Infinite staircases for Hirzebruch surfaces

Abstract

We consider the embedding capacity functions $c_{H_b}(z)$ for symplectic embeddings of ellipsoids of eccentricity $z$ into the family of nontrivial rational Hirzebruch surfaces $H_b$ with symplectic form parametrized by $b\in [0,1)$. This function was known to have an infinite staircase in the monotone cases ($b= 0$ and $ b= 1/3$). It is also known that for each $b$ there is at most one value of $z$ that can be the accumulation point of such a staircase. In this manuscript, we identify three sequences of open, disjoint, blocked $b$-intervals, consisting of $b$-parameters where the embedding capacity function for $H_b$ does not contain an infinite staircase. There is one sequence in each of the intervals $(0,1/5)$, $(1/5,1/3)$, and $(1/3,1)$. We then establish six sequences of associated infinite staircases, one occurring at each endpoint of the blocked $b$-intervals. The staircase numerics are variants of those in the Fibonacci staircase for the projective plane (the case $b=0$). We also show that there is no staircase at the point $b=1/5$, even though this value is not blocked. The focus of this paper is to develop techniques, both graphical and numeric, that allow identification of potential staircases, and then to understand the obstructions well enough to prove that the purported staircases really do have the required properties. A subsequent paper will explore in more depth the set of $b$ that admit infinite staircases.