DETERMINISTIC STATISTICAL INDEPENDENCE, NONPERIODIC SYMBOL SEQUENCES AND GIBBS DISTRIBUTIONS

Abstract

We use maps that generate a hierarchy of partitionings that can be organized onto a complete binary tree to ask the question when does the Gibbs distribution describe the frequency of distribution of iterates of a map over its natural partitioning. We are led to a generalization of Borel’s normal numbers, to symbol sequences where the blocks of symbols of a given length are distributed statistically independently, but with unequal frequencies. This result leads to a new universality principle based upon classes of initial conditions that limits the extent within which one can use an observed f(α) spectrum to infer the underlying dynamical system. An application to a turbulence experiment is discussed as an example. Our method is based upon the generation of symbol sequences by an algorithm and the consequent recovery of the corresponding initial condition by backward iteration of the map. Our analysis illustrates that measure one theorems in mathematics cannot serve as a guide to what should be the outcome of either computation or experiment.