以全空间为D_λ-n-攀援集的系统

Systems with the Whole Spaces being D_λ- n- Scrambled

从全空间的角度来研究Dλ-攀援集.借助Furstenberg族为工具,把分布攀援集的定义推广到Dλ-n-攀援集,把关于全空间的分布攀援集的已有结论推广成Dλ-n-攀援集的情形.对任意实数λ∈[0,1]和任意整数n≥2,证得不存在紧致的动力系统以全空间为Dλ-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Dλ-n-攀援集.

The Dλ-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 Dλ-n-scrambled set. Then the results on distributionally scrambled sets with the whole space obtained are generalized to the case of Dλ-n-scrambled sets. For each real λ∈[0,1] and each integer n≥2, the main conclusions are as follows： （1） there is no compact dynamical system with the whole space being a Dλ-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 Dλ-n-scrambled set can be the whole space.