The μ-topological Hausdoff dimension

Abstract

In 2015, R. Balkaa, Z. Buczolich and M. Elekes introduced the topological Hausdoff dimension which is a combination of the definitions of the topological dimension and the Hausdorff dimension. In our manuscript, we propose to generalize the topological Hausdorff dimension by combining the definitions of the topological dimension and the μ-Hausdorff dimension and we call it the μ-topological Hausdorff dimension. We will present upper and lower bounds for the μ-topological Hausdorff dimension of the Sierpinski carpet valid in a general framework. As an application, we give a large class of measures μ, where the μ-topological Hausdorff dimension of the Sierpinski carpet coincides with the lower and upper bounds.