Open Access
2014 Scaling by $5$ on a $\frac{1}{4}$--Cantor measure
Palle E.T. Jorgensen, Keri A. Kornelson, Karen L. Shuman
Rocky Mountain J. Math. 44(6): 1881-1901 (2014). DOI: 10.1216/RMJ-2014-44-6-1881


Each Cantor measure $\mu$ with scaling factor $\frac{1}{2n}$ has at least one associated orthonormal basis of exponential functions (ONB) for $L^2(\mu)$. In the particular case where the scaling constant for the Cantor measure is $\frac14$ and two specific ONBs are selected for $L^2(\mu_{1/4})$, there is a unitary operator $U$ defined by mapping one ONB to the other. This paper focuses on the case in which one ONB $\Gamma$ is the original Jorgensen-Pedersen ONB for the Cantor measure $\mu_{{1}/{4}}$ and the other ONB is $5\Gamma$. The main theorem of the paper states that the corresponding operator $U$ is ergodic in the sense that only the constant functions are fixed by $U$.


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Palle E.T. Jorgensen. Keri A. Kornelson. Karen L. Shuman. "Scaling by $5$ on a $\frac{1}{4}$--Cantor measure." Rocky Mountain J. Math. 44 (6) 1881 - 1901, 2014.


Published: 2014
First available in Project Euclid: 2 February 2015

MathSciNet: MR3310953
zbMATH: 1329.42006
Digital Object Identifier: 10.1216/RMJ-2014-44-6-1881

Primary: 26A30 , 42A63 , 42A85‎ , 46L45 , 47L60 , 58C40

Keywords: Bernoulli convolution , Cantor measure , decomposition theory , Fourier expansions , fractal measures , Hilbert space , Spectral theory , Unbounded operators

Rights: Copyright © 2014 Rocky Mountain Mathematics Consortium

Vol.44 • No. 6 • 2014
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