Iterated convolutions and endless Riemann surfaces

Abstract

We discuss a version of Ecalle's definition of resurgence, based on the notion of endless continuability in the Borel plane. We relate this with the notion of Ω-continuability, where Ω is a discrete filtered set or a discrete doubly filtered set, and show how to construct a universal Riemann surface X_Ω whose holomorphic functions are in one-to-one correspondence with Ω-continuable functions. We then discuss the Ω-continuability of convolution products and give estimates for iterated convolutions of the form f_1*...*f_n. This allows us to handle nonlinear operations with resurgent series, e.g. substitution into a convergent power series.