Strict Hölder equivalence of self-similar sets
Yanfang Zhang, Xinhui Liu
TL;DR
The paper completely characterizes strict Hölder equivalence for two natural classes of self-similar sets: totally disconnected fractal cubes and two-branch self-similar sets under the strong separation condition. It reduces the problems to symbolic dynamics and leverages known Lipschitz classification results to derive necessary and sufficient conditions, yielding a clean arithmetical criterion for fractal cubes and a precise, case-split dichotomy for two-branch systems. Specifically, fractal cubes $E=K(n,\mathcal{D})$ and $F=K(n',\mathcal{D}')$ are strictly Hölder equivalent iff $\frac{\log N}{\log N'}\in \mathbb{Q}$, where $N=|\mathcal{D}|$, $N'=|\mathcal{D}'|$. For SSC two-branch sets, strict Hölder equivalence depends on the ratio $\log r_1/\log r_2$ being irrational (leading to $\frac{\log r_1}{\log t_1}=\frac{\log r_2}{\log t_2}$) or rational (allowing only the specific pairings $(\frac{\log r_1}{\log r_2},\frac{\log t_1}{\log t_2})=(\frac{2}{3},\frac{1}{5})$ or swapped). These results deepen the understanding of rigidity in Hölder classifications of self-similar fractals.
Abstract
The study of Lipschitz equivalence of fractals is a very active topic in recent years. It is natural to ask when two fractal sets are strictly Hölder equivalent. In the present paper, we completely characterize the strict Hölder equivalence for two classes of self-similar sets: the first class is totally-disconnected fractal cubes and the second class is self-similar sets with two branches which satisfy the strong separation condition.