Iterations of symplectomorphisms and p-adic analytic actions on the Fukaya category
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2020
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866909430882238464 |
|---|---|
| author | Kartal, Yusuf Barış |
| author_facet | Kartal, Yusuf Barış |
| contents | Inspired by the work of Bell on the dynamical Mordell-Lang conjecture, and by family Floer cohomology, we construct p-adic analytic families of bimodules on the Fukaya category of a monotone or negatively monotone symplectic manifold, interpolating the bimodules corresponding to iterates of a symplectomorphism $ϕ$ isotopic to the identity. This family can be thought of as a $p$-adic analytic action on the Fukaya category. Using this, we deduce that the ranks of the Floer cohomology groups $HF(ϕ^k(L),L';Λ)$ are constant in $k\in\mathbb{Z}$, with finitely many possible exceptions. We also prove an analogous result without the monotonicity assumption for generic $ϕ$ isotopic to the identity by showing how to construct a p-adic analytic action in this case. We give applications to categorical entropy and a conjecture of Seidel. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2008_08566 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Iterations of symplectomorphisms and p-adic analytic actions on the Fukaya category Kartal, Yusuf Barış Symplectic Geometry K-Theory and Homology Number Theory Inspired by the work of Bell on the dynamical Mordell-Lang conjecture, and by family Floer cohomology, we construct p-adic analytic families of bimodules on the Fukaya category of a monotone or negatively monotone symplectic manifold, interpolating the bimodules corresponding to iterates of a symplectomorphism $ϕ$ isotopic to the identity. This family can be thought of as a $p$-adic analytic action on the Fukaya category. Using this, we deduce that the ranks of the Floer cohomology groups $HF(ϕ^k(L),L';Λ)$ are constant in $k\in\mathbb{Z}$, with finitely many possible exceptions. We also prove an analogous result without the monotonicity assumption for generic $ϕ$ isotopic to the identity by showing how to construct a p-adic analytic action in this case. We give applications to categorical entropy and a conjecture of Seidel. |
| title | Iterations of symplectomorphisms and p-adic analytic actions on the Fukaya category |
| topic | Symplectic Geometry K-Theory and Homology Number Theory |
| url | https://arxiv.org/abs/2008.08566 |