A counterexample to Lagrangian Poincar\'e recurrence in dimension four

TL;DR

This paper constructs counterexamples to Lagrangian Poincaré recurrence in four-dimensional symplectic manifolds, extending previous results known only in higher dimensions, using techniques involving almost toric fibrations.

Contribution

It provides the first known counterexamples in dimension four, utilizing almost toric fibrations, thus filling a gap in the understanding of recurrence phenomena.

Findings

Counterexamples in dimension four established

Uses almost toric fibrations for construction

Extends previous higher-dimensional results

Abstract

Counterexamples to Lagrangian Poincar\'e recurrence were recently found in dimensions greater than six by Bro\'ci\'c and Shelukhin. We construct counterexamples in dimension four using almost toric fibrations.