Entropy and its variational principle for noncompact metric spaces

TL;DR

This paper extends the concept of topological entropy to noncompact metric spaces, proves a variational principle linking different entropy notions, and applies it to automorphisms of nilpotent Lie groups showing their entropy vanishes.

Contribution

It introduces a natural extension of AKM-topological entropy for noncompact spaces and establishes a variational principle connecting topological, measure-theoretic, and metric entropies.

Findings

Topological entropy of automorphisms of simply connected nilpotent Lie groups always vanishes.

Classical entropy formulas for noncompact Lie group automorphisms are upper bounds.

The variational principle confirms the equality of different entropy measures in this setting.

Abstract

In the present paper, we introduce a natural extension of AKM-topological entropy for noncompact spaces and prove a variational principle which states that the topological entropy, the supremum of the measure theoretical entropies and the minimum of the metric theoretical entropies always coincide. We apply the variational principle to show that the topological entropy of automorphisms of simply connected nilpotent Lie groups always vanishes. This shows that the classical formula for the entropy of an automorphism of a noncompact Lie group is just an upper bound for its topological entropy.