Hausdorff dimension of a Cantor set on \(\mathbb{R}^1\) (Q1429088)

This work is based upon the second author's idea how to calculate the Hausdorff dimension generated by piecewise linear Markov transformations on a bounded interval. The authors consider a Cantor set generated by a piecewise linear and Markov transformation on \(\mathbb{R}^1\) with parameter \(p\). They consider the Hausdorff dimension of this Cantor set as a function of \(p\). If \(0<p<1/4\), the Markov chain is recurrent and the Hausdorff dimension of this Cantor set equals \(1\). If \(1/4<p<1/2\), the Markov chain is transient and the Hausdorff dimension of the Cantor set is lesser than \(1\).