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

Usher, Michael¹

¹ Department of Mathematics, University of Georgia, Athens, GA, United States

Algebraic and Geometric Topology, Tome 19 (2019) no. 4, pp. 1935-2022

Résumé

We study the symplectic embedding capacity function Cβ for ellipsoids E(1,α) ⊂ ℝ4 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.

Détail

DOI : 10.2140/agt.2019.19.1935

Classification : 53D22
Keywords: symplectic embeddings, Cremona moves