Description

• Title:
  Subshifts, rotations and the specification property
• Creator:
  Mazur, Marcin;
  Oprocha, Piotr
• Description:
  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
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2015 Topological Methods in Nonlinear Analysis
• Data:
  2015.12.01
• Typ:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Język:
  English
• Źródło:
  Topological Methods in Nonlinear Analysis; Vol 46, No 2 (December 2015); 799 - 812;
  1230-3429
• Format:
  application/pdf
• Wydawca:
  Nicolaus Copernicus University in Toruń
• Relacja: https://apcz.umk.pl/TMNA/article/view/TMNA.2015.077/7289 ;https://apcz.umk.pl/TMNA/article/view/TMNA.2015.077/7291
• Temat i słowa kluczowe:
  Distributional chaos;
  scrambled set;
  specification property;
  symbolic dynamics;
  circle rotation;
  Primary;
  37B05;
  37B10;
  37B20;
  Secondary;
  54H20
• Identyfikator: https://apcz.umk.pl/TMNA/article/view/TMNA.2015.077
• Dostawca danych:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
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