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2019 Infinite staircases in the symplectic embedding problem for four-dimensional ellipsoids into polydisks
Michael Usher
Algebr. Geom. Topol. 19(4): 1935-2022 (2019). DOI: 10.2140/agt.2019.19.1935

Abstract

We study the symplectic embedding capacity function Cβ for ellipsoids E(1,α)4 into dilates of polydisks P(1,β) as both α and β vary through [1,). For β=1, Frenkel and Müller showed that Cβ has an infinite staircase accumulating at α=3+22, while for integer β2, Cristofaro-Gardiner, Frenkel and Schlenk found that no infinite staircase arises. We show that for arbitrary β(1,), the restriction of Cβ to [1,3+22] is determined entirely by the obstructions from Frenkel and Müller’s work, leading Cβ on this interval to have a finite staircase with the number of steps tending to as β1. On the other hand, in contrast to the results of Cristofaro-Gardiner, Frenkel and Schlenk, for a certain doubly indexed sequence of irrational numbers Ln,k we find that CLn,k has an infinite staircase; these Ln,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+22.

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Michael Usher. "Infinite staircases in the symplectic embedding problem for four-dimensional ellipsoids into polydisks." Algebr. Geom. Topol. 19 (4) 1935 - 2022, 2019. https://doi.org/10.2140/agt.2019.19.1935

Information

Received: 17 March 2018; Revised: 21 December 2018; Accepted: 7 January 2019; Published: 2019
First available in Project Euclid: 22 August 2019

zbMATH: 07121518
MathSciNet: MR3995022
Digital Object Identifier: 10.2140/agt.2019.19.1935

Subjects:
Primary: 53D22

Rights: Copyright © 2019 Mathematical Sciences Publishers

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Vol.19 • No. 4 • 2019
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