Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Different Asymptotic Behavior versus Same Dynamical Complexity   For any dynamical system $T:X\rightarrow X$ 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 $\beta-$shfits ($\beta>1$),
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 {\it Refined Dynamical Structure} for Recurrence Theory and
Multi-fractal Analysis.

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The categories of this paper are: alg-geom
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The categories of this paper are: plasm-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: mtrl-th
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The categories of this paper are: ao-sci
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The categories of this paper are: atom-ph
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The categories of this paper are: eess
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The categories of this paper are: supr-con
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The categories of this paper are: acc-phys
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The categories of this paper are: cmp-lg
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The categories of this paper are: cond-mat
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-ex
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The categories of this paper are: nucl-th
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The categories of this paper are: nucl-ex
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The categories of this paper are: q-fin