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

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.