On the weakness of the vague specification property

TL;DR

This paper investigates the vague specification property in dynamical systems, establishing its equivalence with the asymptotic average shadowing property and exploring its implications for various classes of systems.

Contribution

It demonstrates that the vague specification property is strictly weaker than the weak specification property and establishes new equivalences involving shadowing properties in surjective systems.

Findings

Vague specification property is equivalent to asymptotic average shadowing.

Weak specification implies vague specification, but not vice versa.

Proximal and minimal shift spaces have the vague specification property.

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 [Ergod. Th. \& Dynam. Sys. 44(9) (2024), 2565--2580]. 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 [Ergod. Th. \& Dynam. Sys., 45(2) (2025), 396--426] possess the vague specification property (asymptotic average shadowing property). Our findings also allow us to address a couple of questions from [Fund. Math., 224(3) (2014), 241--278] about the asymptotic average shadowing property.