Iterated function systems, Ruelle operators, and invariant projective measures

Abstract

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 X comes with a finite-to-one endomorphism r : X → X which is onto but not one-to-one. (2) In the case of affine Iterated Function Systems (IFSs) in R d , this harmonic analysis arises naturally as a spectral duality defined from a given pair of finite subsets B, L in R d of the same cardinality which generate complex Hadamard matrices.Our harmonic analysis for these iterated function systems (IFS) (X, µ) 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 Wharmonic functions. The properties of the functions in H are analyzed, and they determine the spectral theory of L 2 (µ). For affine IFSs we establish orthogonal bases in L 2 (µ). These bases are generated by paths with infinite repetition of finite words. We use this in the last section to analyze tiles in R d .

Multifractals, Mumford curves and eternal inflation

“…We first describe how to reinterpret the case without terminal vacua in terms of multifractal measures and operator algebra arising from representations of the Cuntz algebras of [5]. We show, in particular, that the construction of the multiverse fields in the Eternal Symmetree model is closely related to the construction of [7] (see also [14]) of stochastic processes and wavelets on the Cantor sets dual to the maximal abelian subalgebra of the Cuntz algebra. We will then consider the case with terminal vacua, where we focus on pruning of the tree obtained through an admissibility condition on adjacent edges.…”

“…In particular this implies that what plays the role of the proper time in the model, which gives the discrete evolution of the stochastic process, in turn can be seen as depending on an internal notion of time evolution acting on the creation and annihilation operators given by the generators of the noncommutative algebra associated to the graph. The propagators in the correlation functions for the multiverse fields of [9] in turn provide a measure of autocorrelation for wavelets on fractals arising from the construction of the multifractal measure (as in [7], [16]) on the boundary of the tree, which determined the stochastic process. We also show how one can extend the eternal inflation model from the case of the p-adic BruhatTits tree to infinite graphs given by quotients of the Bruhat-Tits tree by a p-adic Schottky group.…”

Harmonic analysis of iterated function systems with overlap

“…For many purposes, it is convenient to normalize the measure µ in (3.3) such that µ (X) = µ (Y ) = 1. Because of applications to harmonic analysis [DJ06b], we will also restrict the weights (p i ) in (3.3) such that p i = 1/N . Definition 3.1.…”

“…Since then, IFSs have found uses in geometry (e.g., [Bar06,BHS05]), in infinite network problems (e.g., [GRS01]), in wavelets (e.g., [BJMP05,BJMP06,Jor05]), and in dynamics and operator theory [Kaw05,Jor06,DJ06c,DJ06b,DJ05].…”

“…Recent references [DJ05,DJ06c,DJ06d,DJ06b,Jor06] deal with wavelet constructions on non-linear attractors X, such as arise from systems of branched mappings, e.g., finite affine systems (affine IFSs), or Julia sets [Bea91] generated by branches of the inverses of given rational mappings of one complex variable. These X come with associated equilibrium measures µ.…”