Twelve Different Statistical Future of Dynamical Orbits: Empty Syndetic Center

Abstract

In the theory of dynamical systems, a fundamental problem is to study the asymptotic behavior of dynamical orbits. Lots of different asymptotic behavior have been learned including different periodic-like recurrence such periodic and almost periodic, the level sets and irregular sets of Birkhoff ergodic avearge, Lyapunov expoents. In present article we use upper and lower natural density, upper and lower Banach density to differ statistical future of dynamical orbits and establish several statistical concepts on limit sets, in particular such that not only different recurrence are classifiable but also different non-recurrence are classifiable. In present paper we mainly deal with dynamical orbits with empty syndetic center and show that twelve different statistical structure over expanding or hyperbolic dynamical systems all have dynamical complexity as strong as the dynamical system itself in the sense of topological entropy. Moreover, multifractal analysis on various non-recurrence and Birkhoff ergodic averages are considered together to illustrate that the non-recurrent set has rich and colorful asymptotic behavior from the statistical perspective, although the non-recurrent set has zero measure for any invariant measure from the probabilistic perspective.Roughly speaking, on one hand our results describe a world in which there are twelve different predictable order in strongly chaotic systems but also there are strong chaos in any fixed predictable order from the viewpoint of dynamical complexity on full topological entropy; and on other hand we find that various asymptotic behavior such as (non-)recurrence and (ir)regularity from differnt perspecitve survive togother and display strong dynamical complexity in the sense of full topological entropy. In this process we obtain two powerful ergodic properties on entropy-dense property and saturated property.

Strongly distributional chaos of irregular orbits that are not uniformly hyperbolic

“…Equivalent statements of recurrence. Let us recall some equivalent statements of recurrence referring to [28,65,66,22] whose proofs are fundamental and standard.…”

“…In uniformly hyperbolic systems, irregular set has strong dynamical complexity as shown by the existing results. In dynamics beyond uniform hyperbolicity, using Katok's approximation of hyperbolic measures by horseshoes, Dong and Tian [22] obtain that the topological entropy of irregular set is bounded from below by metric entropies of ergodic hyperbolic measures.…”

“…Theorem 1.1. [22,Corollary B] Let M be a compact Riemannian manifold of dimension at least 2 and f be a C 1+α diffeomorphism. Then one has h top (f, IR(f )) ≥ sup{h top (f, Λ) : Λ is a transitive locally maximal hyperbolic set} = sup{h µ (f ) : µ is a hyperbolic ergodic measure}.…”

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.…”

Unstable entropies and Dimension Theory of Partially Hyperbolic Systems

“…The general concept of multifractal analysis is to decompose the phase space into subsets of points which have a similar dynamical behavior and to describe the size of these subsets from the geometrical or topological viewpoint. Sets with similar dynamical behavior include the basin set of an invariant measure or general saturated sets [8,30], recurrent and dense sets [42,9], non-dense sets [45,13,47,48], level sets and irregular sets of Birkhoff ergodic average [26,27,3,4,6,5,7,38,30,23,39,19,37], level sets and irregular sets of Lyapunov exponents [2,28,15,41], which have been studied a lot by using various measurements such as Hausdorff dimension, topological entropy or pressure, Lebesgue measure and distributional chaos etc. Here the topological entropy used was introduced by Bowen [8] to characterize the dynamical complexity of arbitrary sets which are not necessarily compact nor invariant from the perspective of "dimensional" nature.…”