Distributionally chaotic C0-semigroups on complex sectors

The uploaded preprint proves exactly the four-way equivalence the candidate addressed (existence of a distributionally semi‑irregular vector; existence of a distributionally irregular vector; distributional chaos; and the existence, on some closed invariant subspace, of a dense Gδ set of distributionally irregular vectors). This is Theorem 3.13 in the paper, with a proof that relies on translation invariance of sector densities (Lemma 2.1) and intermediate structural results (Theorem 3.12) to pass through distributional sensitivity and proximality to build residual scrambled/irregular sets . The candidate supplies a different (and somewhat more direct) proof: (i)⇒(iii) and (ii)⇒(iii) via a scaling argument on an uncountable ray {αx}, (iii)⇒(ii) by citing Albanese–Barrachina–Mangino–Peris (2013), and (ii)⇒(iv) by constructing an invariant orbit-span subspace and showing the set of irregular vectors is dense Gδ. These steps align with the paper’s statements, though the (iii)⇒(ii) step is justified in the paper by an argument adapted to complex sectors (Theorem 3.13) rather than by quoting the R+ case, and the candidate’s openness/Gδ argument could use a more careful topological-measure justification (the paper uses a Baire-category route via DProx and sensitivity; see Theorem 3.12) . Overall: both are correct; the proofs are not the same in structure.