Existence and Distributional Chaos of Points that are Recurrent but Not Banach Recurrent

According to the recurrent frequency, many levels of points are found such as periodic points, almost weakly quasi-weakly and Banach points. In this paper, we consider symbolic dynamics show existence six refined between recurrence general recurrence. Despite fact that these all null-measure under any invariant measure, further they carry strong topological complexity. Each level is dense in whole space contains an uncountable distributionally chaotic subset. For a wide range dynamical systems expansive with shadowing property, also distributional chaos but not recurrent.