Self-similar sets and self-similar measures in the $p$-adics

In this paper we investigate $p$-adic self-similar sets and $p$-adic self-similar measures. We show that $p$-adic self-similar sets are $p$-adic path set fractals, and that the converse is not necessarily true. For $p$-adic self-similar sets and $p$-adic self-similar measures, we show the existence of a unique essential class. We show that, under mild assumptions, the decimation of $p$-adic self-similar sets is maximal. For $p$-adic self-similar measures, we show that many results involving local dimension are similar to those of their real counterparts, with fewer complications. Most of these results use the additional structure of self-similarity, and are not true in general for $p$-adic path set fractals.