Porosity, dimension, and local entropies: a survey

zbMATH Open1241.28008arXiv1110.5682MaRDI QIDQ2888980

Pablo Shmerkin

Publication date: 4 June 2012

Abstract: Porosity and dimension are two useful, but different, concepts that quantify the size of fractal sets and measures. An active area of research concerns understanding the relationship between these two concepts. In this article we will survey the various notions of porosity of sets and measures that have been proposed, and how they relate to dimension. Along the way, we will introduce the idea of local entropy averages, which arose in a different context, and was then applied to obtain a bound for the dimension of mean porous measures.

Full work available at URL: https://arxiv.org/abs/1110.5682

zbMATH Keywords

measuresdimensionporosityfractal sets

Mathematics Subject Classification ID

Entropy and other invariants (28D20) Fractals (28A80)

Related Items (8)

Locally rich compact sets ⋮ Traces of Sobolev spaces on piecewise Ahlfors-David regular sets ⋮ Fractal percolation, porosity, and dimension ⋮ Some porosity-type properties of sets related to the \(d\)-Hausdorff content ⋮ Porosities of Mandelbrot percolation ⋮ Oscillation of functions in the Hölder class ⋮ Dynamics of the scenery flow and geometry of measures ⋮ Porosity properties of perturbed Cantor sets

This page was built for publication: Porosity, dimension, and local entropies: a survey