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

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.