Entropy and Ergodic Measures for Toral Automorphisms

TL;DR

This paper demonstrates that for all linear toral automorphisms, including non-hyperbolic cases, the entropies of ergodic measures densely fill the entire range from zero to the topological entropy.

Contribution

It establishes the density of ergodic measure entropies for all linear toral automorphisms, extending previous results to non-hyperbolic cases.

Findings

Entropies of ergodic measures are dense in [0, topological entropy] for all linear toral automorphisms.

Includes non-hyperbolic automorphisms, broadening the scope of previous density results.

Shows the rich diversity of ergodic measures in terms of entropy for these systems.

Abstract

We show that for every linear toral automorphism, especially the non-hyperbolic ones, the entropies of ergodic measures form a dense set on the interval from zero to the topological entropy.