ar X iv : m at h / 05 02 27 7 v 3 [ m at h . D S ] 9 A pr 2 00 5 HARMONIC ANALYSIS AND DYNAMICS FOR AFFINE ITERATED FUNCTION SYSTEMS

We introduce a harmonic analysis for a class of affine iteration models in R d. Using Hilbert space geometry, and certain complex Hadamard matrices, we develop a new Fourier duality notion for affine and contractive iterated function systems (IFSs). It is known that not all such affine systems admit orthogonal Fourier bases. As a result, our duality is more general than earlier related constructs. Our present Fourier duality is based on a natural transfer operator R W , a weight function W , and on an associated random walk process Px. We introduce an associated dynamical system, and we analyze corresponding finite cycles C on which W attains it maximum. These extreme cycles C generate our Fourier bases. As a byproduct, we get dual IFS systems, and we show for a fixed IFS, that the absolute-square of the Fourier transform of the associated IFS-invariant measure ν λ is the unique solution to a specific functional equation involving Px; and we use this in turn to establish a detailed harmonic analysis of the transfer operator R W. From this we derive our main results on harmonic analysis for affine iterated function systems. We further illustrate our results with a classical one-parameter family of examples.