A Weakly Mixing System With The Whole Space Being Not A Distributionally Scrambled Set

Chaos is regarded as an important description of the complexity of dynamical systems. However, people do not give a clear notion of chaos in a long time. Until 1975, Li and Yorke put forward mathematical definitions of chaos for the first time in their papers “Period three implies chaos”. Since then, experts of different fields have given different definitions of chaos. For example, Schweizer and Smital put forward definition of distributional chaos in the interval by the distance distribution function of the two points' corresponding orbit in 1994. It is very important to study the size and property of scrambled sets. In recent years, some experts devoted themselves to studying the distributional chaos, and they obtained a series of results : the whole space could not be a distributionally scrambled set in a compact dynamical system; but in a noncompact dynamical system, the whole space could being a distributionally scrambled set and so on. Weakly mixing implies distributional chaos in a sequence, and there were examples which are weakly mixing and the whole space is a distributionally scrambled set. Does weakly mixing imply distributional chaos? This paper definitely answered the question.In this paper, at first, we construct a family of finite sequences {P_n}_n=1^? which only have symbols 0 and 1, then we construct a noncompact metric space?Y, d?, the system?Y,?? is weakly mixing and the orbit of every point x?Y is dense in Y. But there are no distributionally chaotic pairs in the whole space. This example points out that weakly mixing and distributional chaos are independent.