Different Statistical Future of Dynamical Orbits over Expanding or Hyperbolic Systems (II): Nonempty Syndetic Center

Abstract

In [30] different statistical behavior of dynamical orbits without syndetic center are considered. In present paper we continue this project and consider different statistical behavior of dynamical orbits with nonempty syndetic center: Two of sixteen cases appear (for which other fourteen cases are still unknown) in transive topologically expanding or hyperbolic systems and are discovered to have full topological entropy for which it is also true if combined with non-recurrence and multifractal analysis such as quasi-regular set, irregular set and level sets. In this process a strong entropy-dense property, called minimal-entropy-dense, is established.In particular, we show that points that are minimal (or called almost periodic), a classical and important concept in the study of dynamical systems, form a set with full topological entropy if the dynamical system satisfies shadowing or almost specification property.

Non-dense orbits of systems with the approximate product property

“…Non-dense orbits are also closely related to irregular behaviours. For hyperbolic systems, Dong and Tian [16,17] have an elaborated classification of the sets exhibiting various statistical behaviours as well as a multifractal analysis on these sets.…”

Non-dense orbits of systems with approximate product property

“…Non-dense orbits are also closely related to irregular behaviors. [15,16] contain an elaborated classification of the sets exhibiting various statistical behaviors as well as a multifractal analysis on them for hyperbolic systems.…”