Hausdorff dimension of certain random self--affine fractals

TL;DR

This paper derives a formula for the Hausdorff dimension of certain random self-affine fractals generated by a finite set of transformations chosen via Bernoulli measure, extending previous work on self-affine fractals.

Contribution

It provides a new dimension formula for random self-affine fractals, generalizing earlier results on deterministic fractals like Sierpinski carpets.

Findings

Derived a Hausdorff dimension formula using variational principles.

Extended known results to random self-affine fractals.

Applicable under specific technical hypotheses.

Abstract

In this work we are interested in the self--affine fractals studied by Gatzouras and Lalley and by the author which generalize the famous general Sierpinski carpets studied by Bedford and McMullen. We give a formula for the Hausdorff dimension of sets which are randomly generated using a finite number of self-affine transformations each one generating a fractal set as mentioned before, with some technical hypotheses. The choice of the transformation is random according to a Bernoulli measure. The formula is given in terms of the variational principle for the dimension.