In this paper, we show how a class of operators used in the analysis of measures from wavelets and iterated function systems may be understood from a special family of representations of Cuntz algebras. Let (X, d) be a compact metric space, and let an iterated function system (IFS) be given on X, i.e., a finite set of continuous maps σi: X → X, i = 0, 1, · · · , N −1. The maps σi transform the ...

We revisit here the notion of discrete scale invariance. Initially defined for signal indexed by the positive reals, we present a generalized version of discrete scale invariant signals relying on a renormalization group approach. In this view, the signals are seen as fixed point of a renormalization operator acting on a space of signal. We recall how to show that these fixed point present disc...

In this paper, we are concerned with spectral-theoretic features of general iterated function systems (IFS). Such systems arise from the study of iteration limits of a finite family of maps τi, i = 1, . . . , N , in some Hausdorff space Y . There is a standard construction which generally allows us to reduce to the case of a compact invariant subset X ⊂ Y . Typically, some kind of contractivity...

A theoretical approach to computing the Hausdorff dimension of the topological boundary of attractors of iterated function systems is developed. The curve known as the Lévy Dragon is then studied in detail and the Hausdorff dimension of its boundary is computed using the theory developed. The actual computation is a complicated procedure. It involves a great deal of combinatorial topology as we...

This paper presents a novel two-dimensional split-vector-radix fast-Fourier-transform (2D svr-FFT) algorithm. The modularizing feature of the 2D svr-FFT structure enables us to explore its characteristics by identifying the local structural property. Each local module is designated as a DFT (non-DFT) block if its output corresponds to DFT (non-DFT) values. The block attribute (DFT or non-DFT) d...

It is proved that every pseudo-self-affine tiling in R is mutually locally derivable with a self-affine tiling. A characterization of pseudo-self-similar tilings in terms of derived Voronöı tessellations is a corollary. Previously, these results were obtained in the planar case, jointly with Priebe Frank. The new approach is based on the theory of graph-directed iterated function systems and su...

The two main theorems of this paper provide a characterization of hyperbolic affine iterated function systems defined on R. Atsushi Kameyama [Distances on topological self-similar sets, in Fractal Geometry and Applications: A Jubilee of Benǒit Mandelbrot. Proceedings of Symposia in Pure Mathematics, Volume 72, Part 1, 2004, pages 117–129] asked the following fundamental question: Given a topolo...

We examine iterated function systems consisting of a countably innnite number of contracting mappings (IIFS). We state results analogous to the well-known case of nitely many mappings (IFS). Moreover, we show that IIFS can be approximated by appropriately chosen IFS both in terms of Hausdorff distance and of Hausdorff dimension. Comparing the descriptive power of IFS and IIFS as mechanisms deen...