Expansive systems on lattices

TL;DR

This paper extends the concept of expansive dynamical systems to the setting of distributive lattices, generalizing classical theorems and analyzing entropy in non-Hausdorff shifts.

Contribution

It generalizes Mañé's and Utz's theorems to lattice-based systems and explores entropy calculations for non-Hausdorff shifts.

Findings

Generalized Mañé's Theorem for lattice systems

Generalized Utz's Theorem on finite-dimensionality

Calculated entropy for non-Hausdorff shifts

Abstract

We study expansive dynamical systems in the setting of distributive lattices and their automorphisms, the usual notion of expansiveness for a homeomorphism of a compact metric space being the particular case when the lattice is the topology of the phase space ordered by inclusion and the automorphism the one induced by the homeomorphism, mapping open sets to open sets. We prove in this context generalizations of Ma\~n\'e's Theorem and Utz's Theorem about the finite dimensionality of the phase space of an expansive system, and the finiteness of a space supporting a positively expansive homeomorphism, respectively. We also discuss the notion of entropy in this setting calculating it for the non-Hausdorff shifts.