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Topological conjugacy of real projective flows. (English) Zbl 1311.37010
The paper gives a topological classification for flows on real projective space induced by linear flows on Euclidean space. An endomorphism \(A\) of a finite-dimensional real vector space \(V\) induces a linear flow \(e^{tA}\) on \(V\). Let \({\mathbb P}(V)\) denote the projective space of \(V\), the quotient of \(V\setminus\{0\}\) by the equivalence relation \(v\sim w\) if and only if \(w=\alpha v\) for some nonzero \(\alpha\in V\). A linear flow \(e^{tA}\) on \(V\) induces a real projective flow on \({\mathbb P}(V)\) because \(e^{tA}\) maps lines through the origin to lines through the origin.
The main theorem of the paper is that for endomorphisms \(A\) and \(B\) of a finite-dimensional real vector space, the projective flows induced by the linear flows \(e^{tA}\) and \(e^{tB}\) on \(V\) are topologically conjugate if and only if, with respect to individual linear coordinates on \(V\), \(A = (\lambda_1 {id} + \sigma_1)\oplus(\lambda_2 {id} + \sigma_2)\oplus\cdots \oplus(\lambda_k {id} + \sigma_k)\), \(B =(\mu_1 {id} + \sigma_1)\oplus(\mu_2 {id} + \sigma_2)\oplus\cdots \oplus(\mu_k {id} + \sigma_k)\), for real numbers \(\lambda_1>\lambda_2>\cdots>\lambda_k\) and \(\mu_1>\mu_2>\cdots> m_k\) and endomorphisms \(\sigma_1,\sigma_2,\dots,\sigma_k\) all of whose eigenvalues are on the imaginary axis.
The existence of a topological conjugacy in the main theorem follows from the fundamental domain method and an adaptation of Kuiper’s topological classification of real projective transformations to the continuous time case. The other direction in the proof of the main theorem relies on the discription in algebraic terms of several dynamical invariants, namely the finest Morse decompositions, the current set, and the dimensions of the stable manifolds. A correction to a formula of Kuiper for the dimension of the stable manifold is given as well.

37C15 Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems
15A21 Canonical forms, reductions, classification
37C10 Dynamics induced by flows and semiflows
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