Hausdorff dimension of asymptotic self-similar sets

TL;DR

This paper extends the concept of self-similar sets to asymptotic self-similar sets in general doubling metric spaces and determines their Hausdorff dimensions, including for Sierpinski gaskets on surfaces.

Contribution

It introduces asymptotic self-similar sets and almost similarity maps, extending prior results to more general metric spaces and geometric constructions.

Findings

Determined Hausdorff dimensions of asymptotic self-similar sets.

Extended results to Sierpinski gaskets on surfaces.

Provided a framework for geometric constructions in metric spaces.

Abstract

In this paper, we introduce the notion of asymptotic self-similar sets on general doubling metric spaces by extending the notion of self-similar sets, and determine their Hausdorff dimensions, which gives an extension of Balogh and Rohner 's result. This is carried out by introducing the notions of almost similarity maps and asymptotic similarity systems. These notions have an advantage of making geometric constructions possible. Actually, as an application, we determined the Hausdorff dimension of general Sierpinski gaskets on complete surfaces constructed by a geometric way in a natural manner.