具有重叠结构的非线性吸引予的维数估计

Lower Estimation of the Hausdorff Dimension for Nonlinear Attactors with Overlaps

利用质量分布原理,给出了不满足任何分离条件从而具有重叠结构的非线性吸引子的Hausdorff维数下界的一个估计,也就是:设J为R中非空紧子集,Si(x)in=1为一簇二次可微的压缩映射,且满足以下条件:1)对任意i∈I,Si(J)J,2)对任意i∈I,x∈J,0<a≤|Si′(x)|≤b<1。K为J在迭代函数系统Si(x)in=1下的非线性吸引子,假如∩i∈ImSi(K)=,则dimHK≥sm,这里sm>0且满足maxA∈Ωm∑i∈A(Si′)dm-sm=1,Ωm为所有m级最大重叠序列的集合,dm满足∑i∈Im(S′i)dm=1,且Si′=minx∈J{Si′(x)}。

Give a lower estimate of the Hausdorff dimension for nonlinear attactors which do not satisfy any separated condition and can be obtained by an overlapping construction using the mass distrbution principle. Namely, let J a nonempty compact subset of R, a group of secondary differentiable contractive mappings satisfy that 1） for any i∈I,Si（J）J;2） for any i∈I,x∈J,0〈a≤｜S′i（x）｜≤b〈1.K is nonlinear attractor of the iterated function systemSi（x）i=1^n,if ∩i∈I^mSi（K）=Ф,then dimHK≥sm.Where sm 〉0 and maxA∈Ωm∑i∈A（Si′）^dm^-sm=1,Ωm is the set of m-order maximal overlapping sequence, dm satisfies ∑i∈Im（S′i）dm=1,and Si′=minx∈J{Si′（x）}