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We can produce such a subshift by a standard hierarchical construction. Let $w_{0,0} = 01$ and $w_{0,1} = 011$. For each $k \geq 0$, define $w_{k+1,0} = w_{k,0} w_{k,0} w_{k,1}$ and $w_{k+1,1} = w_{k, …

Let $f$ be a sublinear function that tends to infinity, such as $f(n) = \sqrt{n}$. Define $X \subset \{0,1,2\}^{\mathbb{N}}$ by forbidding all long enough words $w$ with more than $f(|w|)$ occurrences …

No, even if we assume $\nu$ to be invariant under $\phi$. Let $X = \{0,1\}^\mathbb{Z}$ be the set of two-way infinite binary sequences with the prodiscrete topology, and let $\phi$ be the left shift …

No. Let $\phi$ be the left shift on the set $X = \{0,1\}^\mathbb{Z}$ of bi-infinite binary sequences with the prodiscrete topology, and let $V = \{ x \in X : x_0 = 0 \}$ be the set of sequences that …

Consider the shift space $X \subset \{0,1\}^{\mathbb{Z}}$ obtained by forbidding the words $1 0^m 1^n 0$ for all $m, n \geq 1$, and denote the shift map on $X$ by $\sigma$. Since a point of $X$ can co …