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On the L q spectra of in-homogeneous self-similar measures

  • Shuqin Zhang , Bing Gao and Yingqing Xiao EMAIL logo
Published/Copyright: July 23, 2022

Abstract

The in-homogeneous self-similar measure μ is defined by the relation

μ = i = 1 N p i μ S i - 1 + p ν ,

where ( p 1 , , p N , p ) is a probability vector, each S i : d d , i = 1 , , N , is a contraction similarity, and ν is a compactly supported Borel probability measure on d . In this paper, we study the L q -spectra of in-homogeneous self-similar measures. We obtain non-trivial lower and upper bounds for the L q -spectra of an arbitrary in-homogeneous self-similar measure. Moreover, if the IFS satisfies some separation conditions, the bounds for the L q -spectra can be improved.

MSC 2010: 28A80; 37C45

Communicated by Christopher D. Sogge


Award Identifier / Grant number: 12071118

Award Identifier / Grant number: 2020JJ4163

Funding statement: This work was supported by National Natural Science Foundation of China (Grant No. 12071118) and Natural Science Foundation of Hunan Province (Grant No. 2020JJ4163).

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Received: 2022-05-03
Published Online: 2022-07-23
Published in Print: 2022-09-01

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