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Distributional chaos in multifractal analysis, recurrence and transitivity

Part of: Smooth dynamical systems: general theory Topological dynamics Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  27 August 2019

AN CHEN  and
XUETING TIAN
AN CHEN
Affiliation:
School of Mathematical Sciences, Fudan University, Shanghai200433, PR China email [email protected], [email protected]
XUETING TIAN
Affiliation:
School of Mathematical Sciences, Fudan University, Shanghai200433, PR China email [email protected], [email protected]
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Abstract

There is much research on the dynamical complexity on irregular sets and level sets of ergodic average from the perspective of density in base space, the Hausdorff dimension, Lebesgue positive measure, positive or full topological entropy (and topological 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 various chaos. 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. In this paper, we will show that, for dynamical systems with specification properties, there exist uncountable DC1-scrambled subsets in irregular sets and level sets. Meanwhile, we prove that several recurrent level sets of points with different recurrent frequency have uncountable DC1-scrambled subsets. The major argument in proving the above results is that there exists uncountable DC1-scrambled subsets in saturated sets.

Keywords

irregular set and level setrecurrence and transitivityspecificationdistributional chaosscrambled set

MSC classification

Primary: 37C50: Approximate trajectories (pseudotrajectories, shadowing, etc.) 37B20: Notions of recurrence 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.)
Secondary: 37D45: Strange attractors, chaotic dynamics 37C45: Dimension theory of dynamical systems
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 2 , February 2021 , pp. 349 - 378
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Copyright
© Cambridge University Press, 2019

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