“…The related concept distributional chaotic pair as two points for which the statistical distribution of distances between the orbits does not converge, and Schweizer and Smital (1994) proved that the existence of a single distributional chaotic pair is equivalent to the positive topological entropy (and some other notions of chaos) when restricted to the compact interval case. Since then, distributional chaos has been widely concerned in dynamical system theory (see Smítal and Štefánková, 2004;Balibrea et al, 2005;Martínez-Giménez et al, 2009;Liao et al, 2009;Oprocha, 2009;Li, 2011;Dvorakova, 2011;Wu and Chen, 2013;Shao et al, 2018). Smítal and Štefánková (2004) showed that the two notions of distributional chaos used in the paper, for continuous maps of a compact metric space, are invariants of topological conjugation.…”