School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Quantization dimension for Gibbs-like measures on cookie-cutter sets

Document Type

Article

Publication Date

Summer 2014

Abstract

In this paper using the Banach limit we have determined a Gibbs-like measure μ h supported by a cookie-cutter set E which is generated by a single cookie-cutter mapping f . For such a measure μ h and r ∈ ( 0 , + ∞ ) we have shown that there exists a unique κ r ∈ ( 0 , + ∞ ) such that κ r is the quantization dimension function of the probability measure μ h , and we established its functional relationship with the temperature function of the thermodynamic formalism. The temperature function is commonly used to perform the multifractal analysis, in our context of the measure μ h . In addition, we have proved that the κ r -dimensional lower quantization coefficient of order r of the probability measure is positive.

Comments

© 2014 Kyoto University. Original published version available at https://doi.org/10.1215/21562261-2642377

Publication Title

Kyoto Journal of Mathematics

DOI

10.1215/21562261-2642377

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