Journal of Operator Theory

Volume 58, Issue 2, Fall 2007 pp. 269-310.

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

Authors: Dorin Ervin Dutkay (1) and Palle E.T. Jorgensen (2)
Author institution: (1) University of Central Florida, Department of Mathematics, 4000 Central Florida Blvd., P.O. Box 161364, Orlando, FL 32816-1364, U.S.A.
(2) Department of Mathematics, The Univer sity of Iowa, Iowa City, IA 52242-1419, U.S.A.

Summary: 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.

Keywords: wavelet, Julia set, subshift, Cuntz algebra, iterated function system (IFS), Perron-Frobenius-Ruelle operator, multiresolution, martingale, scaling function, transition probability

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