Strict Hölder equivalence of self-similar sets

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.