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

TL;DR

This paper explores the properties of $p$-adic self-similar sets and measures, revealing their fractal structure, uniqueness of essential classes, and similarities to real self-similar measures, with some distinctions.

Contribution

It establishes the connection between $p$-adic self-similar sets and path set fractals, and analyzes their measure-theoretic properties, including local dimensions and decimation behavior.

Findings

$p$-adic self-similar sets are $p$-adic path set fractals

Existence of a unique essential class for these sets and measures

Decimation of $p$-adic self-similar sets is maximal under mild conditions

Abstract

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.