Iterated function systems, Ruelle operators, and invariant projective measures
Authors:
Dorin Ervin Dutkay and Palle E. T. Jorgensen
Journal:
Math. Comp. 75 (2006), 1931-1970
MSC (2000):
Primary 28A80, 31C20, 37F20, 39B12, 41A63, 42C40, 47D07, 60G42, 60J45
DOI:
https://doi.org/10.1090/S0025-5718-06-01861-8
Published electronically:
May 31, 2006
MathSciNet review:
2240643
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: We introduce a Fourier-based harmonic analysis for a class of discrete dynamical systems which arise from Iterated Function Systems. Our starting point is the following pair of special features of these systems. (1) We assume that a measurable space comes with a finite-to-one endomorphism
which is onto but not one-to-one. (2) In the case of affine Iterated Function Systems (IFSs) in
, this harmonic analysis arises naturally as a spectral duality defined from a given pair of finite subsets
in
of the same cardinality which generate complex Hadamard matrices.
Our harmonic analysis for these iterated function systems (IFS) is based on a Markov process on certain paths. The probabilities are determined by a weight function
on
. From
we define a transition operator
acting on functions on
, and a corresponding class
of continuous
-harmonic functions. The properties of the functions in
are analyzed, and they determine the spectral theory of
. For affine IFSs we establish orthogonal bases in
. These bases are generated by paths with infinite repetition of finite words. We use this in the last section to analyze tiles in
.
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Additional Information
Dorin Ervin Dutkay
Affiliation:
Department of Mathematics, Hill Center-Busch Campus, Rutgers, The State University of New Jersey, 110 Frelinghuysen Rd, Piscataway, New Jersey 08854-8019
Email:
ddutkay@math.rutgers.edu
Palle E. T. Jorgensen
Affiliation:
Department of Mathematics, The University of Iowa, 14 MacLean Hall, Iowa City, Iowa 52242-1419
Email:
jorgen@math.uiowa.edu
DOI:
https://doi.org/10.1090/S0025-5718-06-01861-8
Keywords:
Measures,
projective limits,
transfer operator,
martingale,
fixed-point,
wavelet,
multiresolution,
fractal,
Hausdorff dimension,
Perron-Frobenius,
Julia set,
subshift,
orthogonal functions,
Fourier series,
Hadamard matrix,
tiling,
lattice,
harmonic function
Received by editor(s):
January 5, 2005
Received by editor(s) in revised form:
June 16, 2005
Published electronically:
May 31, 2006
Additional Notes:
This research was supported in part by the National Science Foundation DMS-0139473 (FRG)
Article copyright:
© Copyright 2006
American Mathematical Society