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arXiv · 1511.02081

Quantifying inhomogeneity in fractal sets

Jonathan M. Fraser·Mike Todd
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Abstract

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 $(\times m, \times n)$-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.

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Jonathan M. Fraser, Mike Todd. 2015-11-06. Quantifying inhomogeneity in fractal sets. https://doi.org/10.1088/1361-6544%2Faa9ee6

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