The entropy of co-compact open covers

TL;DR

This paper introduces co-compact entropy as a topological invariant applicable to perfect mappings on Hausdorff spaces, generalizing classical entropy concepts to non-compact, non-metrizable spaces and establishing its properties and bounds.

Contribution

It defines co-compact entropy for perfect mappings on Hausdorff spaces, extending topological entropy beyond compact and metric spaces, and proves it is a lower bound for Bowen's metric-dependent entropy.

Findings

Co-compact entropy is zero for the linear system (R, f) with f(x)=2x.

Co-compact entropy does not exceed the entropy of the whole system.

Co-compact entropy is a lower bound for Bowen's entropy.

Abstract

Co-compact entropy is introduced as an invariant of topological conjugation for perfect mappings defined on any Hausdorff space(compactness and metrizability not necessarily required). This is achieved through the consideration of co-compact covers of the space. The advantages of co-compact entropy include: 1) it does not require the space to be compact, and thus generalizes Adler, Konheim and McAndrew's topological entropy of continuous mappings on compact dynamical systems, and 2) it is an invariant of topological conjugation, compared to Bowen's entropy that is metric-dependent. Other properties of co-compact entropy are investigated, e.g., the co-compact entropy of a subsystem does not exceed that of the whole system. For the linear system (R, f) defined by f(x) = 2x, the co-compact entropy is zero, while Bowen's entropy for this system is at least log 2. More general, it is found that co-compact entropy is a lower bound of Bowen's entropies, and the proof of this result generates the Lebesgue Covering Theorem to co-compact open covers of non-compact metric spaces, too.