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Matrix representations for some self-similar measures on \({\mathbb {R}}^{d}\)

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Abstract

We establish matrix representations for self-similar measures on \({\mathbb {R}}^d\) generated by equicontractive IFSs satisfying the finite type condition. As an application, we prove that the \(L^q\)-spectrum of every such self-similar measure is differentiable on \((0,\infty )\). This extends an earlier result of Feng (J. Lond. Math. Soc. 68(1):102–118, 2003) to higher dimensions.

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Notes

  1. The self-similar set generated by \(\Phi \) is called a golden gasket in [3] if \(\lambda ^{-1}\) is a multinacci number and \(a_1, a_2, a_3\) are vertices of an equilateral triangle.

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Acknowledgements

The author is grateful to his supervisor, De-Jun Feng, for many helpful discussions, suggestions and in particular pointing out Lemma 4.4. He also wish to thank the anonymous referee for his/her suggestions that led to the improvement of the paper.

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The author was partially supported by the HKRGC GRF grant (project 14304119).

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  1. Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong

    Yu-Feng Wu

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  1. Yu-Feng Wu
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Correspondence to Yu-Feng Wu.

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Wu, YF. Matrix representations for some self-similar measures on \({\mathbb {R}}^{d}\). Math. Z. 301, 3345–3368 (2022). https://doi.org/10.1007/s00209-022-03019-2

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