Horofunctions on Sierpi\'nski type triangles

TL;DR

This paper investigates a class of recursively constructed graphs inspired by the Sierpiński gasket, analyzing their properties and describing horofunctions in a specific standard case.

Contribution

It introduces a new family of infinite graphs based on recursive word construction and characterizes horofunctions for the standard example.

Findings

Infinitely many non-isomorphic graphs in the family

Explicit description of horofunctions for the standard graph

Recursive construction method for Sierpiński-inspired graphs

Abstract

We study an infinite set of graphs which are recursively constructed from an infinite word in a finite alphabet. These graphs are inspired by the construction of the Sierpi\'nski gasket. We show that there are infinitely many non-isomorphic such graphs and we describe the horofunctions on the standard case.