Iterated convolutions and endless Riemann surfaces

We discuss a version of Ecalles definition of resurgence, based on the notion of endless continuability in the Borel plane. We relate this with the notion of Omega-continuability, where Omega is a discrete filtered set, and show how to construct a universal Riemann surface X_Omega whose holomorphic functions are in one-to-one correspondence with Omega-continuable functions. We then discuss the Omega-continuability of convolution products and give estimates for iterated convolutions of the form hatphi_1*cdots *hatphi_n. This allows us to handle nonlinear operations with resurgent series, e.g. substitution into a convergent power series.