Iterated function systems, representations, and Hilbert space

Details Iterated function systems, representations, and Hilbert space Iterated function systems, representations, and Hilbert space

2004-02-11

arxiv.org/abs/math/0402175v2

Internat. J. Math. 15 (2004), 813-832

Mathematics

Classical Analysis and ODEs

22 pages, 3 figures containing 7 EPS graphics; LaTeX2e ("elsart" document class); v2 reflects change in Comments only

Scientific paper

10.1142/S0129167X04002569

This paper studies a general class of Iterated Function Systems (IFS). No contractivity assumptions are made, other than the existence of some compact attractor. The possibility of escape to infinity is considered. Our present approach is based on Hilbert space, and the theory of representations of the Cuntz algebras O_n, n=2,3,.... While the more traditional approaches to IFS's start with some equilibrium measure, ours doesn't. Rather, we construct a Hilbert space directly from a given IFS; and our construction uses instead families of measures. Starting with a fixed IFS S_n, with n branches, we prove existence of an associated representation of O_n, and we show that the representation is universal in a certain sense. We further prove a theorem about a direct correspondence between a given system S_n, and an associated sub-representation of the universal representation of O_n.

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