Dynamics of composite symplectic Dehn twists

TL;DR

This paper demonstrates that composite symplectic Dehn twists exhibit nonuniform hyperbolicity with positive entropy, exponential Floer cohomology growth, and potential links to the standard map, advancing understanding in symplectic topology.

Contribution

It introduces new dynamical properties of composite symplectic Dehn twists, including hyperbolicity and Floer cohomology growth, addressing classification questions in higher-dimensional symplectic mapping class groups.

Findings

Positive topological entropy for composite symplectic Dehn twists

Exponential growth of Floer cohomology under iteration

Presence of stable and unstable Lagrangian manifolds

Abstract

This paper appears as the confluence of hyperbolic dynamics, symplectic topology and low dimensional topology, etc. We show that composite symplectic Dehn twists have certain form of nonuniform hyperbolicity: it has positive topological entropy as well as two families of local stable and unstable Lagrangian manifolds, which are analogous to signatures of pseudo Anosov mapping classes. Moreover, we show that the rank of the Floer cohomology group of these compositions grows exponentially under iterations, which partially answers a question of Smith concerning the classification of symplectic mapping class group in higher dimensions. Finally, we propose a conjecture on the positive metric entropy of our model and point out its relationship with the standard map.