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Topological Methods in Nonlinear Analysis

Subshifts, rotations and the specification property
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Subshifts, rotations and the specification property

Authors

  • Marcin Mazur
  • Piotr Oprocha

DOI:

https://doi.org/10.12775/TMNA.2015.077

Keywords

Distributional chaos, scrambled set, specification property, symbolic dynamics, circle rotation

Abstract

Let $X=\Sigma_2$ and let $F\colon X\times \mathbb{S}^1\to X\times \mathbb{S}^1$ be a map given by \[ F(x,t)=(\sigma(x),R_{x_0}(t)), \] where $(\Sigma_2,\sigma)$ denotes the full shift over the alphabet $\{0,1\}$ while $R_0$, $R_1$ are the rotations of the unit circle $\mathbb{S}^1$ by the angles $r_0$ and $r_1$, respectivelly. It was recently proved by X.~Wu and G.~Chen that if $r_0$ and $r_1$ are irrational, then the system $(X\times \mathbb{S}^1,F)$ has an uncountable distributionally $\delta$-scrambled set $S_\delta$ for every positive $\delta\leq \textrm{diam } X\times \mathbb{S}^1=1$. Moreover, each point in $S_\delta$ is recurrent but not weakly almost periodic (this answeres a question from [Wang et al., Ann. Polon. Math. \textbf{82} (2003), 265--272]). We generalize the above result by proving that if $r_0-r_1\in \R\setminus \Q$ and $X\subset \Sigma_2$ is a nontrivial subshift with the specification property, then the system $(X\times \mathbb{S}^1,F)$ also has the specification property. As a consequence, there exist a constant $\delta\ge 0$ and a dense Mycielski distributionally $\delta$-scrambled set for $(X\times \mathbb{S}^1,F)$, in which each point is recurrent but not weakly almost periodic

References

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Vol 46, No 2 (December 2015)

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Published

2015-12-01

How to Cite

1.
MAZUR, Marcin and OPROCHA, Piotr. Subshifts, rotations and the specification property. Topological Methods in Nonlinear Analysis. Online. 1 December 2015. Vol. 46, no. 2, pp. 799 - 812. [Accessed 6 July 2025]. DOI 10.12775/TMNA.2015.077.
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