In this paper,we get the formulas of upper(lower) pointwise dimensions of some Moran measures on Moran sets in Rd under the strong separation condition.We also obtain formulas for the dimension of the Moran measures.Our results extend the known results of some self-similar measures and Moran measures studied by Cawley and Mauldin.
In this paper, we discuss the Lipschitz equivalence of self-similar sets with triangular pattern. This is a generalization of {1, 3, 5}-{1, 4, 5} problem proposed by David and Semmes. It is proved that if two such self-similar sets are totally disconnected, then they are Lipschitz equivalent if and only if they have the same Hausdorff dimension.
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 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)).
