Gap property of Bi-Lipschitz constants of Bi-Lipschitz automorphisms on self-similar sets

Abstract

For a given self-similar set E ⊂ ℝd satisfying the strong separation condition, let Aut(E) be the set of all bi-Lipschitz automorphisms on E. The authors prove that “f ∈ Aut(E): blip(f) = 1” is a finite group, and the gap property of bi-Lipschitz constants holds, i.e., inf“blip(f) ≠ 1: f ∈ Aut(E)” > 1, where lip(g) = Open image in new window and blip(g) = max(lip(g), lip(g−1)).