On unconventional limit sets of contractive functions on $\mathbb Z_p$

TL;DR

This paper investigates the metric properties of unconventional limit sets generated by contractive functions on the p-adic integers, establishing their topological features and equivalence to symbolic Cantor sets.

Contribution

It introduces the concept of unconventional limit sets in the p-adic setting and proves their compactness, perfectness, and uniform disconnectedness, along with an example of their quasi-symmetric equivalence to Cantor sets.

Findings

Unconventional limit sets are compact and perfect.

These sets are uniformly disconnected.

They can be quasi-symmetrically equivalent to symbolic Cantor sets.

Abstract

In the present paper, we are going to study metric properties of unconventional limit set of a semigroup $G$ generated by contractive functions $\{f_{i}\}_{i=1}^N$ on the unit ball $\mathbb Z_p$ of $p$-adic numbers. Namely, we prove that the unconventional limit set is compact, perfect and uniformly disconnected. Moreover, we provide an example of two contractions for which the corresponding unconventional limiting set is quasi-symmetrically equivalent to the symbolic Cantor set.