Conley Index Theory and the Attractor-Repeller Decomposition for Differential Inclusions

The paper’s Theorem 3.1 states the attractor–repeller decomposition for a multiflow on a compact X with closed invariant S: S = A ∪ R ∪ C(A,R) with disjoint parts, R is a repeller, C(A,R) = {x | ω_S(x) ⊂ A and α_S(x) ⊂ R}, and A is the dual attractor of R. The paper proves item (2) by choosing U with ω_S(U)=A, showing there exists t*>0 with Φ_S([t*,∞),U) ⊂ U, setting U* = S \ Φ_S([t*,∞),U), and establishing R = α_S(U*) (via Lemma 3.2 and an invariance argument), then deriving (3) and (4) from this setup . The candidate solution arrives at the same four conclusions but via a slightly different route: it constructs a forward-invariant V by taking a sufficiently late image K_T(U) and closing it in S, proves ω_S(V)=A, sets U* = S\V, and then shows α_S(U*) equals the dual repeller R := {x | ω_S(x) ⊄ A}, identifies C(A,R), and proves A is the dual attractor directly. Both arguments are valid under the paper’s standing assumptions (S closed invariant; Φ upper semicontinuous with compact values) . Minor presentational gaps (e.g., justifying the existence of t* in the paper, or the closure and nestedness steps in the model) can be made rigorous with standard compactness and USC arguments. Hence, both are correct, with different proofs.