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Interpolating Between Dimensions

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Fractal Geometry and Stochastics VI

Part of the book series: Progress in Probability ((PRPR,volume 76))

Abstract

Dimension theory lies at the heart of fractal geometry and concerns the rigorous quantification of how large a subset of a metric space is. There are many notions of dimension to consider, and part of the richness of the subject is in understanding how these different notions fit together, as well as how their subtle differences give rise to different behaviour. Here we survey a new approach in dimension theory, which seeks to unify the study of individual dimensions by viewing them as different facets of the same object. For example, given two notions of dimension, one may be able to define a continuously parameterised family of dimensions which interpolates between them. An understanding of this ‘interpolation function’ therefore contains more information about a given object than the two dimensions considered in isolation. We pay particular attention to two concrete examples of this, namely the Assouad spectrum, which interpolates between the box and (quasi-)Assouad dimension, and the intermediate dimensions, which interpolate between the Hausdorff and box dimensions.

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Acknowledgements

The author thanks Stuart Burrell, Kenneth Falconer, Kathryn Hare, Kevin Hare, Antti Käenmäki, Tom Kempton, Sascha Troscheit, and Han Yu for many interesting discussions relating to dimension interpolation. He was financially supported in part by the EPSRC Standard Grant EP/R015104/1. He is also grateful to the Leverhulme Trust for funding his project New Perspectives in the dimension theory of fractals (2019–2023), which is largely focused on the concept of dimension interpolation. Finally, he thanks Stuart Burrell and Kenneth Falconer for making several helpful comments on an earlier version of this article.

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Authors and Affiliations

  1. School of Mathematics and Statistics, The University of St Andrews, St Andrews, Scotland

    Jonathan M. Fraser

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  1. Jonathan M. Fraser
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Correspondence to Jonathan M. Fraser .

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Editors and Affiliations

  1. Fakultät für Mathematik, Technische Universität Chemnitz, Chemnitz, Germany

    Uta Freiberg

  2. Mathematical Institute, University of Oxford, Oxford, UK

    Ben Hambly

  3. Fakultät für Mathematik, Universität Bielefeld, Bielefeld, Germany

    Michael Hinz

  4. Institut für Stochastik, Karlsruher Institut für Technologie, Karlsruhe, Germany

    Steffen Winter

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Fraser, J.M. (2021). Interpolating Between Dimensions. In: Freiberg, U., Hambly, B., Hinz, M., Winter, S. (eds) Fractal Geometry and Stochastics VI. Progress in Probability, vol 76. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-59649-1_1

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