Dynamics of composite symplectic Dehn twists

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, and provide a classification of the symplectic mapping class group of the $A^2_m$ configuration, 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.