n方体连续自映射混沌集合的Hausdorff维数

Hausdorff Dimension of Chaotic Sets Caused by a Continuous Self-map on I^n

把线段、方体自映射混沌集合的Hausdorff维数的有关结果推广到n方体上,证明在C0(I^n)中存在一个剩余集R,使对每一f∈R,如果集合C■I^n对f是Li-Yorke混沌的,则dimH(C)≤n-1.对于高维笛卡尔积的情形,也得到类似的结果,即在C^0(I^ni,I^ni)中存在一个剩余集Ri,使得对于每个fi∈Ri,i=1,2,若集合Ci■I^ni对于fi而言是Li-Yorke混沌的,则dimH(C1×C2)≤n-1.

This paper extends the results of Hausdorff dimension of chaotic sets caused by continuous self-maps on I and I^2 to n-dimensional cube.We prove that there is a residual set R in C^0(I^n),if set C■I^n is chaotic for any given f∈R in the sense of Li-Yorke,then dimH(C)≤n-1.Similarly way,the results on high dimensional Cartesian product can be obtained.That is,there is residual sets Ri in C^0(I^ni,I^n i)such that for any fi∈Ri,i=1,2,if set Ci■I^ni is chaotic in the sense of Li-Yorke,then dimH(C1×C2)≤n-1.