A continuous linear flow on a trivial vector bundle B×RdB\times\mathbb{R}^d over a compact metric base flow — for instance the control flow of a bilinear control system x˙=(A0+∑iuiAi)x\dot{x}=(A_0+\sum_i u_i A_i)x — induces a projective flow on B×Pd−1B\times\mathbb{P}^{d-1}. Ayala, Colonius and Kliemann (2007) asked for a complete characterization of when two such induced projective flows are topologically skew-conjugate, i.e. conjugate by a fiber-preserving homeomorphism over a conjugacy of the base flows, and observed that the question was open even for single matrices. The single-matrix case has since been solved completely by Ayala and Kawan (2014): the projective flows of two endomorphisms are conjugate exactly when their Jordan structures coincide except for the values of the real parts of the eigenvalues, whose order and multiplicities must agree. For general base flows, including the projectivized control flows of bilinear systems, only partial results are known: preservation of Morse decompositions and Lyapunov-dimension data as necessary conditions, a hyperbolic-case classification at the bundle level, and a Grassmann-graph invariant that captures only the ordered block dimensions. The open problem is a complete classification in this general setting.

Background

A continuous linear flow on a vector bundle is a skew-product flow Φ=(θ,φ)\Phi=(\theta,\varphi) on V=B×Rd\mathcal{V}=B\times\mathbb{R}^d, where θ\theta is a flow (the base flow) on a compact metric space BB and each fiber map φ(t,⋅,b)\varphi(t,\cdot,b) is linear, with φ(t+s,⋅,b)=φ(t,⋅,θsb)∘φ(s,⋅,b)\varphi(t+s,\cdot,b)=\varphi(t,\cdot,\theta_s b)\circ\varphi(s,\cdot,b). Such flows are the natural state space for nonautonomous, random, and control-driven linear systems. The central example here is the control flow of a bilinear control system x˙=(A0+∑i=1muiAi)x\dot{x}=(A_0+\sum_{i=1}^m u_i A_i)x with admissible controls u∈U=L∞(R,U)u\in\mathcal{U}=L^\infty(\mathbb{R},U) for a compact convex control range UU: the base is the shift flow on U\mathcal{U} and the fiber maps are the solution operators x↦φ(t,x,u)x\mapsto\varphi(t,x,u) (Colonius and Kliemann 2000).

Because norms are irrelevant for qualitative behavior, one studies the induced projective flow PΦ\mathbb{P}\Phi on B×Pd−1B\times\mathbb{P}^{d-1}, which records the direction, but not the magnitude, of the fiber dynamics. Selgrade's theorem states that PΦ\mathbb{P}\Phi admits a finest Morse decomposition, i.e. a finite family of disjoint compact invariant sets (Morse sets), ordered so that every orbit outside them moves down the order, and these Morse sets lift to a unique decomposition of the bundle into invariant subbundles (Selgrade or Lyapunov subbundles) whose exponential growth rates form the Morse spectral intervals. The ordered list of subbundle dimensions is called the short Lyapunov index. The natural equivalence relation in this setting is skew conjugacy: a fiber-preserving homeomorphism h=(f,g)h=(f,g), whose first component ff conjugates the base flows and which commutes with the flows themselves, h∘Φt=Ψt∘hh\circ\Phi_t=\Psi_t\circ h (a skew equivalence, by contrast, only maps trajectories to trajectories).

For autonomous systems x˙=Ax\dot{x}=Ax on Rd\mathbb{R}^d the topological conjugacy classification is classical: two such linear flows are conjugate precisely when the Jordan structures of the generators coincide, except that the values of the real parts of the eigenvalues may differ as long as their order and multiplicities agree (see the historical account and elementary proof in Berger and Wynne 2020, and the statement in Ayala and Kawan 2014). Kuiper (1976) treated the discrete-time projective analogue, classifying transformations of real projective space induced by invertible matrices up to topological conjugacy, with a residual restriction on the periods of periodic points.

Ayala, Colonius and Kliemann (2007) extended this circle of ideas to bundle flows and bilinear control systems. They proved that hyperbolic linear bundle flows (those whose bundle splits as a direct sum of an exponentially stable and an exponentially unstable invariant subbundle) are skew conjugate exactly when the base flows are conjugate and the stable subbundle dimensions coincide; that a C1C^1 skew conjugacy forces fiberwise linear conjugacy; that a skew equivalence of projective flows necessarily maps finest Morse components onto each other, preserving their order, the number and order of spectral intervals, and the Lyapunov subbundle dimensions; and that the Grassmann graph — a finite directed graph built from the Morse decompositions on the Grassmann bundles (manifolds of kk-dimensional subspaces) — determines and is determined by the short Lyapunov index. However, they did not characterize when two linear flows have topologically skew-conjugate projective flows, and noted that this question was unresolved even for single matrices. The single-matrix case was subsequently resolved by Ayala and Kawan (2014): two projective flows induced by endomorphisms AA and BB of a finite-dimensional real vector space are topologically conjugate if and only if, up to individual linear coordinates, A=⨁i(λi id+σi)A=\bigoplus_i(\lambda_i\,\mathrm{id}+\sigma_i) and B=⨁i(μi id+σi)B=\bigoplus_i(\mu_i\,\mathrm{id}+\sigma_i) with the same blocks σi\sigma_i (all of whose eigenvalues lie on the imaginary axis) and strictly decreasing λi\lambda_i and μi\mu_i — that is, the Jordan structures coincide except for the values of the real parts of the eigenvalues, whose order and multiplicities must agree. Notably, the fine Jordan structure of the imaginary-axis blocks, invisible to Morse-spectral data, is essential. The general case over a nontrivial base — the paper's own main setting, e.g. for bilinear control systems — remains without any such characterization.

Problem Statement

Let Φ=(θ,φ)\Phi=(\theta,\varphi) and Ψ=(ϑ,ψ)\Psi=(\vartheta,\psi) be continuous linear flows on trivial vector bundles B×RdB\times\mathbb{R}^d and C×RdC\times\mathbb{R}^d over compact metric base flows, and let PΦ\mathbb{P}\Phi and PΨ\mathbb{P}\Psi be the induced projective flows on B×Pd−1B\times\mathbb{P}^{d-1} and C×Pd−1C\times\mathbb{P}^{d-1}. Give a complete characterization of the pairs (Φ,Ψ)(\Phi,\Psi) for which PΦ\mathbb{P}\Phi and PΨ\mathbb{P}\Psi are topologically skew-conjugate, i.e. for which there exists a homeomorphism h=(f,g)h=(f,g) with f:B→Cf:B\to C conjugating θ\theta to ϑ\vartheta and h∘PΦt=PΨt∘hh\circ\mathbb{P}\Phi_t=\mathbb{P}\Psi_t\circ h for all t∈Rt\in\mathbb{R}. The characterization should cover the general non-hyperbolic case over a nontrivial compact base, including the projectivized control flows of bilinear control systems, completing the classification program of Ayala, Colonius and Kliemann (2007), which classifies linear flows via their Selgrade/Morse (exponential) subbundle decompositions and resolves the hyperbolic case. The question retains the generality of the source: no restriction on base flows, fiber spectra, or dimension is imposed beyond the source setting itself; the already-resolved single-matrix case (trivial base) is part of the boundary of knowledge, not the target.

References

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