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 periodiclike recurrences and find that they all carry full 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 multifractal analyses for all continuous observable functions. These
results are suitable for all β-shifts (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.