Recurrence properties for linear dynamical systems: an approach via invariant measures

The paper proves, in the Hilbert-space setting, exactly what the problem asks: span(E(T)) = URec(T) = RRecbo(T), plus the equivalence of the eight “existence” statements and the eight “density/measure” statements (Theorem 1.7) . Its proof uses: (i) constructing an invariant probability measure from a reiteratively recurrent bounded-orbit vector (Theorem 2.3, invoked in the proof of Theorem 1.7) ; (ii) a Gaussianization step and support analysis (Lemma 4.4) forcing the support to lie in the span of unimodular eigenvectors ; and (iii) the easy inclusion span(E(T)) ⊂ ∆*Rec ⊂ IP*Rec ⊂ URec ⊂ RRecbo (Proposition 4.1 provides the key Δ*-recurrence of finite linear combinations of unimodular eigenvectors) . The candidate solution reproduces these ingredients: the direct “easy” implications from unimodular eigenvectors; construction of invariant Gaussian measures from eigen-expansions; and then it explicitly invokes the Hilbert-space theorem of Grivaux–López-Martínez for the hard direction and for the equivalences. Where it sketches density implications via Birkhoff from a full-support invariant measure, the paper instead closes the equivalence circle using the set equalities and a standard measure-combination argument to obtain full-support measures (still consistent with the candidate’s approach) . No logical gaps or contradictions were found; the model’s write-up is essentially a streamlined summary of the paper’s proof.