集值离散动力系统的拓扑遍历性、拓扑熵与混沌

设(X,d)为紧致度量空间, f: X→X连续, (K(X),H)是X所有非空紧致子集构成的紧致度量空间. 通过研究点运动与点集运动的关系, 证明了集值映射拓扑遍历与f拓扑双重遍历等价并构造一个零拓扑熵且不具有任何混沌性质的紧致系统, 其诱导的集值映射有无穷拓扑熵且分布混沌, 表明集值离散动力系统的拓扑复杂性可以远远大于原系统.

Topological Ergodicity, Entropy and Chaos of Set-valued Discrete Systems

Let (X,d)be a compact metric space, f: X→X a continuous map, and (K(X),H) a compact metric space consisting of all non empty compact subsets of X. It has been proved that the topological ergodicity of set valued map is equivalent to the topological double ergodicityof f by studying the relation between the motion of points and the motion of sets; moreover, a compactsystem has been constructed which has zero topological entropy and no chaotic property, but the inducedset valued map of which has infinite to pological entropy and distributional chaos, this implies that the topological complexity of could be far greater than that of f.