A note concerning a property of symplectic matrices

TL;DR

This paper presents a counterexample that challenges a previously claimed property of certain 4x4 symplectic matrices, highlighting a need to revise the understanding of their neighborhood structure.

Contribution

It provides a specific counterexample to a known proposition about the neighborhood of 4x4 symplectic matrices, correcting a misconception in the literature.

Findings

Counterexample disproves the previous proposition.

Reveals limitations in the current understanding of symplectic matrix neighborhoods.

Suggests the need for revised theoretical frameworks.

Abstract

This note provides a counterexample to a proposition stated in [J. Differ. Equ. 261.4 (2016) 2528--2551] regarding the neighborhood of certain $4\times 4$ symplectic matrices.