Variational principles for Hausdorff and packing dimensions of fractal percolation on self-affine sponges

TL;DR

This paper develops variational principles to determine the Hausdorff and packing dimensions of self-affine fractal sets, including percolation models, by analyzing inhomogeneous Mandelbrot measures and digit frequency structures.

Contribution

It introduces a novel approach to compute dimensions of self-affine sponges using specialized Mandelbrot measures and localized digit frequency analysis, extending existing methods.

Findings

Established variational principles for self-affine sponge dimensions.

Computed dimensions of inhomogeneous Mandelbrot measures on random sets.

Provided sharp bounds for Hausdorff and packing dimensions.

Abstract

We establish variational principles for the Hausdorff and packing dimensions of a class of statistically self-affine sponges, including in particular fractal percolation sets obtained from Bara\'nski and Gatzouras-Lalley carpets and sponges. Our first step is to compute the Hausdorff and packing dimensions of non-degenerate inhomogeneous Mandelbrot measures supported on the associated random limit sets. This is not a straightforward combination of the existing approaches for the deterministic inhomogeneous Bernoulli measures and the Mandelbrot measures on random Sierpi\'nski sponges; it reveals new structural features. The variational principles rely on a specific subclass of inhomogeneous Mandelbrot measures, which are connected to localized digit frequencies in the underlying coding space. This connection makes it possible to construct effective coverings of the random limit set, leading to sharp upper bounds for its Hausdorff and packing dimensions.