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On entropy and Hausdorff dimension of measures defined through a non-homogeneous Markov process

Athanasios Batakis — 2006

Colloquium Mathematicae

We study the Hausdorff dimension of measures whose weight distribution satisfies a Markov non-homogeneous property. We prove, in particular, that the Hausdorff dimensions of this kind of measures coincide with their lower Rényi dimensions (entropy). Moreover, we show that the packing dimensions equal the upper Rényi dimensions. As an application we get a continuity property of the Hausdorff dimension of the measures, when viewed as a function of the distributed weights under the ℓ ∞ norm.

Dimension of the harmonic measure of non-homogeneous Cantor sets

Athanasios Batakis — 2006

Annales de l’institut Fourier

We prove that the dimension of the harmonic measure of the complementary of a translation-invariant type of Cantor sets is a continuous function of the parameters determining these sets. This results extends a previous one of the author and do not use ergotic theoretic tools, not applicables to our case.

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