The Hausdorff Measure Of A Class Of Non-Homogeneous Moran Sets

The Hausdorff measure of self-similar sets is not only a focus but also a more difficult quesstion. At present, only a little of hausdrff measure of regular fractal are given (such as tri-cantor set). Even for Koch curve so typical fractal, its hausdorff measure is only estimated. Recently, people pay more attention to homogeneous Cantor set and Moran set, Cheng-Qin Qu, Zuo-ling Zhou discussed that the hausdorff measure of a class of homogeneous Moran sets, gave its precise expression. In this paper, it is mainly discussed that the hausdorff measures, given an equivalent defination of self-similar set satisfied open set condition and a hausdorff measure of a class of non-homogeneous Moran sets.The main research work is described :First, show that there exist a measureμ,such that the R-N derivative of hausdorff measure forμexist by Radon-Nikodym derivative and absolute continuous.based on it, get a equivalent definition of hausdorff measure. the conclusions as follow :Theorem2.4.1 Let E is a self-similar set satisfied open set condition, s=dim_H E , then:1)there exist a measureμ, such that (dH^s)/(dμ) exist;2)H^s(E)=(?)[sum from k=0 to (n2ⁿ-1) k/2ⁿμ(k/2ⁿ≤(dH^s)/(dμ)<(k+1)/2ⁿ)+nμ((dH^s)/(dμ)≥n)]。 Especially, if(dH^s)/(dμ)=k , then H^s(E)=kμ(E)。Secondly, we extend a class of homogeneous Moran sets defined in[18] to Non-homogeneous. Prove hausdorff measure of this kind of non-homogeneous Moran sets.Theorem4.2.1 suppose E(?)Rⁿ(n≥2) is a s-dimension non-homogeneousMoran set determined by {n_k}_k≥1, {Φ_k}_k≥1,{m_k}_k≥1.let(?)=t_k,t₀=max{t_k},n_k→∞,m_kc_k,jk→0,(k→∞), if (?){t_k}>3ⁿε, then:H^s(E)=(?).Thirdly, discussed hausdorff measure a kind of "square flower" ,get its estimate:Theorem5.2 let F is a "square flower" fractal, for n≥1,1≤k_n≤5ⁿ,let F_?ⁿ,F_?ⁿ…F(?)ⁿ∈{F_?ⁿ:j∈D_n} ,μis a probability measure on F,let b_kn=(?),if thereexist constant A>0, such that a_n≥A(n=1,2…), then H^s(F)≥A.Theorem5.3 for n≥1, the a_n decreases and (?)a_n=H^s(E)Theorem5.4 Hausdorff measure of "square flower" fractal satisfies the estimation:...