as the core of the proof.

Abstract

We prove that all ergodic automorphisms of the $N$-dimensional torus with two dimensional center are stably ergodic. This includes all ergodic automorphisms in dimension $N\leq 5$ or $N=7$. This generalizes a previous result of Rodriguez-Hertz, that required an additional algebraic condition on the carachteristic polynomial of the linear automorphism. The core of the proof is a minimality criterion.