The equivalence of asymptotic average shadowing and vague specification properties and its consequences

We establish the equivalence between the asymptotic average shadowing and the vague specification properties and we use this equivalence to answer a question posed by Downarowicz and Weiss [Ergod. Th. \& Dynam. Sys. 44 (9) (2024), 2565-2580]. Additionally, we prove that 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 latter result to prove that proximal and minimal shift spaces from [Minimal and proximal examples of $\bar{d}$-stable and $\bar{d}$-approachable shift spaces, Ergod. Th. \& Dynam. Sys., (2024) First View] possess the asymptotic average shadowing property.