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PreprintCC BY 4.0arXiv

Robustness of topological entropy under small area deformations

  • Marcelo R. R. Alves
  • Matthias Meiwes
  • Beomjun Sohn
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Abstract

In this paper, we establish a new type of stability phenomenon for the topological entropy of Hamiltonian diffeomorphisms of closed surfaces. For a closed surface endowed with an area form (Σ,ω) and a Hamiltonian diffeomorphism ϕ of (Σ,ω), we show that for every ε>0 there exists A=A(ϕ,ε)>0 such that htop(ϕ') > htop(ϕ)−ε for every Hamiltonian diffeomorphism ϕ' obtained from ϕ by a deformation supported in a disjoint union of disks, each of area less than A. In particular, if htop(ϕ)>0, then ϕ cannot be made to have zero entropy by an area-preserving deformation supported in disks of small area. This follows from the new braid stability result established in this paper with respect to the spectral distance recently introduced by Connery-Grigg.

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Cite

BibTeXRIS
BibTeX
@misc{alves2026robustness,
  title = {{Robustness of topological entropy under small area deformations}},
  author = {Marcelo R. R. Alves and Matthias Meiwes and Beomjun Sohn},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.31132},
  eprint = {2608.31132},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.31132},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Robustness of topological entropy under small area deformations
AU  - Marcelo R. R. Alves
AU  - Matthias Meiwes
AU  - Beomjun Sohn
PY  - 2026
DA  - 2026/08/31
PB  - arXiv
DO  - 10.48550/arxiv.2608.31132
UR  - https://arxiv.org/abs/2608.31132
AB  - In this paper, we establish a new type of stability phenomenon for the topological entropy of Hamiltonian diffeomorphisms of closed surfaces. For a closed surface endowed with an area form (Σ,ω) and a Hamiltonian diffeomorphism ϕ of (Σ,ω), we show that for every ε>0 there exists A=A(ϕ,ε)>0 such that h_top(ϕ') > h_top(ϕ)−ε for every Hamiltonian diffeomorphism ϕ' obtained from ϕ by a deformation supported in a disjoint union of disks, each of area less than A. In particular, if h_top(ϕ)>0, then ϕ cannot be made to have zero entropy by an area-preserving deformation supported in disks of small area. This follows from the new braid stability result established in this paper with respect to the spectral distance recently introduced by Connery-Grigg.
ER  -