Equivalence of Variants of Shadowing of Free Semigroup Actions

Abstract

We prove that for finitely generated free semigroup actions the average shadowing property, the weak asymptotic average shadowing property, the mean ergodic shadowing property, the almost asymptotic average shadowing property, the asymptotic average shadowing property and the $M_{\alpha}$-shadowing property for every $\alpha\in (0,1)$, are equivalent. This gives an affirmative answer to an open question asked in Question 10.3 [M. Kulczycki, D. Kwietniak, P. Oprocha, On almost specification and average shadowing properties, Fundamenta Mathematicae, 224 (2014)].