A note on the three versions of distributional chaos

The concept of distributional chaos was introduced by Schweizer et al. [Schweizer B, Sklar A, Smítal J. Measures of chaos and a spectral decomposition of dynamical systems on the interval. Tran Amer Math Soc 1994;344:737–854.] for a continuous selfmap on an interval. However, it turns out that, for a continuous selfmap on a compact metric space, three mutually nonequivalent versions of distributional chaos, DC1–DC3, can be discussed. In this paper, we consider a continuous map f : X → X, where X is a compact metric space, and show that DC1 (resp. DC2) is an iteration invariant, that is, for any integer N > 0, f is DC1 (resp. DC2) if and only if fN is also DC1(resp. DC2). As applications, we show that the following statements hold: be a graph and f : G → G a continuous map. Then f is DC1 if and only if f is DC2. continuous selfmap f on a tree T, these three versions of distributional chaos, DC1 − DC3 are mutually equivalent. rmore, we present two examples which show that DC3 may be an iteration invariant. We will also discuss and partly solve the problem.