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Ergodic scales in fractal measures


Author: Palle E. T. Jorgensen
Journal: Math. Comp. 81 (2012), 941-955
MSC (2010): Primary 42A15, 43A10, 47A35, 60G10
DOI: https://doi.org/10.1090/S0025-5718-2011-02517-2
Published electronically: July 7, 2011
MathSciNet review: 2869044
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Abstract: We will consider a family of fractal measures on the real line $ \mathbb{R}$ which are fixed, in the sense of Hutchinson, under a finite family of contractive affine mappings. The maps are chosen such as to leave gaps on $ \mathbb{R}$. Hence they have fractal dimension strictly less than $ 1$. The middle-third Cantor construction is one example. Depending on the gaps and the scaling factor, it is known that the corresponding Hilbert space $ L^{2}(\mu)$ exhibits strikingly different properties. In this paper we show that when $ \mu$ is fixed in a certain class, there are positive integers $ p$ such that multiplication by $ p$ modulo $ 1$ induces an ergodic automorphism on the measure space (support($ \mu$), $ \mu$).


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Additional Information

Palle E. T. Jorgensen
Affiliation: Department of Mathematics, The University of Iowa, Iowa City, Iowa 52242-1419
Email: jorgen@math.uiowa.edu

DOI: https://doi.org/10.1090/S0025-5718-2011-02517-2
Keywords: Fourier expansion, fractal measure, bases in Hilbert space, ergodic automorphism.
Received by editor(s): October 6, 2009
Received by editor(s) in revised form: January 16, 2011
Published electronically: July 7, 2011
Additional Notes: This work was supported in part by a grant from the National Science Foundation.
Article copyright: © Copyright 2011 American Mathematical Society