Symbolic Systems with Prescribed Partial Digit Density

Abstract

In this paper, we present a class of subshifts on $\{0,1\}^{{\mathbb {N}}}$ defined with partial digit density. Fixed a real number $p_c\in [0,1]$, initially we consider the sets $P(p_c)$ that satisfy the following rules of transition: $0\rightarrow 0$ and $0\rightarrow 1$ are always possible; $1\rightarrow 0$ is allowed if density of ones (ratio between amount of ones and length of the word) is bigger than $p_c$; and $1\rightarrow 1$ is allowed if density of ones is less or equal than $p_c$. Those subsets are not shift invariant; in order to get a set with this property, define $\Sigma (p_c)$ as the closure of all images of the $P(p_c)$ by the shift; we show that, for every $p_c \in (0, 1)$, $\Sigma (p_c)$ is not a subshift of finite type. Moreover, it is possible to show that $\Sigma (p_c)$ have positive topological entropy for all $p_c\in [0,1)$.