Characterization of spacing shifts with positive topological entropy

Abstract

Suppose P Í N and let (SPsP) be the spacing shift defined by P. We show that if the topological entropy h(sP) of a spacing shift is equal zero, then (SPsP) is proximal. Also h(sP) = 0 if and only if P = N - E. where E is an intersective set. Moreover, we show that h(sP) > 0 implies that P is a D*-set; and by giving a class of examples, we show that this is not a sufficient condition. Using these results we solve question 5 given in [J. Banks et al., Dynamics of Spacing Shifts, Discrete Contin. Dyn. Syst., to appear].