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

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

2004-07-19

arxiv.org/abs/math/0407330v5

J. Operator Theory 58 (2007), no. 2, 269--310.

Mathematics

Classical Analysis and ODEs

44 pages, LaTeX2e ("jotart" document class); v2: A few opening paragraphs were added to the paper; an addition where a bit of

Scientific paper

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.

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