SHADOWING, TOPOLOGICAL ENTROPY AND RECURRENCE OF INDUCED MORSE-SMALE DIFFEOMORPHISM

The paper proves two claims: (i) for Morse–Smale diffeomorphisms on the circle S^1, the induced continuum map C(f) does not have the shadowing property (Theorem A), and (ii) the entropy of C(f) is either 0 (on S^1) or ∞ (in dimensions ≥2), with Theorem 11 giving h(C(f))=0 on S^1 and Theorem 13 giving h(C(f))=∞ when dim M≥2. All three results appear explicitly and are argued within the paper, including the bi-infinite pseudoorbit construction for non-shadowing on S^1 and the separated-set construction for ∞ entropy in higher dimensions . The candidate solution derives the same conclusions: it gives an alternative non-shadowing construction on S^1 (using arcs near a source and sink and a crossing argument) and independently constructs m^n separated families in higher dimension via local product structure. Small technical gaps in the model’s pseudo-orbit stitching can be repaired by a minor reindexing; similarly, the paper’s limit assertions in the S^1 case could be justified with brief lemmas on forward/backward limits in C(S^1). Overall, both are correct, with different proof details. Preliminaries in the paper also confirm shadowing invariance under powers, a reduction used by the model , and the paper’s Lemma 5 underpins the circle-entropy claim .