Metric compactification of infinite Sierpi\'nski carpet graphs

TL;DR

This paper constructs a sequence of finite graphs approximating the Sierpiński carpet from infinite words, analyzes their metric compactification, and classifies boundary points, revealing a typical boundary structure with four Busemann points.

Contribution

It provides an explicit description of the metric boundary of Sierpiński carpet graphs and classifies Busemann and non-Busemann points in the boundary.

Findings

Boundary typically has four Busemann points.

Countably many non-Busemann points are present.

Almost all graphs share this boundary structure under Bernoulli measure.

Abstract

We associate, with every infinite word over a finite alphabet, an increasing sequence of rooted finite graphs, which provide a discrete approximation of the famous Sierpi\'nski carpet fractal. Each of these sequences converges, in the Gromov-Hausdorff topology, to an infinite rooted graph. We give an explicit description of the metric compactification of each of these limit graphs. In particular, we are able to classify Busemann and non-Busemann points of the metric boundary. It turns out that, with respect to the uniform Bernoulli measure on the set of words indexing the graphs, for almost all the infinite graphs, the boundary consists of four Busemann points and countably many non-Busemann points.