Completion of $S/I$

We study completion of $S =$ the Sierpinski gasket minus $I =$ the unit interval = the one of the segment of outer triangle of the SG. In the Euclidean distance, the completion is just the SG itself. But if we consider an intrinsic metric on $S/I$, we have di fferent space. In fact, if we consider the Brownian motion on $S/I$, it is "equivalent" to a random walk on a tree and we will get the ternary Cantor set as the Martin boundary. This fact is closely related to the study of a trace of the Brownian motion on the SG to the unit interval.