Analyticity, scaling and renormalization for some complex analytic dynamical systems

We review some results about the analytic structure of Lindstedt series for some complex analytic dynamical systems: in particular, we consider Hamilt onian maps (like the standard map and its generalizations), the semi-standa rd map and Siegel's problem of the linearization of germs of holomorphic di ffeomorphisms of (C, 0). The analytic structure of those series can be stud ied numerically using Pade approximants, and one can show the existence of natural boundaries for real, diophantine values of the rotation number; by complexifying the rotation number, we show how these natural boundaries ari se from the accumulation of singularities due to resonances, providing a ne w intuitive insight into the mechanism of the break-down of invariant KAM c urves. Moreover, vie study the Lindstedt series at resonances, i.e. for rat ional values of the rotation number, by suitably rescaling to 0 the value o f the perturbative parameter, and a simple analytic structure emerges. Fina lly, we present some proofs for the simplest models and relate these result s to renormalization ideas. (C) 1998 Elsevier Science Ltd. All rights reser ved.