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

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 $3+2\sqrt{2}$.