Title

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 dbyd matrix over ℤ. Our starting point is a given pair (A,D) with the matrix A assumed expansive, D and a chosen complete digit set, i.e., in bijective correspondence with the points in ℤd/ ATℤd. We give an explicit geometric representation and encoding with infinite words in letters from D. We show that the attractor X(AT,D) for an affine Iterated Function System (IFS) based on (A,D) is a set of fractions for our digital representation of points in ℝd. 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,D). The intricate part (Theorem6.15) is played by the cycles in ℤd for the initial (A,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 D-representation. We show how some wavelet representations can be realized on the solenoid, and on symbolic spaces. © 2009 Cambridge University Press.

Publication Date

12-1-2009

Publication Title

Ergodic Theory and Dynamical Systems

Volume

29

Issue

6

Number of Pages

1815-1852

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1017/S0143385708000904

Socpus ID

74349109625 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/74349109625

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