For any dynamical system T : X → X of a compact metric space X with g−almost product property and uniform separation property, under the assumptions that the periodic points are dense in X and the periodic measures are dense in the space of invariant measures, we distinguish various periodic-like recurrences and find that they all carry full topological topological entropy and so do their gap-sets. In particular, this implies that any two kind of periodic-like recurrences are essentially different. Moreover, we coordinate periodic-like recurrences with (ir)regularity and obtain lots of generalized multi-fractal analysis for all continuous observable functions. These results are suitable for all β−shfits (β > 1), topological mixing subshifts of finite type, topological mixing expanding maps or topological mixing hyperbolic diffeomorphisms, etc.Roughly speaking, we combine many different "eyes" (i.e., observable functions and periodic-like recurrences) to observe the dynamical complexity and obtain a Refined Dynamical Structure for Recurrence Theory and Multi-fractal Analysis.

We prove that when f is a continuous self-map acting on a compact metric space (X, d) which satisfies the shadowing property, then the set of irregular points (i.e. points with divergent Birkhoff averages) has full entropy.Using this fact we prove that in the class of C 0 -generic maps on manifolds, we can only observe (in the sense of Lebesgue measure) points with convergent Birkhoff averages. In particular, the time average of atomic measures along orbit of such points converges to some SRB-like measure in the weak * topology. Moreover, such points carry zero entropy. In contrast, irregular points are nonobservable but carry infinite entropy.

The topological entropy of various gap-sets on periodic-like recurrence and Birkhoff regularity were considered in [69] but some Banach recurrence and Lyapunov regularity are not considered. In this paper we introduce five new levels on Banach recurrence and show they all carry full topological entropy, and simultaneously combine with Lyapunov regularity to get some refined theory on mixed multifractal analysis of [8,29].In this process, we strengthen Pfister and Sullivan's result of [58] from saturated property to transitively-saturated property (and from single-saturated property to transitively-convex-saturated property).

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.

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