Gap Property of Bi-Lipschitz Constants of Bi-Lipschitz Automorphisms on Self-similar Sets

For a given self-similar set E subset of 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 is an element of Aut(E) : blip(f) = 1} is a finite group, and the gap property of bi-Lipschitz constants holds, i.e., inf{blip(f) not equal 1: f is an element of Aut(E)} > 1, where lip(g) = sup(x,y is an element of Ex not equal y)vertical bar g(x)-g(y)/vertical bar x-y vertical bar and blip(g) = max(lip(g),lip(g(-1))).