GENERIC AND DENSE DISTRIBUTIONAL CHAOS WITH SHADOWING

The paper establishes two equivalence theorems (Theorems 1.1 and 1.2) for systems with shadowing, using a clean chain-structure characterization (Lemma 3.2) and carefully crafted implications via sensitivity/regionally proximal pairs (Lemma 3.4) for guC and property S for guDC1 (Lemmas 3.7 and 3.8), with a correct gC⇒non-singleton chain-stable component argument (Lemma 3.5) . The candidate solution reproduces the intended equivalences but has serious gaps: (i) it misidentifies the Gδ description of the proximal set by using Prox = ⋂m ⋃n{d(fn x, fn y) < 1/m}, which does not enforce arbitrarily large return times and is not equal to liminf-distance zero; the correct Gδ form requires an additional intersection over tail indices (cf. the paper’s correct Gδ constructions for Cδ and DC1δ) ; (ii) key density/residual claims (residual proximality and uniform limsup-separation) are proved only with “chain recurrence” and shadowing, but they actually require the chain mixing part of Lemma 3.2, which the candidate does not invoke, leading to unaddressed period/equal-length issues in the equalization arguments . By contrast, the paper’s proofs explicitly leverage the chain-mixing characterization and avoid these pitfalls.