Random affine code tree fractals: Hausdorff and affinity dimensions and pressure

TL;DR

This paper establishes that for random affine code tree fractals, the affinity, Hausdorff, packing, and box counting dimensions all coincide with the zero of a pressure function, without extra assumptions.

Contribution

It proves the equality of various fractal dimensions to the pressure zero for random affine code tree fractals without requiring Falconer-Sloan conditions.

Findings

Dimensions equal to pressure zero almost surely

No need for additional assumptions like Falconer-Sloan

Results apply to a broad class of random affine fractals

Abstract

We prove that for random affine code tree fractals the affinity dimension is almost surely equal to the unique zero of the pressure function. As a consequence, we show that the Hausdorff, packing and box counting dimensions of such systems are equal to the zero of the pressure. In particular, we do not presume the validity of the Falconer-Sloan condition or any other additional assumptions which have been essential in all the previously known results.