Journal article
Unitary representations of wavelet groups and encoding of iterated function systems in solenoids
Ergodic Theory and Dynamical Systems, Vol.29(6), pp.1815-1852
2009
DOI: 10.1017/S0143385708000904
Abstract
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 โค. Our starting point is a given pair ( A ,๐) with the matrix A assumed expansive, and ๐ a chosen complete digit set, i.e., in bijective correspondence with the points in โค d / A T โค d . We give an explicit geometric representation and encoding with infinite words in letters from ๐. We show that the attractor X ( A T ,๐) for an affine Iterated Function System (IFS) based on ( A ,๐) is a set of fractions for our digital representation of points in โ d . Moreover our positional โnumber representationโ is spelled out in the form of an explicit IFS-encoding of a compact solenoid ๐ฎ A associated with the pair ( A ,๐). The intricate part (Theorem6.15) is played by the cycles in โค d for the initial ( A ,๐)-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 ๐-representation. We show how some wavelet representations can be realized on the solenoid, and on symbolic spaces.
Details
- Title: Subtitle
- Unitary representations of wavelet groups and encoding of iterated function systems in solenoids
- Creators
- Dorin Ervin DutkayPalle E.T JorgensenGabriel Picioroaga
- Resource Type
- Journal article
- Publication Details
- Ergodic Theory and Dynamical Systems, Vol.29(6), pp.1815-1852
- DOI
- 10.1017/S0143385708000904
- ISSN
- 1469-4417
- eISSN
- 1469-4417
- Language
- English
- Date published
- 2009
- Academic Unit
- Mathematics
- Record Identifier
- 9983985999702771
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