Quantifying inhomogeneity in fractal sets
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An inhomogeneous fractal set is one which exhibits different scaling behaviour at different points. The Assouad dimension of a set is a quantity which finds the ‘most difficult location and scale’ at which to cover the set and its difference from box dimension can be thought of as a first-level overall measure of how inhomogeneous the set is. For the next level of analysis, we develop a quantitative theory of inhomogeneity by considering the measure of the set of points around which the set exhibits a given level of inhomogeneity at a certain scale. For a set of examples, a family of ( ×m, ×n )-invariant subsets of the 2-torus, we show that this quantity satisfies a Large Deviations Principle. We compare members of this family, demonstrating how the rate function gives us a deeper understanding of their inhomogeneity.
Fraser , J & Todd , M J 2018 , ' Quantifying inhomogeneity in fractal sets ' , Nonlinearity , vol. 31 , no. 4 , pp. 1313-1330 . https://doi.org/10.1088/1361-6544/aa9ee6
© 2018, IOP Publishing Ltd & London Mathematical Society. This work has been made available online in accordance with the publisher’s policies. This is the author created, accepted version manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.1088/1361-6544/aa9ee6
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