动力系统球族的勒贝格稠密定理

Lebesgue Density Theorem for a Family of Dynamical Balls

【目的】研究由离散动力系统生成的动力系统球族并证明它对应的勒贝格稠密定理。【方法】利用系统的可扩性研究了动力系统球族的几何及测度性质,通过有界形变性等动力系统工具,对经典勒贝格稠密定理需要收缩集族具有良收缩性这一条件进行了改进。【结果】将动力系统球族代替经典的欧几里得球族,验证了勒贝格稠密定理。并且通过举例说明了动力系统球族不具备良收缩性。【结论】这种动力系统观点下的勒贝格稠密定理,对处理一些带测度迭代估计的动力学问题有重要的理论意义。

[Purposes]Dynamical balls generated by discrete dynamical system is studied and then Lebesgue density theorem for this kind of dynamical balls is proved.[Methods]By expanding behavior of the system,the geometric and measure properties of the dynamical balls is investigated.Then by borrowing some ideas from dynamical systems such as bounded distortion argument,the well shrinkable condition for the family of shrinking sets that required for the classic Lebesgue density theorem is proved.[Findings]The Lebesgue density theorem for the family of dynamical balls is given.Furthermore,it gives a simple example which illustrates that the family of dynamical balls may not well shrinkable.[Conclusions]For this kind of Lebesgue density theorem that established in the viewpoint of dynamical systems,it has its important theoretical significance when dealing with dynamical problems in terms of the estimates of iterated Lebesgue measures.