Assume that \({\mathscr{B}}=\{b_1,b_2,\ldots\}\subset
\{2,3,\ldots\}\) is such that
$$
(b_i,b_j)=1\text{ whenever } i\neq j\text{ and }\sum_{i\geq1}1/b_i<+\infty.
\label{f1}
$$
For example, we can take \({\mathscr{B}}=\{p_i^2: i\geq1\}\), where
\(p_i\in\mathscr{P}\) stands for the \(i\)th prime number. To
\(\mathscr{B}\) we associate a two-sided sequence
\(\eta\in\{0,1\}^{\mathbb{Z}}\) by setting
\[
\eta(n):=\begin{cases}
1& \text{if }b_i\nmid n \text{ for all }i\geq 1,\\
0& \text{otherwise.}
\end{cases}
\]
Let
\[
X_\eta:=\{y\in\{0,1\}^\mathbb{Z} : \text{each block occurring on }y \text{
occurs on }\eta\}
\]
and let \(S\) stand for the shift transformation on
\(\{0,1\}^\mathbb{Z}\). Notice that \(X_\eta\) is closed and
\(S\)-invariant, i.e.\ \(X_\eta\) is a subshift. We call \(X_\eta\)
the \(\mathscr{B}\){\em-free subshift}. During my talk I will
provide a description of the set of all invariant measures on
\(X_\eta\).
|
