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For any dynamical system $T:X\rightarrow X$ of a compact metric space $X$with $g-$almost product property and uniform separation property, under theassumptions that the periodic points are dense in $X$ and the periodic measuresare dense in the space of invariant measures, we distinguish variousperiodic-like recurrences and find that they all carry full topologicaltopological entropy and so do their gap-sets. In particular, this implies thatany two kind of periodic-like recurrences are essentially different.  Moreover, we coordinate periodic-like recurrences with (ir)regularity andobtain lots of generalized multi-fractal analysis for all continuous observablefunctions. These results are suitable for all $\beta-$shfits ($\beta>1$),topological mixing subshifts of finite type, topological mixing expanding mapsor topological mixing hyperbolic diffeomorphisms, etc.  Roughly speaking, we combine many different "eyes" (i.e., observablefunctions and periodic-like recurrences) to observe the dynamical complexityand obtain a {\it Refined Dynamical Structure} for Recurrence Theory andMulti-fractal Analysis.

Different Asymptotic Behavior versus Same Dynamical Complexity