On the weakness of the vague specification property
Melih Emin Can, Alexandre Trilles · Nonlinearity · 2025
Abstract We show that the vague specification property is strictly weaker than most of the specification-like properties, by establishing its equivalence with the asymptotic average shadowing property. In particular, we see that the weak specification property implies the vague specification property, but the converse does not hold, answering the question posed by Downarowicz and Weiss in (2024 Ergod. Theory Dyn. Syst. 44 2565–80). Additionally, we prove that, for surjective systems, the asymptotic average shadowing property is equivalent to the average shadowing property if the phase space is complete with respect to the dynamical Besicovitch pseudometric. We use the combination of both results to prove that the proximal and minimal shift spaces from (Can et al 2025 Ergod. Theory Dyn. Syst. 45 396–426) possess the vague specification property (asymptotic average shadowing property). Our findings also allow us to address a couple of questions from (Marcin et al 2014 Fund. Math. 224 241–278) about the asymptotic average shadowing property.