Topologically stable and β-persistent points of group actions

Part A of the candidate solution matches Theorem 3.9 in the paper and is essentially correct (constructs h on the Ψ-orbit via a uniquely shadowing y, uses expansivity for well-definedness, and proves continuity; compare the paper’s proof and Lemma 3.8-based continuity argument) . Part B, however, contains a key inequality error: it tries to deduce d(Φ_g(x), Ψ_g(y_p)) ≤ ε from topological stability at p and equicontinuity, but replaces the needed bound on d(Φ_g(p), Ψ_g(y_p)) with a bound on d(Φ_g(y_p), Ψ_g(p)), which is not interchangeable. The paper’s Theorem 4.14 avoids this pitfall by showing each equicontinuous topologically stable point x is already β-persistent with y = x, and then promotes pointwise β-persistence to global β-persistence via measure-theoretic lemmas .