Unitary Representations of Wavelet Groups and Encoding of Iterated Function Systems in Solenoids

Abstract

For points in $d$ real dimensions, we introduce a geometry for general digit sets. We introduce a positional number system where the basis for our representation is a fixed $d$ by $d$ matrix over $\bz$. Our starting point is a given pair $(A,\mathcal{D})$ with the matrix $A$ assumed expansive, and $\mathcal{D}$ a chosen complete digit set, i.e., in bijective correspondence with the points in $\bz^d/A^T\bz^d$. We give an explicit geometric representation and encoding with infinite words in letters from $\mathcal{D}$. We show that the attractor $X(A^T,\mathcal{D})$ for an affine Iterated Function System (IFS) based on $(A,\mathcal{D})$ is a set of fractions for our digital representation of points in $\br^d$. Moreover our positional "number representation" is spelled out in the form of an explicit IFS-encoding of a compact solenoid $\sa$ associated with the pair $(A,\mathcal{D})$. The intricate part (Theorem \ref{thenccycl}) is played by the cycles in $\bz^d$ for the initial $(A,\mathcal{D})$-IFS. Using these cycles we are able to write down formulas for the two maps which do the encoding as well as the decoding in our positional $\mathcal{D}$-representation. We show how some wavelet representations can be realized on the solenoid, and on symbolic spaces.

References in corpus (1)

• Hilbert spaces built on a similarity and on dynamical renormalization

Cited by in corpus (1)

• Orthonormal dilations of Parseval wavelets