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 Hamiltonian maps (like the standard map and its generalizations), the semi-standard map and Siegelʹs problem of the linearization of germs of holomorphic diffeomorphisms of (C,0). The analytic structure of those series can be studied numerically using Padé 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 arise from the accumulation of singularities due to resonances, providing a new intuitive insight into the mechanism of the break-down of invariant KAM curves. Moreover, we study the Lindstedt series at resonances, i.e. for rational values of the rotation number, by suitably rescaling to 0 the value of the perturbative parameter, and a simple analytic structure emerges. Finally, we present some proofs for the simplest models and relate these results to renormalization ideas.