Barcode entropy of geodesic flows

TL;DR

This paper introduces the concept of barcode entropy for geodesic flows on closed Riemannian manifolds, establishing its relationship with topological entropy and providing new insights into the complexity of geodesic dynamics.

Contribution

It defines barcode entropy for geodesic flows and proves its bounds relative to topological entropy, including equality on surfaces, with a novel crossing energy theorem as a key tool.

Findings

Barcode entropy bounds topological entropy from below.

On surfaces, barcode entropy equals topological entropy.

Crossing energy theorem for gradient flow lines is established.

Abstract

We introduce and study the barcode entropy for geodesic flows of closed Riemannian manifolds, which measures the exponential growth rate of the number of not-too-short bars in the Morse-theoretic barcode of the energy functional. We prove that the barcode entropy bounds from below the topological entropy of the geodesic flow and, conversely, bounds from above the topological entropy of any hyperbolic compact invariant set. As a consequence, for Riemannian metrics on surfaces, the barcode entropy is equal to the topological entropy. A key to the proofs and of independent interest is a crossing energy theorem for gradient flow lines of the energy functional.