A Multifractal Decomposition for Self-similar Measures with Exact Overlaps
We study self-similar measures in $\mathbb{R}$ satisfying the weak separation condition along with weak technical assumptions which are satisfied in all known examples. For such a measure $\mu$, we show that there is a finite set of concave functions $\{\tau_1,\ldots,\tau_m\}$ such that the $L^q$-sp...
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Main Author | |
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Format | Journal Article |
Language | English |
Published |
14.04.2021
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Subjects | |
Online Access | Get full text |
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Summary: | We study self-similar measures in $\mathbb{R}$ satisfying the weak separation
condition along with weak technical assumptions which are satisfied in all
known examples. For such a measure $\mu$, we show that there is a finite set of
concave functions $\{\tau_1,\ldots,\tau_m\}$ such that the $L^q$-spectrum of
$\mu$ is given by $\min\{\tau_1,\ldots,\tau_m\}$ and the multifractal spectrum
of $\mu$ is given by $\max\{\tau_1^*,\ldots,\tau_m^*\}$, where $\tau_i^*$
denotes the concave conjugate of $\tau_i$. In particular, the measure $\mu$
satisfies the multifractal formalism if and only if its multifractal spectrum
is a concave function. This implies that $\mu$ satisfies the multifractal
formalism at values corresponding to points of differentiability of the
$L^q$-spectrum. We also verify existence of the limit for the $L^q$-spectra of
such measures for every $q\in\mathbb{R}$. As a direct application, we obtain
many new results and simple proofs of well-known results in the multifractal
analysis of self-similar measures satisfying the weak separation condition. |
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DOI: | 10.48550/arxiv.2104.06997 |