From Barcode Entropy to Metric Entropy

TL;DR

This paper links barcode entropy with metric entropy, showing that barcode entropy provides a lower bound for metric entropy in certain invariant measures, refining previous bounds via the variational principle.

Contribution

It introduces pseudo-chord measures and establishes a lower bound of metric entropy by barcode entropy, refining the relationship between these entropies.

Findings

Barcode entropy bounds metric entropy from below.

The inequality refines the upper bound of barcode entropy by topological entropy.

Introduction of pseudo-chord measures for invariant measures.

Abstract

We establish a connection between barcode entropy and metric entropy. Namely, we show that the barcode entropy bounds the metric entropy from below for a measure from a specific class of invariant measures associated with a pair of Lagrangian or Legendrian submanifolds, which we call pseudo-chord measures. This inequality refines, via the variational principle, the previously known upper bound of barcode entropy by topological entropy.