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

Abstract

We study the symplectic embedding capacity function $C_{\beta }$ for ellipsoids $E\left(1,\alpha \right)\subset {ℝ}^4$ into dilates of polydisks $P\left(1,\beta \right)$ as both $\alpha$ and $\beta$ vary through $\left[1,\infty \right)$. For $\beta =1$, Frenkel and Müller showed that $C_{\beta }$ has an infinite staircase accumulating at $\alpha =3+2\sqrt{2}$, while for integer $\beta \ge 2$, Cristofaro-Gardiner, Frenkel and Schlenk found that no infinite staircase arises. We show that for arbitrary $\beta \in \left(1,\infty \right)$, the restriction of $C_{\beta }$ to $\left[1,3+2\sqrt{2}\right]$ is determined entirely by the obstructions from Frenkel and Müller’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 results of Cristofaro-Gardiner, Frenkel and Schlenk, 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 $3+2\sqrt{2}$.