Characterization of spacing shifts with positive topological entropy (Q2913252)

Let \(P\) be a subset of the set \(\mathbb N\) of positive integers and \(\Sigma _P\) be the set of all infinite binary sequences \(s_i\) for which \(s_i=s_j= 1\) and \(i\neq j\) imply \(| i-j| \in P\). The spacing shift defined by \(P\) is the pair \((\Sigma _P, \sigma _P)\), where \(\sigma _P\) is the shift map restricted to \(\Sigma _P\). The topological entropy of \((\Sigma , \sigma)\) is defined as \(h(\sigma)=\lim _{n\to \infty }\frac {1}{n} \log | L_n(\Sigma)| \), where \(| L_n(\Sigma)| \) denotes the cardinality of the set of all words of length \(n\). It was shown that if the topological entropy \(h(\sigma _P)=0\), then \((\Sigma _P, \sigma _P)\) is proximal. Further, a characterization of spacing shifts with zero topological entropy using the following notion is given. A set \(E\subset \mathbb N\) is density intersective if for any \(A\subset \mathbb N\) with positive upper Banach density \(E\cap (A-A)\neq \emptyset \). It is shown that the topological entropy of a spacing shift \((\Sigma _P, \sigma _P)\) is zero if and only if \(P=\mathbb N\setminus E\) where \(E\) is a density intersective set. In addition, other properties of spacing shifts in relation to zero entropy are studied and the results obtained enable to solve Question 5 given in [\textit{J. Banks}, \textit{T. T. Ding Nguyen}, \textit{P. Oprocha} and \textit{B. Trotta}, ``Dynamics of spacing shift'', Discrete Contin. Dyn. Syst. (to appear)].