On the \(L^q\)-spectrum of planar self-affine measures. (English) Zbl 1342.28014
In this paper, the author studies the dimension theory of a class of planar self-affine multifractal measures. These are the Bernoulli measures supported on box-like self-affine sets. These measures were introduced by the author in previous work, and they are the attractors of iterated systems consisting of contracting affine maps which take the unit square to rectangles with sides parallel to the axes. The \(L^q\)-spectrum of a measure is a quantitative description of the global fluctuations of this measure. Under some technical condition called “the rectangular open set condition”, the author computes the \(L^q\)-spectrum of the measures he studies, by means of a \(q\)-modified singular value function. There are connections between the \(L^q\)-spectrum of a measure and many of its geometric characteristics. They usually are concerning dimension. In the case where there are no mappings that switch the coordinate axes, the author obtains a closed form expression for the \(L^q\)-spectrum. He gives applications to the computation of the Hausdorff, packing and entropy dimensions of the measure, as well as of the Hausdorff and packing dimensions of the support.
Reviewer: Athanase Papadopoulos (Strasbourg)
MSC:
| 28A80 | Fractals |
| 37C45 | Dimension theory of smooth dynamical systems |
| 28A78 | Hausdorff and packing measures |
| 15A18 | Eigenvalues, singular values, and eigenvectors |
| 26A24 | Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems |
Keywords:
\(L^q\) spectrum; self-affine measure; modified singular value function; Hausdorff dimensionReferences:
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