skip to main content
US FlagAn official website of the United States government
dot gov icon
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
https lock icon
Secure .gov websites use HTTPS
A lock ( lock ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.

Explore Research Products in the PAR
It may take a few hours for recently added research products to appear in PAR search results.


Title: On the growth of the Floer barcode
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 partic- ular, we introduce sequential barcode entropy which has properties similar to barcode entropy, bounds it from above and is more sensitive to the bar- code 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 behav- ior of the spectral norm, aka the γ-norm, under iterations. We show that the γ-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 ∞-generically in dimension two. We also touch upon properties of the barcode entropy of pseudo-rotations and, more generally, γ-almost periodic maps.  more » « less
Award ID(s):
2304206 1454342 2304207
PAR ID:
10628190
Author(s) / Creator(s):
; ;
Publisher / Repository:
American Institute of Mathematical Sciences
Date Published:
Journal Name:
Journal of Modern Dynamics
Volume:
20
Issue:
0
ISSN:
1930-5311
Page Range / eLocation ID:
275 to 298
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. ; ; (, Mathematische Zeitschrift)
    We study topological entropy of compactly supported Hamiltonian diffeomorphisms from a perspective of persistent homology and Floer theory. We introduce barcode entropy, a Floer-theoretic invariant of a Hamiltonian diffeomorphism, measuring exponential growth under iterations of the number of not-too-short bars in the barcode of the Floer complex. We prove that the barcode entropy is bounded from above by the topological entropy and, conversely, that the barcode entropy is bounded from below by the topological entropy of any hyperbolic invariant set, e.g., a hyperbolic horseshoe. As a consequence, we conclude that for Hamiltonian diffeomorphisms of surfaces the barcode entropy is equal to the topological entropy. 
    more » « less
  2. ; ; (, Journal of the European Mathematical Society)
    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. 
    more » « less
  3. ; ; ; (, Selecta Mathematica)
    In this paper we continue investigating connections between Floer theory and dynamics of Hamiltonian systems, focusing on the barcode entropy of Reeb flows. Barcode entropy is the exponential growth rate of the number of not-too-short bars in the Floer or symplectic homology persistence module. The key novel result is that the barcode entropy is bounded from below by the topological entropy of any hyperbolic invariant set. This, combined with the fact that the topological entropy bounds the barcode entropy from above, established by Fender, Lee and Sohn, implies that in dimension three the two types of entropy agree. The main new ingredient of the proof is a variant of the Crossing Energy Theorem for Reeb flows. 
    more » « less
  4. ; (, The Astrophysical Journal)
    Abstract The parametric decay instability of Alfvén waves has been widely studied, but few investigations have examined wave packets of finite size and the effect of different boundary conditions on the growth rate. In this paper, we perform a linear analysis of circular and arc-polarized wave trains and wave packets in periodic and open boundary systems in a low-βplasma. We find that both types of wave are 3–5 times more stable in open boundary conditions compared to periodic. Additionally, once the wave packet widthℓbecomes smaller than the system sizeL, the growth rate decreases nearly with a power lawγ∝ℓ/L. This study demonstrates that the stability of a pump wave cannot be separated from the laboratory settings, and that the growth rate of daughter waves depends on the conditions downstream and upstream of the pump wave and on the fraction of volume it fills. Our results can explain simulations and experiments of localized Alfvén waves. They also suggest that Alfvénic fluctuations in the solar wind, including sharp impulses known as switchbacks, can be more stable than traditional theory suggests depending on wind conditions. 
    more » « less
  5. ; ; (, Pacific Journal of Mathematics)
    Our main result is that for any closed symplectic manifold, the spectral norm of the iterates of a Hamiltonian diffeomorphism is locally uniformly bounded away from zero C-infinity generically. 
    more » « less
  • Citation Formats
  • MLA
  • APA
  • Chicago
  • BibTeX
  • Save / Share this Record
  • Send to Email