Equivalence of Variants of Shadowing of Free Semigroup Actions
Pramod Kumar Das, Priyabrata Bag
TL;DR
The paper addresses whether multiple variants of shadowing coincide for finitely generated free semigroup actions. It develops a rigorous framework of pseudo-orbits, densities, and $w$-shadowing notions, and then establishes the equivalence of $w$-average, $w$-weak asymptotic average, $w$-mean ergodic, $w$-almost asymptotic average, $w$-$M_\alpha$-shadowing for all $\alpha\in(0,1)$, and $w$-asymptotic average shadowing. This result extends known autonomous-case relationships to nonautonomous, semigroup-driven dynamics and affirms open questions in the literature. By showing these properties are interchangeable under the free semigroup action framework, the work provides a unified view of shadowing phenomena and a toolkit for analyzing long-term behavior in multi-rule dynamical systems.
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 $M_α$-shadowing property for every $α\in (0,1)$, 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)].