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

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

2007-06-11

arxiv.org/abs/0706.1483v3

Mathematics

Number Theory

Scientific paper

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.

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