Positive upper density points and chaos

In this work, we mainly investigate the problem of complexity for a topologically dynamical system (X, f). We prove that f has a full measure center if there exists a countable base of X satisfying that, for any i, there is y in X such that N (y, Ui) is a positive Banach upper density set. Moreover, we consider the chaotic property of (X, f). We show that such a system is chaotic in the sense of Takens-Ruelle if it is transitive but not minimal.