On the growth of the Floer barcode

Erman Çineli; Viktor L. Ginzburg; Başak Z. Gürel

doi:10.3934/jmd.2024007

Article Contents

2024, Volume 20: 275-298. Doi: 10.3934/jmd.2024007

This volume Previous Article Invariant measures of the topological flow and measures at infinity on hyperbolic groups Next Article The strong closing lemma and Hamiltonian pseudo-rotations

On the growth of the Floer barcode

1.
Institut de Mathématiques de Jussieu - Paris Rive Gauche (IMJ-PRG), 4 place Jussieu, Boite Courrier 247, 75252 Paris Cedex 5, France
2.
Department of Mathematics, UC Santa Cruz, Santa Cruz, CA 95064, USA
3.
Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA

EÇ: Partially supported by Starting Grant 851701 via a postdoctoral fellowship
VLG: Partially supported by Simons Foundation Collaboration Grant 581382
BZG: Partially supported by NSF CAREER award DMS-1454342 and Simons Foundation Collaboration Grant 855299

Received: July 18, 2022

Revised: March 19, 2024

Published online: June 5, 2024

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Abstract

This paper is a follow-up to the authors' recent work on barcode entropy. We study the growth of the barcode of the Floer complex for the iterates of a compactly supported Hamiltonian diffeomorphism. In particular, we introduce sequential barcode entropy which has properties similar to barcode entropy, bounds it from above and is more sensitive to the barcode growth. In the same vein, we explore another variant of barcode entropy based on the total persistence growth and revisit the relation between the growth of periodic orbits and topological entropy. We also study the behavior of the spectral norm, aka the $ \gamma $-norm, under iterations. We show that the $ \gamma $-norm of the iterates is separated from zero when the map has sufficiently many hyperbolic periodic points and, as a consequence, it is separated from zero $ C^\infty $-generically in dimension two. We also touch upon properties of the barcode entropy of pseudo-rotations and, more generally, $ \gamma $-almost periodic maps.

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Mathematics Subject Classification: Primary: 53D40; Secondary: 37J11, 37J46.

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