Volume 19, issue 4 (2019)

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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

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 β 2, 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 β 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 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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symplectic embeddings, Cremona moves
Mathematical Subject Classification 2010
Primary: 53D22
Received: 17 March 2018
Revised: 21 December 2018
Accepted: 7 January 2019
Published: 16 August 2019
Michael Usher
Department of Mathematics
University of Georgia
Athens, GA
United States