We study, from the point of view of the multifractal analysis, iterated function systems on totally disconnected spaces, namely, the boundaries of homogeneous trees. In particular, we study in this setting the ＂weak quasi-Bernoulli property introduced by Testud [3, 4]. After projection on R or R2, we get new examples of self-similar measures which being WQB, obey the multifractal formalism for positive q,s.
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)).
Suppose C r = (r C r ) ∪ (r C r + 1 ? r) is a self-similar set with r ∈ (0, 1/2), and Aut(C r ) is the set of all bi-Lipschitz automorphisms on C r . This paper proves that there exists f* ∈ Aut(C r ) such that $$ blip(f*) = inf\{ blip(f) > 1:f \in Aut(C_r )\} = min\left[ {\frac{1} {r},\frac{{1 - 2r^3 - r^4 }} {{(1 - 2r)(1 + r + r^2 )}}} \right], $$ where $ lip(g) = sup_{x,y \in C_r ,x \ne y} \frac{{\left| {g(x) - g(y)} \right|}} {{\left| {x - y} \right|}} $ and blip(g) = max(lip(g), lip(g ?1)).
