Floer-theoretic entropy of exact symplectomorphisms

Abstract

We introduce the notion of a Penner-type class of an exact symplectomorphism on a Liouville domain with an $A_k$-configuration of Lagrangian spheres for $k\geqslant 2$, and prove that such a class has positive Floer-theoretic entropy. As a corollary, we construct infinitely many smoothly trivial symplectic isotopy classes of exact symplectomorphisms with positive Floer-theoretic entropy on any $4n$-dimensional Liouville domain that admits an $A_2$-configuration of Lagrangian spheres. We also prove that the Floer-theoretic entropy of an exact symplectomorphism on a Liouville domain provides a lower bound for its topological entropy.