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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Probability measures on solenoids corresponding to fractal wavelets


Authors: Lawrence W. Baggett, Kathy D. Merrill, Judith A. Packer and Arlan B. Ramsay
Journal: Trans. Amer. Math. Soc. 364 (2012), 2723-2748
MSC (2010): Primary 42C40; Secondary 22D30, 28A80
Published electronically: January 6, 2012
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Abstract: The measure on generalized solenoids constructed using filters by Dutkay and Jorgensen (2007) is analyzed further by writing the solenoid as the product of a torus and a Cantor set. Using this decomposition, key differences are revealed between solenoid measures associated with classical filters in $ \mathbb{R}^d$ and those associated with filters on inflated fractal sets. In particular, it is shown that the classical case produces atomic fiber measures, and as a result supports both suitably defined solenoid MSF wavelets and systems of imprimitivity for the corresponding wavelet representation of the generalized Baumslag-Solitar group. In contrast, the fiber measures for filters on inflated fractal spaces cannot be atomic, and thus can support neither MSF wavelets nor systems of imprimitivity.


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Additional Information

Lawrence W. Baggett
Affiliation: Department of Mathematics, Campus Box 395, University of Colorado, Boulder, Colorado 80309-0395
Email: baggett@colorado.edu

Kathy D. Merrill
Affiliation: Department of Mathematics, Colorado College, Colorado Springs, Colorado 80903-3294
Email: kmerrill@coloradocollege.edu

Judith A. Packer
Affiliation: Department of Mathematics, Campus Box 395, University of Colorado, Boulder, Colorado 80309-0395
Email: packer@colorado.edu

Arlan B. Ramsay
Affiliation: Department of Mathematics, Campus Box 395, University of Colorado, Boulder, Colorado 80309-0395
Email: ramsay@colorado.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05584-X
PII: S 0002-9947(2012)05584-X
Keywords: Fractals, wavelets, solenoids, probability measures
Received by editor(s): January 1, 2010
Published electronically: January 6, 2012
Additional Notes: This research was supported in part by a grant from the National Science Foundation DMS–0701913
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.