Chaoticity of generic points for ergodic measures in hyperbolic systems and beyond

In this paper, we search the chaotic behavior in the set of generic points of ergodic measures (called the Birkhoff basin) and find several types of chaoticity stronger than Li-Yorke chaos. More precisely, we consider nonuniformly hyperbolic systems first. On one hand, the Birkhoff basin of every ergodic hyperbolic measure with positive metric entropy exhibits a type of distributional chaos property between DC1 and Li-Yorke chaos, called Banach DC1. On the other hand, the Birkhoff basin of every totally ergodic hyperbolic measure with nondegenerate support exhibits a type of distributional chaos property between DC1 and DC2, called almost DC1. For hyperbolic systems, the Birkhoff basin of every ergodic measure with nondegenerate support from an elementary part of an Axiom A system exhibits both almost DC1 and Banach DC1, and the Birkhoff basin of any trivial ergodic measure supported on some fixed point exhibits Banach DC1 but no almost DC1.

In this process, Katok's shadowing and horseshoe approximation motivate us to obtain two types of weak specification property as useful techniques to reach our results. Such weak specifications are also valid to symbolic systems like sofic subshifts and β-shifts, so we put them as abstract frameworks in the proof part. Compared with Chen-Tian's result [1] considering the ergodic measures whose Birkhoff basin has a distal pair, we need to overcome general ergodic measures without the assumption of a distal pair and overcome the nonuniform difficulties from the weak specification property.