Projections of the natural measure for percolation fractals

TL;DR

This paper proves that for percolation fractals, all orthogonal projections of the natural measure are absolutely continuous, with most having Hölder continuous densities, highlighting regularity properties of these projections.

Contribution

It establishes the almost sure absolute continuity and Hölder regularity of projections of the natural measure on percolation fractals, a novel result in fractal measure theory.

Findings

All projections are absolutely continuous.

Most projections have Hölder continuous densities.

Results hold with probability 1.

Abstract

We prove that, with probability 1, all orthogonal projections of the natural measure on a percolation fractal are absolutely continuous and (except for the horizontal and vertical projection) have H\"older continuous density.