A note on the wild symplectic ellipsoids

TL;DR

This paper explores symplectic geometry, demonstrating how certain star-shaped domains relate to ellipsoids and constructing open sets with boundaries of specified Hausdorff dimension, advancing understanding of symplectic embeddings.

Contribution

It introduces a method to relate star-shaped domains to ellipsoids via symplectomorphisms and constructs open sets with boundaries of controlled Hausdorff dimension.

Findings

Symplectic 2-product of star-shaped domains is symplectomorphic to an ellipsoid.

Any open subset with smooth boundary can be symplectomorphically transformed to include a boundary set with specified Hausdorff dimension.

The construction generalizes symplectic embedding techniques to higher dimensions.

Abstract

We show that the symplectic $2$-product of $n$ two-dimensional star-shaped domains has an interior symplectomorphic to that of a symplectic ellipsoid. Adapting this construction, given $0<\alpha \leq 1$, we obtain that every open subset of $\mathbb{R}^{2n}$ with a smooth boundary is symplectomorphic to an open set whose boundary contains a set of Hausdorff dimension $2n-1+\alpha$.