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

Usher, Michael

doi:10.2140/agt.2019.19.1935

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

Michael Usher

Algebraic & Geometric Topology 19 (2019) 1935–2022

DOI: 10.2140/agt.2019.19.1935

Abstract

We study the symplectic embedding capacity function $C_{β}$ for ellipsoids $E (1, α) \subset ℝ^{4}$ into dilates of polydisks $P (1, β)$ as both $α$ and $β$ vary through $[1, \infty)$ . For $β = 1$ , Frenkel and Müller showed that $C_{β}$ has an infinite staircase accumulating at $α = 3 + 2 \sqrt{2}$ , while for integer $β \geq 2$ , Cristofaro-Gardiner, Frenkel and Schlenk found that no infinite staircase arises. We show that for arbitrary $β \in (1, \infty)$ , the restriction of $C_{β}$ to $[1, 3 + 2 \sqrt{2}]$ is determined entirely by the obstructions from Frenkel and Müller’s work, leading $C_{β}$ on this interval to have a finite staircase with the number of steps tending to $\infty$ as $β \to 1$ . 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 $3 + 2 \sqrt{2}$ .

Keywords

symplectic embeddings, Cremona moves

Mathematical Subject Classification 2010

Primary: 53D22

References

Publication

Received: 17 March 2018

Revised: 21 December 2018

Accepted: 7 January 2019

Published: 16 August 2019

Authors

Michael Usher
Department of Mathematics University of Georgia Athens, GA United States

Volume 19, issue 4 (2019)