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

Abstract

In this paper, we investigate $p$-adic self-similar sets and $p$-adic self-similar measures. We introduce a condition (C) under which $p$-adic self-similar sets can be shown to have a number of nice properties. It is shown that $p$-adic self-similar sets satisfying condition (C) are $p$-adic path set fractals. This allows us to easily compute the Hausdorff dimension of these sets. We further show that the set of $p$-adic path set fractals is strictly larger than this set of $p$-adic self-similar sets. The directed graph associated to $p$-adic self-similar sets satisfying condition (C) is shown to have a unique essential class. Moreover, it is shown that almost all points are eventually in the essential class. For $p$-adic self-similar measures satisfying this condition, we show that many results involving local dimension are similar to those of their real counterparts, with fewer complications. We next study the more general $p$-adic path set fractals, first showing that the existence of an interior point is equivalent to the set having Hausdorff dimension $1$. We further show that often the decimation of $p$-adic path set fractals results in a set with maximal Hausdorff dimension.