An upper bound for the intermediate dimensions of Bedford–McMullen carpets
Abstract
The intermediate dimensions of a set Λ\LambdaΛ, elsewhere denoted by dimθΛ\dim_{\theta}\LambdadimθΛ, interpolate between its Hausdorff and box dimensions using the parameter θ∈[0,1]\theta\in[0,1]θ∈[0,1]. For a Bedford–McMullen carpet Λ\LambdaΛ with distinct Hausdorff and box dimensions, we show that dimθΛ\dim_{\theta}\LambdadimθΛ is strictly less than the box dimension of Λ\LambdaΛ for every θ<1\theta<1θ<1. Moreover, the derivative of the upper bound is strictly positive at θ=1\theta=1θ=1. This answers a question of Fraser; however, determining a precise formula for dimθΛ\dim_{\theta}\LambdadimθΛ still remains a challenging problem.
Citation
Kolossváry , I 2022 , ' An upper bound for the intermediate dimensions of Bedford–McMullen carpets ' , Journal of Fractal Geometry , vol. 9 , no. 1-2 , pp. 151-169 . https://doi.org/10.4171/jfg/118
Publication
Journal of Fractal Geometry
Status
Peer reviewed
ISSN
2308-1309Type
Journal article
Rights
Copyright ©2022 European Mathematical Society. Published by EMS Press. Open Access. This work is licensed under a CC BY 4.0 license
Description
The author was supported by the Leverhulme Trust Research Project Grant RPG-2019-034.Collections
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