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

Volume 104 / 2006

Athanasios Batakis Colloquium Mathematicum 104 (2006), 193-206 MSC: 28A78, 28A80, 60J60. DOI: 10.4064/cm104-2-3


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 $\ell^{\infty}$ norm.


  • Athanasios BatakisMAPMO
    Université d'Orléans
    BP 6759
    45067 Orléans Cedex 2, France

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