Topological Stability of Semigroup Actions and Shadowing

The paper proves that expansive + pseudo-orbit tracing property (POTP) implies topological stability for monoid actions (Theorem 1.1), by first establishing semigroup topological semistability (Theorem 3.4) and then using the identity element to get closeness to the identity map; the construction of h, the semiconjugacy α_m ∘ h = h ∘ β_m, and continuity all follow the standard shadowing/expansivity scheme, relying on uniqueness of tracers and uniform expansivity (Propositions 2.12 and 2.7). The candidate solution implements the same scheme with slightly different entourage bookkeeping (choosing W ⊂ U and cl(W)∘cl(W) ⊂ E). Aside from a minor misphrasing that POTP directly gives uniqueness (the paper proves uniqueness via expansivity), the model’s argument matches the paper’s structure and is correct. See Theorem 1.1 and its proof, building on Theorem 3.4, Proposition 2.12 (uniqueness of tracers), and Proposition 2.7 (uniform expansivity) in the uploaded paper .