平均熵

Mean Entropy

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).