Subshifts, rotations and the specification property

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. 82 (2003), 265-272]). We generalize the above result by proving that if $r_0-r_1\in \mathbb R\setminus \mathbb Q$ 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.