Barcode entropy of geodesic flows

Viktor L. Ginzburg (1) , Basak Z. Gurel (2) , Marco Mazzucchelli (3)

1 UC Santa Cruz - University of California [Santa Cruz]
2 UCF - University of Central Florida [Orlando]
3 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

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.

Domaines: Géométrie symplectique [math.SG]; Géométrie différentielle [math.DG]; Systèmes dynamiques [math.DS]

Viktor L. Ginzburg, Basak Z. Gurel, Marco Mazzucchelli. Barcode entropy of geodesic flows. Journal of the European Mathematical Society, In press. ⟨ensl-03895619⟩