Abstract

Every symplectic automorphism on a symplectic manifold induces an auto-equivalence on the (derived) Fukaya category, which gives rise to a categorical dynamical system. In this talk, I will first give a brief review of various Fukaya categories of symplectic manifolds with boundaries. Then, for a given symplectic automorphism, I will discuss how the categorical entropies of auto-equivalences induced by f on different Fukaya categories are compared. Then I will explain that in the case when M is a Weinstein domain and f is an exact symplectic automorphism on M that equals the identity map near the boundary of M, the topological entropy of f is greater than or equal to the categorical entropy of the corresponding auto-equivalence on the wrapped Fukaya category of M. This is based on joint work with D. Choa-W. Jeong-D. Karabas- S. Lee and Sangjin Lee.