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Mathematics > Dynamical Systems

arXiv:1804.02830 (math)
[Submitted on 9 Apr 2018 (v1), last revised 8 May 2018 (this version, v2)]

Title:Distributional chaos in multifractal analysis, recurrence and transitivity

Authors:An Chen, Xueting Tian
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Abstract:There are lots of results to study dynamical complexity on irregular sets and level sets of ergodic average from the perspective of density in base space, Hausdorff dimension, Lebesgue positive measure, positive or full topological entropy (and topological pressure) etc.. However, it is unknown from the viewpoint of chaos. There are lots of results on the relationship of positive topological entropy and various chaos but it is known that positive topological entropy does not imply a strong version of chaos called DC1 so that it is non-trivial to study DC1 on irregular sets and level sets. In this paper we will show that for dynamical system with specification property, there exist uncountable DC1-scrambled subsets in irregular sets and level sets. On the other hand, we also prove that several recurrent levels of points with different recurrent frequency all have uncountable DC1-scrambled subsets. The main technique established to prove above results is that there exists uncountable DC1-scrambled subset in saturated sets.
Comments: 23 pages
Subjects: Dynamical Systems (math.DS)
Cite as: arXiv:1804.02830 [math.DS]
  (or arXiv:1804.02830v2 [math.DS] for this version)
  https://doi.org/10.48550/arXiv.1804.02830
Journal reference: Ergodic theory and Dynamical systems, 2019

Submission history

From: Xueting Tian [view email]
[v1] Mon, 9 Apr 2018 06:05:08 UTC (35 KB)
[v2] Tue, 8 May 2018 03:05:12 UTC (40 KB)
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