摘要
设T为紧度量空间X上的连续自映射,m为X上的Borel概率测度,通过把测度(拓扑)摘局部化,引入了T关于m的平均测度(拓扑)熵的概念,它们分别为相应m-测度(拓扑)混沌吸引子熵的加权平均,从而T关于m的平均测度(拓扑)熵大于零当且仅当T有m-测度(拓扑)混沌吸引子.证明了线段I上关于Lebesgue测度平均拓扑熵大于C与等于零的连续自映射都在C0(I,I)中稠密.
Let X be a compact metric space, T:X- X be a continuous transformation, andm be a Borel measure on X. The mean topological entropy H* (T,m) and mean measuretheoretical entropy H (T,m) of T respect to m are defined via the localization of topological entropy and measure-theoretical entropy of T. H* (T,m) (resp. H. (T,m) ) is theweight of topological (resp. measure-theoretical) entropies of corresponding m-topological(resp. measure-theoretical) chaotic attractors. So H(T,m) (resp. H* (T,m) ) is positiveif and only if T has an m-topological (resp. measure-theoretical) chaotic attractor. For interval map f:I-I, the mean topological entropy repect to Lebesgue measure of f is denoted by H(f). It is proved that both {f:I-I: H(f) > c} and {f:I-I: H(f)=0} aredense in C0(I, I).
出处
《数学物理学报(A辑)》
CSCD
北大核心
1999年第4期397-404,共8页
Acta Mathematica Scientia
基金
国家自然科学基金!69874039
关键词
平均拓扑熵
吸引子
紧度量空间
平均熵
动力系统
Mean topological entropy, Mean measure-theoretic entropy, m-attractor, mtopological chaotic attractor, m-measure-theoretic chaotic attractor.