On entropy and Hausdorff dimension of measures defined through a non-homogeneous Markov process
Volume 104 / 2006
Colloquium Mathematicum 104 (2006), 193-206
MSC: 28A78, 28A80, 60J60.
DOI: 10.4064/cm104-2-3
Abstract
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.
