More counterexamples to Lagrangian Poincar\'e recurrence in dimension four

TL;DR

This paper expands the class of known counterexamples to Lagrangian Poincaré recurrence in four-dimensional toric symplectic manifolds, demonstrating that all non-monotone cases also lack recurrence.

Contribution

It extends previous counterexamples to include all non-monotone toric symplectic four manifolds, broadening understanding of recurrence phenomena.

Findings

Counterexamples now include all non-monotone cases

Lagrangian Poincaré recurrence fails in these manifolds

Supports conjectures about recurrence in symplectic geometry

Abstract

In earlier work, we constructed counterexamples to Lagrangian Poincar\'e recurrence for many toric symplectic four manifolds. Here we provide a few more examples extending the family of counterexamples to include all non-monotone toric symplectic four manifolds. This work has been incorporated into arXiv:2506.23754v2 Section 3.3