D_lambda-n-攀援集的一点注记

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 $\lambda\in0,1]$ 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.

A Note on D_lambda-n-Scrambled Sets

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 $\lambda\in0,1]$ 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.