Lipschitz Equivalence Class, Ideal Class and the Gauss Class Number Problem

TL;DR

This paper establishes a connection between the classification of self-similar fractal sets under bi-Lipschitz maps and algebraic number theory, linking Lipschitz invariants to ideal classes in rings.

Contribution

It introduces a bi-Lipschitz invariant based on ideal classes, bridging fractal geometry classification with the Gauss class number problem.

Findings

Lipschitz equivalence classes correspond to ideal classes in a related ring

The invariant provides a new approach to classify self-similar sets

Reveals a deep link between fractal geometry and algebraic number theory

Abstract

In this paper, we study the question of classifying self-similar sets under bi-Lipschitz mappings and obtain an important bi-Lipschitz invariant, which is an ideal of a ring related to IFS. Roughly speaking, different Lipschitz equivalence classes of self-similar sets correspond to different ideal classes of a related ring. This result reveals an interesting relationship between the Lipschitz classification problem in fractal geometry and the Gauss class number problem in algebraic number theory.