Iterated function systems, Ruelle operators, and invariant projective measures

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 $X$ comes with a finite-to-one endomorphism $r\colon X\rightarrow X$ which is onto but not one-to-one. (2) In the case of affine Iterated Function Systems (IFSs) in $\mathbb{R}^d$, this harmonic analysis arises naturally as a spectral duality defined from a given pair of finite subsets $B, L$ in $\mathbb{R}^d$ of the same cardinality which generate complex Hadamard matrices.

Our harmonic analysis for these iterated function systems (IFS) $(X, \mu)$ is based on a Markov process on certain paths. The probabilities are determined by a weight function $W$ on $X$. From $W$ we define a transition operator $R_W$ acting on functions on $X$, and a corresponding class $H$ of continuous $R_W$-harmonic functions. The properties of the functions in $H$ are analyzed, and they determine the spectral theory of $L^2(\mu)$. For affine IFSs we establish orthogonal bases in $L^2(\mu)$. These bases are generated by paths with infinite repetition of finite words. We use this in the last section to analyze tiles in $\mathbb{R}^d$.