Equivalence of Variants of Shadowing of Free Semigroup Actions

The paper proves that, for a fixed itinerary w of a finitely generated free semigroup action on a compact metric space, six shadowing variants (a)–(f) are equivalent (Theorem 1.1) via the chain (b)⇒(d)⇒(c)⇒(b), (c)⇔(e), (a)⇒(b), (c)⇒(f), and (f)⇒(a) . It builds on (i) a mean-ergodic reformulation (Lemma 3.1) , (ii) a density-zero ‘surgery’ converting ergodic to average pseudo-orbits (Lemma 3.2) , and (iii) a Cesàro–density lemma (Remark 3.3) , and uses a standard concatenation argument for (b)⇒(d) , direct estimates for (d)⇒(c) and (c)⇒(b) , density counting for (c)⇔(e) , a compactness/diagonal step for (c)⇒(f) , and an external IFS result for (f)⇒(a) . The candidate solution reproduces the same implication chain with the same key lemmas and techniques (mean-ergodic/density reformulation, density-zero surgery, Cesàro–density lemma, concatenation, and compactness), and cites the same external IFS implication. Minor notational differences (e.g., “Chebyshev-type” counting) do not affect correctness. Hence both the paper and the model solution are correct and essentially follow the same proof structure.