Work

Barcode entropy of geodesic flows

Year: 2024

Type: article

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.

Source: Journal of the European Mathematical Society

DOI: 10.4171/jems/1572

Authors: Viktor L. Ginzburg, Başak Z. Gürel, Marco Mazzucchelli

Institutions: University of California, Santa Cruz, University of Central Florida, École Normale Supérieure de Lyon

Language: English

Topic: Mathematical Dynamics and Fractals

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: diamond

Funders: National Science Foundation, École Normale Supérieure, Agence Nationale de la Recherche, École Normale Supérieure de Lyon