Abstract
We generalize theorems of Peres and Solomyak about the abso- lute continuity resp. singularity of Bernoulli convolutions ([19], [16], [17]) to a broader class of self-similar measures on the real line. Using the dimension the- ory of ergodic measures (see [11] and [2]) we find a formula for the dimension of certain self-affine measures in terms of the dimension of the above mentioned self- similar measures. Combining these results we show the identity of Hausdorff and box-counting dimension of a special class of self-affine sets.
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Neunhauserer, J. Properties of Some Overlapping Self-Similar and Some Self-Affine Measures. Acta Mathematica Hungarica 92, 143–161 (2001). https://doi.org/10.1023/A:1013716430425
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DOI: https://doi.org/10.1023/A:1013716430425