On infinite staircases in toric symplectic four-manifolds

An influential result of McDuff and Schlenk asserts that the function thatencodes when a four-dimensional symplectic ellipsoid can be embedded into afour-dimensional ball has a remarkable structure: the function has infinitelymany corners, determined by the odd-index Fibonacci numbers, that fit togetherto form an infinite staircase. This work has recently led to considerable interest in understanding when theellipsoid embedding function for other symplectic 4-manifolds is partlydescribed by an infinite staircase. We provide a general framework foranalyzing this question for a large family of targets, called finite typeconvex toric domains, which we prove generalizes the class of closed toricsymplectic 4-manifolds. When the target is of finite type, we prove that anyinfinite staircase must have a unique accumulation point a_0, given as thesolution to an explicit quadratic equation. Moreover, we prove that theembedding function at a_0 must be equal to the classical volume lower bound. Inparticular, our result gives an obstruction to the existence of infinitestaircases that we show is strong. In the special case of rational convex toric domains, we can say more. Weconjecture a complete answer to the question of existence of infinitestaircases, in terms of six families that are distinguished by the fact thattheir moment polygon is reflexive. We then provide a uniform proof of theexistence of infinite staircases for our six families, using two tools. For thefirst, we use recursive families of almost toric fibrations to find symplecticembeddings. For the second tool, we find recursive families of convex latticepaths that provide obstructions to embeddings. We conclude by reducing ourconjecture that these are the only infinite staircases among rational convextoric domains to a question in number theory related to a classic work of Hardyand Littlewood.