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

In this work 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 R'enyi dimensions (entropy). Moreover, we show that the Tricot dimensions (packing dimension) equal the upper R'enyi dimensions. As an application we get a continuity property of the Hausdorff dimension of the measures, when it is seen as a function of the distributed weights under the ${\mathrm{\ell }}^{\mathrm{\infty }}$ norm.