Braid Stability and a Floer theory of Braid Isotopies
Abstract
We build a Floer theory on the space of braids of periodic orbits of a Hamiltonian flow on a closed symplectic surface. The differential and continuation maps are defined by counting Floer isotopies of braids: tuples of Floer cylinders with pairwise disjoint graphs. With this new perspective, we prove a quantitative braid stability result that shows the persistence of braids under possibly large Hamiltonian perturbations. As an application, we show that for every α≥ 0 there is a sequence of Hamiltonian diffeomorphisms ϕk on the two-torus T2 such that dH(Ent≤ α(T2,ω),ϕk)→ ∞ (k → ∞), where Ent≤ α(T2,ω)⊂ Ham(T2,ω) denotes the set of Hamiltonian diffeomorphisms with topological entropy at most α and dH the Hofer metric. We prove this result by studying only contractible periodic orbits, whereas analogous higher-genus statements were previously obtained using non-contractible orbits.
Related
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BibTeX
@misc{grunder2026braid,
title = {{Braid Stability and a Floer theory of Braid Isotopies}},
author = {Nicolas Grunder},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.34961},
eprint = {2609.34961},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.34961},
}
RIS
TY - UNPB
TI - Braid Stability and a Floer theory of Braid Isotopies
AU - Nicolas Grunder
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2609.34961
UR - https://arxiv.org/abs/2609.34961
AB - We build a Floer theory on the space of braids of periodic orbits of a Hamiltonian flow on a closed symplectic surface. The differential and continuation maps are defined by counting Floer isotopies of braids: tuples of Floer cylinders with pairwise disjoint graphs. With this new perspective, we prove a quantitative braid stability result that shows the persistence of braids under possibly large Hamiltonian perturbations. As an application, we show that for every α≥ 0 there is a sequence of Hamiltonian diffeomorphisms ϕ_k on the two-torus T^2 such that d_H(Ent_≤ α(T^2,ω),ϕ_k)→ ∞ (k → ∞), where Ent_≤ α(T^2,ω)⊂ Ham(T^2,ω) denotes the set of Hamiltonian diffeomorphisms with topological entropy at most α and d_H the Hofer metric. We prove this result by studying only contractible periodic orbits, whereas analogous higher-genus statements were previously obtained using non-contractible orbits.
ER -