Martingales, endomorphisms, and covariant systems of operators in Hilbert space

TL;DR

This paper develops a framework linking dynamical systems defined by finite-to-one mappings to covariant operator systems in Hilbert space, with applications in wavelets, rational dynamics, and symbolic systems, through a martingale construction.

Contribution

It introduces a novel martingale-based method to analyze operator systems induced by dynamical systems, encompassing various complex systems like Julia sets and subshifts.

Findings

Complete description of the induced martingale systems

Application to wavelets and rational function dynamics

Framework applicable to Julia sets and symbolic dynamics

Abstract

We show that a class of dynamical systems induces an associated operator system in Hilbert space. The dynamical systems are defined from a fixed finite-to-one mapping in a compact metric space, and the induced operators form a covariant system in a Hilbert space of L^2-martingales. Our martingale construction depends on a prescribed set of transition probabilities, given by a non-negative function. Our main theorem describes the induced martingale systems completely. The applications of our theorem include wavelets, the dynamics defined by iterations of rational functions, and sub-shifts in symbolic dynamics. In the theory of wavelets, in the study of subshifts, in the analysis of Julia sets of rational maps of a complex variable, and, more generally, in the study of dynamical systems, we are faced with the problem of building a unitary operator from a mapping r in a compact metric space X. The space X may be a torus, or the state space of subshift dynamical systems, or a Julia set. While our motivation derives from some wavelet problems, we have in mind other applications as well; and the issues involving covariant operator systems may be of independent interest.