Four-periodic infinite staircases for four-dimensional polydisks

TL;DR

The paper proves a new four-periodic infinite staircase for the ellipsoid embedding function of the four-dimensional polydisk $P(1,\beta)$, establishing that for $\beta=(6+5\sqrt{30})/12$ the function $c_{\beta}(z)$ has an infinite staircase accumulating at $\mathrm{acc}(\beta)=(54+11\sqrt{30})/14$ with $c_{\beta}(\mathrm{acc}(\beta))=\operatorname{vol}(\beta)$. The construction hinges on two families of quasi-perfect Diophantine obstructions ${\bf E}_k$ and $\hat{\bf E}_k$, whose centers $p_k/q_k$ and $\hat{p}_k/\hat{q}_k$ yield outer corners that converge to the accumulation point, and inner corners arising from their intersections via explicit ATF mutation sequences. The authors integrate obstructions (via weights and ECH capacities) with almost toric fibrations to produce explicit embeddings at critical points, and they show the accumulation is four-periodic in the continued fraction data, supporting a broader Usher-type framework and conjectures about fractal-like structures in the space of polydisk parameters. The work provides a concrete, computable pathway to extend Usher’s staircase analysis to polydisks and suggests deep connections between polydisk staircases, Hirzebruch surfaces, and potentially universal fractal phenomena in symplectic embedding problems.

Abstract

The ellipsoid embedding function of a symplectic four-manifold measures the amount by which its symplectic form must be scaled in order for it to admit an embedding of an ellipsoid of varying eccentricity. This function generalizes the Gromov width and ball packing numbers. In the one continuous family of symplectic four-manifolds that has been analyzed, one-point blowups of the complex projective plane, there is an open dense set of symplectic forms whose ellipsoid embedding functions are completely described by finitely many obstructions, while there is simultaneously a Cantor set of symplectic forms for which an infinite number of obstructions are needed. In the latter case, we say that the embedding function has an infinite staircase. In this paper we identify a new infinite staircase when the target is a four-dimensional polydisk, extending a countable family identified by Usher in 2019. Our work computes the function on infinitely many intervals and thereby indicates a method of proof for a conjecture of Usher.

Figures (16)

• Figure 1.0.1: In blue, the graph of the embedding capacity function for a ball $X=B^4(1)$ is shown on the domain indicated. The graph in red is the volume lower bound established in Proposition \ref{['prop:cXprops']}(i). The point marked O is an outer corner and the point marked I is an inner corner. This target has an ascending infinite staircase, first identified by McDuff and Schlenk ball and called the Fibonacci staircase in the literature. The green point is the accumulation point.
• Figure 1.1.1: This figure shows the parameterized curve $(\mathrm{acc}(\beta),\operatorname{vol}(\beta))$ in red. The point on the curve at $\beta$ represents a point at which an infinite staircase for $c_\beta$ must accumulate, if it exists. The red dot is the accumulation point of the Pell stairs of Frenkel-Müller; the blue dots are the $L_{n,0}$ staircases of Usher; and the black $\bm{\times}$s indicate values of $\beta$ without infinite staircases, proved by Cristofaro-Gardiner--Frenkel--Schlenk. The accumulation points of the new infinite staircases of Theorem \ref{['thm:main']} and Conjecture \ref{['thm:generalb']} are indicated by green dots.
• Figure 1.1.2: This figure depicts the infinite staircase $c_\beta$ of Theorem \ref{['thm:main']}. In both figures, with $\beta$ as in Theorem \ref{['thm:main']}, the orange curve is $\operatorname{vol}_\beta(z)$ and $c_\beta$ is in blue. The accumulation point curve $(\mathrm{acc}(\beta),\operatorname{vol}(\beta))$ is in red -- for this curve, $\beta$ varies. Thus the accumulation point of $c_\beta$ occurs at the intersection of these three curves. In (b), we have zoomed in; the obstructions from ${\bf{E}}_0, \hat{{\bf{E}}}_1$, and ${\bf{E}}_2$ are visible. See sections §\ref{['ssec:ECdefs']} and §\ref{['sec:proofs']} for these definitions.
• Figure 1.1.3: This figure uses the same color scheme as Figure \ref{['fig:polydisk-accum']}. More detail near the infinite staircase of Theorem \ref{['thm:main']} is shown. The new staircase's accumulation point is the green dot, while the two blue dots are Usher's staircases with $\beta=L_{2,0}$ and $L_{3,0}$.
• Figure 2.3.1: With $\Omega$ the region outlined in red and $\nu$ in blue, the black point is $p_{{\Omega},\nu}$.

Theorems & Definitions (72)

• Proposition 1.0.1: cghmp
• Theorem 1.1.1
• Conjecture 1.1.2
• Conjecture 1.2.1
• Definition 2.1.1
• Theorem 2.1.2: cghmp
• Lemma 2.1.3
• proof
• Proposition 2.1.4: cghmp
• Example 2.2.1