Ergodic Theory and Dynamical Systems

Research Article

Sufficient conditions under which a transitive system is chaotic

E. AKINa1, E. GLASNERa2, W. HUANGa3, S. SHAOa3 and X. YEa3

a1 Mathematics Department, The City College, 137 Street and Convent Avenue, New York City, NY 10031, USA (email: ethanakin@earthlink.net)

a2 Department of Mathematics, Tel Aviv University, Tel Aviv, Israel (email: glasner@math.tau.ac.il, eli.glasner@gmail.com)

a3 Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, P.R. China (email: wenh@mail.ustc.edu.cn, songshao@ustc.edu.cn, yexd@ustc.edu.cn)

Abstract

Let (X,T) be a topologically transitive dynamical system. We show that if there is a subsystem (Y,T) of (X,T) such that (X×Y,T×T) is transitive, then (X,T) is strongly chaotic in the sense of Li and Yorke. We then show that many of the known sufficient conditions in the literature, as well as a few new results, are corollaries of this statement. In fact, the kind of chaotic behavior we deduce in these results is a much stronger variant of Li–Yorke chaos which we call uniform chaos. For minimal systems we show, among other results, that uniform chaos is preserved by extensions and that a minimal system which is not uniformly chaotic is PI.

(Received January 27 2009)

(Revised June 17 2009)