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Unitary representations of wavelet groups and encoding of iterated function systems in solenoids

Published online by Cambridge University Press:  12 March 2009

DORIN ERVIN DUTKAY ,
PALLE E. T. JORGENSEN  and
GABRIEL PICIOROAGA
DORIN ERVIN DUTKAY
Affiliation:
Department of Mathematics, University of Central Florida, 4000 Central Florida Blvd., PO Box 161364, Orlando, FL 32816-1364, USA (email: ddutkay@mail.ucf.edu)
PALLE E. T. JORGENSEN
Affiliation:
Department of Mathematics, University of Iowa, 14 MacLean Hall, Iowa City, IA 52242-1419, USA (email: jorgen@math.uiowa.edu)
GABRIEL PICIOROAGA
Affiliation:
Department of Mathematical Sciences, Binghamton University, Binghamton, NY 13902-6000, USA (email: gabriel@math.binghamton.edu)
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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 ℤ. Our starting point is a given pair (A,𝒟) with the matrix A assumed expansive, and 𝒟 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 (Theorem 6.15) 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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 6 , December 2009 , pp. 1815 - 1852
Copyright
Copyright © Cambridge University Press 2009

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