Porosities of Mandelbrot Percolation

TL;DR

This paper investigates the porosity properties of the Mandelbrot percolation set, demonstrating that at almost all points, the mean porosities of the set and measure coincide, with lower porosities being zero and upper porosities reaching maximum values.

Contribution

It establishes the almost sure equality of mean porosities of the set and measure in Mandelbrot percolation for all but countably many parameters.

Findings

Mean porosities of set and measure are equal almost surely.

Lower porosities are zero at almost all points.

Upper porosities reach maximum values almost surely.

Abstract

We study porosities in the Mandelbrot percolation process. We show that, almost surely at almost all points with respect to the natural measure, the mean porosities of the set and the natural measure exist and are equal to each other for all parameter values outside of a countable exceptional set. As a corollary, we obtain that, almost surely at almost all points, the lower porosities of the set and the natural measure are equal to zero, whereas the upper porosities obtain their maximum values.