Asymptotic order of the quantization errors for a class of self-affine measures

Zhu, S.

Proceedings of the American Mathematical Society 146(2): 637-651

2017


ISSN/ISBN: 0002-9939
Accession: 081972100

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Abstract
Let E E be a Bedford-McMullen carpet determined by a set of affine mappings ( f i j ) ( i , j ) G (f_{ij})_{(i,j)\in G} and μ \mu a self-affine measure on E E associated with a probability vector ( p i j ) ( i , j ) G (p_{ij})_{(i,j)\in G} . We prove that, for every r ( 0 , ) r\in (0,\infty ) , the upper and lower quantization coefficient are always positive and finite in its exact quantization dimension s r < mml:annotation encoding="application/x-tex">s_r . As a consequence, the n n th quantization error for μ \mu of order