Measures of the full Hausdorff dimension for a general Sierpi\'{n}ski carpet

TL;DR

This paper establishes criteria for the measure of full Hausdorff dimension in general Sierpiński carpets, extending previous results and providing conditions for existence, uniqueness, and properties of projection measures.

Contribution

It introduces new criteria for the measure of full Hausdorff dimension and analyzes projection measures in Sierpiński carpets, extending prior work without certain conditions.

Findings

Provides a checkable condition for existence and uniqueness of the full Hausdorff dimension measure.

Extends previous results by removing condition (H).

Estimates the number of steps for Markov projection measures.

Abstract

The measure of the full dimension for a general Sierpi\'{n}ski carpet is studied. In the first part of this study, we give a criterion for the measure of the full Hausdorff dimension of a Sierpi\'{n}ski carpet. Meanwhile, it is the conditional equilibrium measure of zero potential with respect to some Gibbs measure $\nu_{\alpha}$ of matrix-valued potential $\alpha\mathbf{N}$ (defined later). On one hand, this investigation extends the result of [17] without condition \textbf{(H)}. On the other hand, it provides a checkable condition to ensure the existence and uniqueness of the measure of the full Hausdorff dimension for a general Sierpi\'{n}ski carpet. In the second part of this paper we give a criterion for the Markov projection measure and estimate its number of steps by means of the induced matrix-valued potential. The results enable us to answer some questions which arise from [1] and [4] on the projection measure and factors.