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

Abstract

For a given self-similar set E ⊂ ℝ ^d 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) = and blip(g) = max(lip(g), lip(g ⁻¹)).

Keywords

Fractal / Bi-Lipschitz automorphism / Self-similar set

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