Holomorphic Dynamics, Small Divisors and Related Topics

University Of California-Los Angeles, Los Angeles CA

Investigators

Abstract

Proposal Number: DMS-0202494 PI: Ricardo Perez-Marco ABSTRACT Research will be conducted on several projects on the theory of Dynamical Systems, and more specifically, in Holomorphic Dynamics and Small Divisor theory. Using heuristic ideas from Holomorphic Dynamics it is proposed to investigate an effective selection of polynomials as iterators in Pollard rho method of factorization. Factorization algorithms are a central topic in Cryptography. Numerical explorations show that there is room for improvement in this field. In Holomorphic Dynamics and Small Divisors, it is proposed to extend the general theory of dynamics on Hedgehogs. The focus of the proposed research is on weak forms of stability when arithmetic conditions, typical of Small Divisors, fail. It is proposed to investigate Siegel disks and their boundary, and to attack the hard problem of classifying invariant annular continua. It is anticipated that the techniques of tube-log Riemann surfaces should play an important role. Another project consists in extending geometric techniques to higher dimensional Small Divisors problems. In particular, a new geometric construction of invariant tori is envisioned. In the theory of Renormalization in Holomorphic Dynamics it is proposed to develop the new theory of renormalization. This involves a better understanding of Degenerate Quasi-Conformal theory (i.e. for non bounded Beltrami coefficients). A last topic in Complex Analysis on which research is proposed is the theory of Borel Monogenic functions. We project to develop a theory that is flexible enough to apply to problems in Small Divisors. Dynamical Systems is a branch of Mathematics with ancient roots linked to Celestial Mechanics. It was founded as a distinct branch of Mathematics more than a century ago by H. Poincare. It is a rich field with multiple interactions with other parts of Mathematics and Science. The main goal is the study of the long time behavior of evolution processes. Some of the most important problems come from fields as Biology, Physical Sciences and Chemistry. One of the central questions is the Problem of Stability. For example, is the Solar System stable according to Newton laws ? K.A.M. theory (also called Small Divisor theory) is one of the major branches in Dynamical Systems. It was founded in the second half of the XXth century and had an important impact in Dynamical Systems and related fields. It is the very first theory that provides stability results in non-linear conservative problems. It continues its expansion to fields as diverse as Partial Differential Equations, the theory of foliations, Holomorphic Dynamics, etc. In the proposed projects some of the most difficult open questions in KAM theory will be explored. What happens when stability fails? Can we still recover some traces of stability that may be useful in the applications?