Sharp Lipschitz constant of bi-Lipschitz automorphism on Cantor set

Suppose Cr = (rCr) ∪ (rCr + 1 - r) is a self-similar set with r ∈ (0, 1/2), and Aut(Cr) is the set of all bi-Lipschitz automorphisms on Cr. This paper proves that there exists f* ∈ Aut(Cr) such that blip(f*) = inf{blip(f) > 1 : f ∈ Aut(Cr)} = min 1r , (1 -1 2 -r) 2(r13 + - r r +4 r2) , where lip(g) = supx,y∈Cr, x=y |g(x|x)--yg(| 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)).