The Morse Spectrum for Linear Random Dynamical Systems

The paper proves, under the bounded growth hypothesis (3.8), that the weak Morse spectrum equals the non-uniform dichotomy spectrum, i.e., Theorem 3.18: Ξ_w = Σ′ . The proof is split into: (i) Ξ_w ⊂ Σ′ (Proposition 3.22), using invariant projectors at a resolvent growth rate and the NUED bounds (3.10)–(3.11) to contradict any limiting finite-time exponent outside Σ′ ; the definitions of NUED and the dichotomy spectrum used here are in Definitions 2.11–2.12, and the passage to a weak attractor–repeller pair via projectivisation is Theorem 2.13 . (ii) Σ′ ⊂ Ξ_w (Proposition 3.24), by constructing a measurable projector splitting a direct sum of Morse-set subspaces and showing that, if a number in [min Σ′, max Σ′] were missing from Ξ_w, one could force a NUED at that number using temperedness—contradicting spectral membership . Auxiliary ingredients include the boundary property for sums of Morse subspaces (Lemma 3.21) , the normalization of bounded growth on a θ-invariant full-measure set (Lemma 3.7) , that Ξ(R^d) = [min Σ′, max Σ′] (Lemma 3.23) , and that each Ξ(P^{-1}M)(ω) is a closed interval (Theorem 3.13) . The candidate solution establishes the same equivalence. Its forward inclusion mirrors Proposition 3.22 by using NUED bounds and tempered constants. Its reverse inclusion proceeds via thresholds a_i,b_i per Morse set and a concatenation argument to realise all exponents in [a_i,b_i], aligning with the interval realisation in the paper. The only substantive discrepancy is a notational swap of which side (range vs. null space of the projector) is called the attractor, whereas Theorem 2.13 uses A = PN(Pγ) and R = PR(Pγ); this is cosmetic and does not affect correctness .