A note on chaos via Furstenberg family couple

The concepts of the first type of distributional chaos in the Tan-Xiong sense (Abbrev. DC1 in the Tan-Xiong sense), the second type of strong-distributional chaos (Abbrev. strong DC2) and the third type of strong-distributional chaos (Abbrev. strong DC3) were introduced by Tan et al. [F. Tan, J. Xiong. Chaos via Furstenberg family couple, Topology Appl. (2008), doi:10.1016/j.topol.2008.08.006] for continuous maps of a metric space. However, it turns out that, for continuous maps of a compact metric space, the three mutually nonequivalent versions of distributional chaos can be discussed. Let X be a compact metric space and f:X→X a continuous map. In this paper, we show that for any integer N>0, f is strong DC2 (resp. strong DC3) if and only if fN is strong DC2 (resp. strong DC3). We also show that the above three versions of distributional chaos are topological conjugacy invariant. In addition, as an application, we present an example.