For a given self-similar set ERd satisfying the strong separation condition,let Aut(E) be the set of all bi-Lipschitz automorphisms on E.The authors prove that {fAut(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)=sup x,y∈E x≠y(|g(x)-g(y)|)/|x-y| and blip(g)=max(lip(g),lip(g-1)).
In this paper, it is proved that any self-affine set satisfying the strong separation condition is uniformly porous. The author constructs a self-affine set which is not porous, although the open set condition holds. Besides, the author also gives a C^1 iterated function system such that its invariant set is not porous.
