Intertwined Synchronized Systems

An asymmetric-RLL$(d_1,k_1,d_0,k_0)$ system is a subshift of $\{0,1{\}}^{\mathbb{Z}}$ with run of $1$ and $0$ restricted to $S=[d_1,k_1]\subseteq {\mathbb{N}}_0=\mathbb{N}\cup \{0\}$ and $S^{\mathrm{\prime }}=[d_0,k_0]\subseteq {\mathbb{N}}_0$ respectively. We extend this concept to the case when $S$ and $S^{\mathrm{\prime }}$ are arbitrary subsets of ${\mathbb{N}}_0$ and we call it a $(S,S^{\mathrm{\prime }})$-gap shift. Moreover, for $i=1,2$, if $X_i$ is a synchronized system generated by $V_{i}={v^{i}\alpha_{i}:\alpha_i v^{i}\alpha_i\in\mathcal B(X_i),\alpha_i\not\subseteq v^{i}}$where$ \alpha_i $ is a synchronizing word for $X_i$, then a natural generalization of $(S,S^{\mathrm{\prime }})$-gap shifts is a coded system $Z$ generated by ${v^{1}\alpha_1 v^{2}\alpha_2:v^{i}\alpha_i\in V_{i}, i=1,2}$ and called the intertwined system. We investigate the dynamical properties of $Z$ with respect to $X_1$ and $X_2$.