Abstract
According to the recurrent frequency, many levels of recurrent points are found such as periodic points, almost periodic points, weakly almost periodic points, quasi-weakly almost periodic points and Banach recurrent points. In this paper, we consider symbolic dynamics and show the existence of six refined levels between Banach recurrence and general recurrence. Despite the fact that these refined levels are all null-measure under any invariant measure, we further show they carry strong topological complexity. Each refined level of recurrent points is dense in the whole space and contains an uncountable distributionally chaotic subset. For a wide range of dynamical systems such as expansive systems with the shadowing property, we also show the distributional chaos of the points that are recurrent but not Banach recurrent.
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Acknowledgements
The authors would like to thank the referees for their careful reading and valuable suggestions and also thank Xiaobo Hou for some comments and Prof. Xiaoyi Wang for providing some references to us.
Funding
Y. Jiang was partially supported by Fudan project FDUROP (Fudan’s Undergraduate Research Opportunities Program), and the EPSRC Centre for Doctoral Training in Mathematics of Random Systems: Analysis, Modelling, and Simulation (EP/S023925/1). X. Tian is the corresponding author and was supported by National Natural Science Foundation of China (grant No.12071082, 11790273) and in part by Shanghai Science and Technology Research Program (grant No.21JC1400700)
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Jiang, Y., Tian, X. Existence and Distributional Chaos of Points that are Recurrent but Not Banach Recurrent. J Dyn Diff Equat 36, 497–514 (2024). https://doi.org/10.1007/s10884-022-10158-x
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DOI: https://doi.org/10.1007/s10884-022-10158-x