The growth rate of Floer homology and symplectic zeta function

TL;DR

This paper investigates the growth rate of symplectic Floer homology for surface symplectomorphisms, linking it to Nielsen numbers, Lefschetz numbers, dilatation, and topological entropy, and establishing a connection to 3-manifold geometry.

Contribution

It proves the equivalence of the Floer homology growth rate with Nielsen, Lefschetz, and dilatation invariants, and relates the symplectic zeta function to topological entropy.

Findings

Growth rate equals asymptotic Nielsen number

Growth rate equals asymptotic Lefschetz number

Growth rate equals largest dilatation of pseudo-Anosov components

Abstract

The main theme of this paper is to study for a symplectomorphism of a compact surface, the asymptotic invariant which is defined to be the growth rate of the sequence of the total dimensions of symplectic Floer homologies of the iterates of the symplectomorphism. We prove that the asymptotic invariant coincides with asymptotic Nielsen number and with asymptotic absolute Lefschetz number. We also show that the asymptotic invariant coincides with the largest dilatation of the pseudo-Anosov components of the symplectomorphism and its logarithm coincides with the topological entropy. This implies that symplectic zeta function has a positive radius of convergence.This also establishes a connection between Floer homology and geometry of 3-manifolds.