On multifractal formalism for self-similar measures with overlaps
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Mathematische Zeitschrift
On multifractal formalism for self-similar measures with overlaps Julien Barral1 · De-Jun Feng2 Received: 18 May 2020 / Accepted: 29 July 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Let μ be a self-similar measure generated by an IFS Φ = {φi }i=1 of similarities on Rd (d ≥ 1). When Φ is dimensional regular (see Definition 1.1), we give an explicit formula for the L q -spectrum τμ (q) of μ over [0, 1], and show that τμ is differentiable over (0, 1] and the multifractal formalism holds for μ at any α ∈ [τμ (1), τμ (0+)]. We also verify the validity of the multifractal formalism of μ over [τμ (∞), τμ (0+)] for two new classes of overlapping algebraic IFSs by showing that the asymptotically weak separation condition holds. For one of them, the proof appeals to the recent result of Shmerkin (Ann. Math. (2) 189(2):319–391, 2019) on the L q -spectrum of self-similar measures. Keywords Multifractal formalism · Self-similar measures · Hausdorff dimension · Asymptotically weak separation condition Mathematics Subject Classification 28A80 · 37C45
1 Introduction Self-similar sets and measures are natural and important objects in fractal geometry, at the interface of geometric measure theory, ergodic theory, number theory and harmonic analysis (see the recent surveys [14,17,33]). Spectacular and influential advances in the dimension theory of these objects have been achieved in the recent period [3,12,15,18,19,34,38,39], in particular in connection with the resolutions of Furstenberg’s conjectures on the Hausdorff
The research of both authors was supported in part by University of Paris 13, the HKRGC GRF grants (projects CUHK14301218, CUHK14304119), the Direct Grant for Research in CUHK, and the France/Hong Kong joint research scheme PROCORE (33160RE, F-CUHK402/14).
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De-Jun Feng [email protected] Julien Barral [email protected]
1
Laboratoire de Géométrie, Analyse et Applications, Université Sorbonne Paris Nord, CNRS, UMR 7539, 93430 Villetaneuse, France
2
Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong
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J. Barral, D.-J. Feng
dimension of the sums and intersections of ×2- and ×3-invariant sets on the 1-dimensional torus. In this paper, we present some new results on the multifractal analysis of self-similar measures. One of them concerns the precise value of their L q -spectra over [0, 1] and the validity of the multifractal formalism, while the other ones provide sufficient conditions under which the underlying iterated function system (IFS) satisfies the asymptotically weak separation condition (AWSC), see Definition 2.1. (This separation condition guarantees the validity of the multifractal formalism in the range of q > 0 [9].) Most of these results rely on the achievements in [1,15,34]. Before giving the backgrounds and precise formulations of our results, below we first introduce some necessary notation and definitions. Recall that for a finite Borel measure η on Rd with compact support, the L q
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