Scaling Function Dynamics

Abstract

In September 1979, P. Hohenberg gave me a picture which showed the first preliminary results of A. Libchaber’s experiment on liquid helium [1], the power spectrum of a measured signal (Fig. 1). It was immediately clear that the picture had something to do with period doubling, but how it was that one was supposed to understand a one-dimensional theory for a discrete dynamics in order to learn what a fluid was doing was in no way very clear. Over a period of a few months, I tried to understand the picture, and, in the end, was lead to an idea that I have called the scaling function [2]. In these lectures, I shall try to explain what the idea is that came out of this observation and while doing so, discuss the idea that goes under the name of “presentation functions” [3]. I will indicate what these notions mean and explain how from that picture you can determine what is actually the most interesting part of the dynamics.