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

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.