Symplectic capacities on surfaces

We classify capacities on the class Sym (o) (2) of connected symplectic surfaces with at most countably many nonplanar ends. To obtain the classification we study diffeomorphism types of surfaces in Sym (o) (2) of infinite genus with nonplanar ends; it turns out that these types are in bijective correspondence with countable successor ordinals of the form omega (alpha) center dot d + 1, where alpha is an ordinal and d a parts per thousand yen 0 is an integer. It also turns out that if S (1) and S (2) are two open surfaces of infinite genera with at most countably many nonplanar ends, then each of the surfaces embeds into the other. Our classification implies that every capacity on the class of symplectic surfaces in Sym (o) (2) of infinite genus differs from the Hofer-Zehnder capacity by a non-negative finite or infinite constant.