Julia集及其Hausdorff维数的连续性

对于d≥2,考虑多项式族Pc=Zd+c,c∈C.Kc={z∈C|{Pcn(z)}n≥0有界}为Pc的填充Julia集,Jc=(?)Kc为其Julia集.HD(Jc)为Jc的Hausdorff维数.设ω(0)为Pc0的临界点0的轨道的聚点集.我们假定Pc0在ω(0)上是扩张的,且O∈Jc0,|c0|>ε>0.如果一序列Cn→c0,则Jcn→Jc0,Kcn→Jc0,在Hausdorff拓扑下.如果存在一常数C1>0和一序列cn→c0,使得d(cn,Jc0)≥C1|cn-c0|1+1/d,则HD(Jcn)→HD(Jc0).这里d(cn,Jc0)为cn与Jc0间距离.

On the Continuity of Julia Sets and Hausdorff Dimension

Given d≥2 consider the family of monic polynomials Pc(z) = zd + c, for c ∈C. Let Kc={z∈C|{Pcn(z)}n≥0 is bounded} be the filled-in Julia set of Pc and Jc=(?)Kc be the Julia set of Pc. Denote by HD(Jc) the Hausdorff dimension of Jc.Let ω(0) be the set of accumulation points of the orbit of critical point 0, and Pc0 be expanding in ω(0). Suppose that 0∈Jc0 and |c0| > ε>0. If a sequence of cn→c0, then Jcn→Jc0 and Kcn→Jc0, in the Hausdorff topology. We also prove that if there is C1>0 such that for a sequence cn→c0, dist(cn, Jc0)≥C1|cn-c0|1+1/d then HD(Jcn)→ HD(Jc0).