Generalize the nonresonance hypothesis to an invariant subbundle without a complementary invariant subbundle

Prove that the invariant-manifold existence and uniqueness results remain valid when the Sacker–Sell subbundle associated with the spectral subset does not arise from an isolated spectral subset of the full Sacker–Sell spectrum, assuming only that the restricted linear dynamics on the invariant subbundle have spectrum Ω and satisfy the stated nonresonance conditions.

Background

The main theorem assumes that Ω is an isolated compact subset of the Sacker–Sell spectrum. This assumption supplies an invariant complementary subbundle, which is used implicitly in the construction.

The authors conjecture that the complementary invariant subbundle is unnecessary. The proposed generalization would require only an invariant subbundle whose restricted linear dynamics have Sacker–Sell spectrum Ω and satisfy the same nonresonance conditions. The assertion is not proved in the paper.

References

We suspect that, in place of assumption \ref{P2}, it would suffice to assume the existence of an invariant subbundle $\mathscr{N}\Omega$ such that the linear dynamics restricted to $\mathscr{N}\Omega$ have Sacker-Sell spectrum $\Omega$ for which the nonresonance conditions eq:intro_nonres_1-eq:intro_nonres_2 hold. In particular, the existence of an invariant complementary subbundle, which we implicitly assume by requesting that $\Omega$ be isolated in \ref{P2}, is unnecessary (c.f.\ Remark~4). We do not prove this assertion herein, however.

— An application of the parameterization method to evolution flows associated to nonautonomous systems  (2610.18078 - Buza et al., 13 Oct 2026) in Remark \ref{remark:P2_generalization}, Section 1.2, Statement of the main results