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.
