Topological pressure and fractal dimensions of cookie-cutter-like sets

TL;DR

This paper establishes that for cookie-cutter-like sets, the topological pressure function exists and determines all key fractal dimensions, which are equal to its unique zero, with finite positive measures at that dimension.

Contribution

It proves the existence of the topological pressure function and links it to the fractal dimensions and measures of cookie-cutter-like sets, extending previous results to this class.

Findings

Fractal dimensions equal the zero of the pressure function

Hausdorff and packing measures are finite and positive at the critical dimension

Topological pressure function exists for cookie-cutter-like sets

Abstract

The cookie-cutter-like set is defined as the limit set of a sequence of classical cookie-cutter mappings. For this cookie-cutter set it is shown that the topological pressure function exists, and that the fractal dimensions such as the Hausdorff dimension, the packing dimension and the box-counting dimension are all equal to the unique zero $h$ of the pressure function. Moreover, it is shown that the $h$-dimensional Hausdorff measure and the $h$-dimensional packing measure are finite and positive.