Asymptotic order of the quantization errors for a class of self-affine measures
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Abstract:
Let $E$ be a Bedford-McMullen carpet determined by a set of affine mappings $(f_{ij})_{(i,j)\in G}$ and $\mu$ a self-affine measure on $E$ associated with a probability vector $(p_{ij})_{(i,j)\in G}$. We prove that, for every $r\in (0,\infty )$, the upper and lower quantization coefficient are always positive and finite in its exact quantization dimension $s_r$. As a consequence, the $n$th quantization error for $\mu$ of order $r$ is of the same order as $n^{-\frac {1}{s_r}}$. In sharp contrast to the Hausdorff measure for Bedford-McMullen carpets, our result is independent of the horizontal fibres of the carpets.References
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Additional Information
- Sanguo Zhu
- Affiliation: School of Mathematics and Physics, Jiangsu University of Technology, Changzhou 213001, People’s Republic of China
- MR Author ID: 662721
- Email: sgzhu@jsut.edu.cn
- Received by editor(s): December 21, 2016
- Received by editor(s) in revised form: March 22, 2017
- Published electronically: September 7, 2017
- Additional Notes: The author was supported by NSFC 11571144
- Communicated by: Jeremy Tyson
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 637-651
- MSC (2010): Primary 28A80, 28A78; Secondary 94A15
- DOI: https://doi.org/10.1090/proc/13756
- MathSciNet review: 3731698