Different asymptotic behavior versus same dynamical complexity: Recurrence & (ir)regularity

Xueting Tian Fudan University

Dynamical Systems mathscidoc:2103.11002

Advances in Mathematics, 288, 464-526, 2016
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.
Recurrence; Entropy
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  title={Different asymptotic behavior versus same dynamical complexity: Recurrence & (ir)regularity},
  author={Xueting Tian},
  booktitle={Advances in Mathematics},
Xueting Tian. Different asymptotic behavior versus same dynamical complexity: Recurrence & (ir)regularity. 2016. Vol. 288. In Advances in Mathematics. pp.464-526. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20210323125413555382750.
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