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.
There is much research on the dynamical complexity on irregular sets andlevel sets of ergodic average from the perspective of density in base space, theHausdorff dimension, Lebesgue positive measure, positive or full topological entropy (andtopological pressure), etc. However, this is not the case from the viewpoint of chaos.There are many results on the relationship of positive topological entropy and variouschaos. However, positive topological entropy does not imply a strong version of chaos,called DC1. Therefore, it is non-trivial to study DC1 on irregular sets and level sets. Inthis paper, we will show that, for dynamical systems with specification properties, thereexist uncountable DC1-scrambled subsets in irregular sets and level sets. Meanwhile, weprove that several recurrent level sets of points with different recurrent frequency haveuncountable DC1-scrambled subsets. The major argument in proving the above results isthat there exists uncountable DC1-scrambled subsets in saturated sets.