Ergodic Renormalization and Universal Strange Attractors

Abstract

Renormalization has emerged as an important tool in a variety of problems in physics. It was originally invented in quantum field theory. In recent years, it has been used extensively in the theory of critical phenomenon. In the last few years it has been proved to be an important technique in the study of transition to chaos in dynamical systems theory. In simple cases, under renormalization, system converges to a fixed point, i.e., the system is invariant under renormalization. The significance of a fixed point is that all systems attracted to it under successive renormalization will have the same large scale behavior, so that the behavior is universal. In critical phenomenon, the fixed point corresponds to the fact that the renormalized system looks the same at every length scale corresponding to an infinite correlation length. The infinite correlation length is the case of second order phase transition. In dynamical systems theory, renormalization operation corresponds to looking at a system on longer time scales and smaller spatial scales. Universally self-similar period-doubling sequences in 1-d logistic map¹ and 2-d area preserving maps² have been explained in terms of fixed points of renormalization. Another important application of renormalization ideas has been the breakdown of invariant circles.²