D_lambda-n-攀援集的一点注记

基金项目:

国家自然科学基金;国家自然科学青年基金;广东高校优秀青年创新人才培养计划项目

详细信息

A Note on D_lambda-n-Scrambled Sets

摘要: 从全空间的角度来研究 $\mathcal{D}_\lambda$ -攀援集. 借助 Furstenberg 族为工具, 把分布攀援集的定义推广到 $\mathcal{D}_\lambda$ - $n$ -攀援集, 把关于全空间的分布攀援集的已有结论推广成 $\mathcal{D}_\lambda$ - $n$ -攀援集的情形. 对任意实数 $\lambda\in[0,1]$ 和任意整数 $n\geqslant2$ , 证得不存在紧致的动力系统以全空间为 $\mathcal{D}_\lambda$ - $n$ -攀援集; 并且构造出了只含可数多个点的非紧致的可逆系统, 以全空间为 $\mathcal{D}_\lambda$ - $n$ -攀援集.
- 全空间
Abstract: The $\mathcal{D}_\lambda$ -scrambled sets are to be analysed in the aspect of whole space. By means of Furstenberg families, the definition of distributionally scrambled set is extended to define a $\mathcal{D}_\lambda$ - $n$ -scrambled set. Then the results on distributionally scrambled sets with the whole space obtained are generalized to the case of $\mathcal{D}_\lambda$ - $n$ -scrambled sets. For each real $\lambda\in[0,1]$ and each integer $n\geqslant2$ , the main conclusions are as follows: (1) there is no compact dynamical system with the whole space being a $\mathcal{D}_\lambda$ - $n$ -scrambled set; (2) based on the example provided by WANG et al, an invertible noncompact dynamical system consisting of countable many points are constructed, whose $\mathcal{D}_\lambda$ - $n$ -scrambled set can be the whole space.
- whole space