Equivariant symplectic homology of Anosov contact structures

TL;DR

This paper demonstrates that certain symplectic homology differentials vanish for non-degenerate Reeb flows with invariant Lagrangian subbundles, leading to new obstructions and results on periodic orbits in Anosov contact structures.

Contribution

It establishes the vanishing of the differential in equivariant symplectic homology for specific Reeb flows and explores implications for contact geometry and dynamical systems.

Findings

Differential vanishes for non-degenerate Reeb flows with invariant Lagrangian subbundles.

Provides obstructions to the existence of Anosov Reeb flows.

Shows abundance of periodic orbits in contact forms with Anosov structures.

Abstract

We show that the differential in positive equivariant symplectic homology or linearized contact homology vanishes for non-degenerate Reeb flows with a continuous invariant Lagrangian subbundle (e.g. Anosov Reeb flows). Several applications are given, including obstructions to the existence of these flows and abundance of periodic orbits for contact forms representing an Anosov contact structure.