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Continuity of the multifractal spectrum of a random statistically self-similar measure. (English) Zbl 0977.37024

Let \(E_\alpha\) be a set of the points where a measure \(\mu\) possesses a local Hölder exponent equal to \(\alpha\) and denote by \(\dim E_\alpha\) the Hausdorff dimension of \(E_\alpha\); then there exists a deterministic open subinterval of \(\mathbb{R}_+\) and \(\tau\) a convex function on \(\mathbb{R}\), such that for every \(\alpha \in I\), with probability 1, \(\dim E_\alpha=\inf_{q\in\mathbb{R}} \alpha q+\tau(q)=\tau^\ast(\alpha)>0\). However, if this statement is precise for a given \(\alpha\in I\) with probability 1, it is not satisfying because it does not give with probability 1 the Hausdorff dimension of \(E_\alpha\) for all \(\alpha\in I\). The main result is: With probability 1, \(\dim E_\alpha=\tau^\ast(\alpha)\) for all \(\alpha\in I\).
The author also studies another problem if \(\alpha=\text{mf}(I)\) or \(\alpha=\sup (I)\), one does not know whether \(E_\alpha\) is empty or not. Under suitable assumptions it is shown that \(E_\alpha\neq \Theta\) and \(\dim E_\alpha\) is calculated.

MSC:

37F35 Conformal densities and Hausdorff dimension for holomorphic dynamical systems
28A80 Fractals
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