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Invariant scrambled sets, uniform rigidity and weak mixing

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Abstract

We show that for a non-trivial transitive dynamical system, it has a dense Mycielski invariant strongly scrambled set if and only if it has a fixed point, and it has a dense Mycielski invariant δ-scrambled set for some δ > 0 if and only if it has a fixed point and is not uniformly rigid. We also provide two methods for the construction of completely scrambled systems which are weakly mixing, proximal and uniformly rigid.

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Authors and Affiliations

  1. Faculty of Applied Mathematics, AGH University of Science and Technology, al. A. Mickiewicza 30, 30-059, Kraków, Poland

    Magdalena Foryś & Piotr Oprocha

  2. Department of Mathematics, Sichuan University, Chengdu, Sichuan, 610064, China

    Wen Huang

  3. School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui, 230026, China

    Wen Huang

  4. Department of Mathematics, Shantou University, Shantou, Guangdong, 515063, China

    Jian Li

Authors
  1. Magdalena Foryś
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  2. Wen Huang
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  3. Jian Li
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  4. Piotr Oprocha
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Corresponding authors

Correspondence to Magdalena Foryś or Jian Li.

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Foryś, M., Huang, W., Li, J. et al. Invariant scrambled sets, uniform rigidity and weak mixing. Isr. J. Math. 211, 447–472 (2016). https://doi.org/10.1007/s11856-015-1278-1

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  • DOI: https://doi.org/10.1007/s11856-015-1278-1

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