Infinite staircases in the symplectic embedding problem for four-dimensional ellipsoids into polydisks

TL;DR

This paper investigates the symplectic embedding capacity function for ellipsoids into polydisks, revealing conditions under which infinite staircases occur and how they depend on the parameters, especially for irrational versus integer beta.

Contribution

It demonstrates that for arbitrary beta, the capacity function has a finite staircase on a key interval, and identifies specific irrational parameters where infinite staircases occur, extending previous results.

Findings

Finite staircase with increasing steps as beta approaches 1.

Existence of infinite staircases for certain irrational parameters.

Obstructions from prior work fully determine the capacity function on a key interval.

Abstract

We study the symplectic embedding capacity function $C_{\beta}$ for ellipsoids $E(1,\alpha)\subset R^4$ into dilates of polydisks $P(1,\beta)$ as both $\alpha$ and $\beta$ vary through $[1,\infty)$. For $\beta=1$ Frenkel and Mueller showed that $C_{\beta}$ has an infinite staircase accumulating at $\alpha=3+2\sqrt{2}$, while for integer $\beta\geq 2$ Cristofaro-Gardiner, Frenkel, and Schlenk found that no infinite staircase arises. We show that, for arbitrary $\beta\in (1,\infty)$, the restriction of $C_{\beta}$ to $[1,3+2\sqrt{2}]$ is determined entirely by the obstructions from Frenkel and Mueller's work, leading $C_{\beta}$ on this interval to have a finite staircase with the number of steps tending to $\infty$ as $\beta\to 1$. On the other hand, in contrast to the case of integer $\beta$, for a certain doubly-indexed sequence of irrational numbers $L_{n,k}$ we find that $C_{L_{n,k}}$ has an infinite staircase; these $L_{n,k}$ include both numbers that are arbitrarily large and numbers that are arbitrarily close to $1$, with the corresponding accumulation points respectively arbitrarily large and arbitrarily close to $3+2\sqrt{2}$.