  In 2012 Lau and Ngai, motivated by the work of Denker and Sato, gave an
example of an isotropic Markov chain on the set of finite words over a three
letter alphabet, whose Martin boundary is homeomorphic to the Sierpi\'nski
gasket. Here, we extend the results of Lau and Ngai to a class of non-isotropic
Markov chains. We determine the Martin boundary and show that the minimal
Martin boundary is a proper subset of the Martin boundary. In addition, we give
a description of the set of harmonic functions.
