Ergodic scales in fractal measures
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- by Palle E. T. Jorgensen PDF
- Math. Comp. 81 (2012), 941-955 Request permission
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($\mu$), $\mu$).References
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Additional Information
- Palle E. T. Jorgensen
- Affiliation: Department of Mathematics, The University of Iowa, Iowa City, Iowa 52242-1419
- MR Author ID: 95800
- ORCID: 0000-0003-2681-5753
- Email: jorgen@math.uiowa.edu
- 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.
- © Copyright 2011 American Mathematical Society
- 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
- MathSciNet review: 2869044