Iterations of symplectomorphisms and p-adic analytic actions on the Fukaya category

TL;DR

This paper constructs p-adic analytic families of bimodules on the Fukaya category, revealing invariance properties of Floer cohomology ranks under iterates of symplectomorphisms and applying these results to entropy and conjectures.

Contribution

It introduces a novel p-adic analytic action on the Fukaya category and extends invariance results to non-monotone cases, advancing understanding of symplectic dynamics.

Findings

Floer cohomology ranks are constant in iterates with finitely many exceptions

Constructed p-adic analytic families of bimodules interpolating symplectomorphism iterates

Applications to categorical entropy and Seidel's conjecture

Abstract

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 $\phi$ 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 $HF(\phi^k(L),L';\Lambda)$ are constant in $k\in\mathbb{Z}$, with finitely many possible exceptions. We also prove an analogous result without the monotonicity assumption for generic $\phi$ 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.