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

Abstract: A theorem of J. Bell states that given a complex affine variety $X$ with an automorphism $\phi$, and a subvariety $Y\subset X$, the set of numbers $k$ such that $\phi^k(x)\in Y$ is a union of finitely many arithmetic progressions and finitely many numbers. Motivated by this statement, Seidel asked whether there is a symplectic analogue of this theorem. In this talk, we give an answer to a version of this question in the case $M$ is monotone, non-degenerate and $\phi$ is symplectically isotopic to identity. The main tool is analogous to the main tool in Bell's proof: namely we interpolate the powers of $\phi$ by a p-adic arc, constructing an analytic action of $\mathbb{Z}_p$ on the Fukaya category.

algebraic geometrydifferential geometrygeometric topologysymplectic geometry

Audience: researchers in the topic

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