2019 On the Dimension and Measure of Inhomogeneous Attractors
Stuart A. Burrell
Real Anal. Exchange 44(1): 199-216 (2019). DOI: 10.14321/realanalexch.44.1.0199


A central question in the field of inhomogeneous attractors has been to relate the dimension of an inhomogeneous attractor to the condensation set and associated homogeneous attractor. This has been achieved only in specific settings, with notable results by Olsen, Snigireva, Fraser and Käenmäki on inhomogeneous self-similar sets, and by Burrell and Fraser on inhomogeneous self-affine sets. This paper is devoted to filling a significant gap in the dimension theory of inhomogeneous attractors, by studying those formed from arbitrary bi-Lipschitz contractions. We show that the maximum of the dimension of the condensation set and a quantity related to pressure, which we term upper Lipschitz dimension, forms a natural and general upper bound on the dimension. Additionally, we begin a new line of enquiry; the methods developed are used to classify the Hausdorff measure of inhomogeneous attractors. Our results have applications for affine systems with affinity dimension less than or equal to one and systems satisfying bounded distortion, such as conformal systems in dimensions greater than one.


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Stuart A. Burrell. "On the Dimension and Measure of Inhomogeneous Attractors." Real Anal. Exchange 44 (1) 199 - 216, 2019. https://doi.org/10.14321/realanalexch.44.1.0199


Published: 2019
First available in Project Euclid: 27 June 2019

zbMATH: 07088971
MathSciNet: MR3951342
Digital Object Identifier: 10.14321/realanalexch.44.1.0199

Primary: 28A78 , 28A80

Keywords: bounded distortion , box dimension , Hausdorff measure , inhomogeneous attractor , self-affine set , self-conformal set

Rights: Copyright © 2019 Michigan State University Press


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Vol.44 • No. 1 • 2019
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