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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.

MSC:
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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References:
[1] Ayala, Dynamical characterization of the Lyapunov form of matrices, Linear Algebra Appl. 402 pp 272– (2005) · Zbl 1076.15013 · doi:10.1016/j.laa.2005.01.019
[2] Batterson, Structurally stable Grassmann transformations, Trans. Amer. Math. Soc. 231 pp 385– (1977) · Zbl 0379.58013 · doi:10.1090/S0002-9947-1977-0448443-6
[3] Cappell, Nonlinear similarity, Ann. of Math. 113 ((2)) pp 315– (1981) · Zbl 0477.57022 · doi:10.2307/2006986
[4] Cappell, Non-linear similarity and linear similarity are equivariant below dimension 6, in: Tel Aviv Topology Conference: Rothenberg Festschrift (1998) pp 59– (1999) · Zbl 0935.15011 · doi:10.1090/conm/231/03352
[5] Cappell, Nonlinear similarity and differentiability, Comm. Pure Appl. Math. 38 pp 697– (1985) · Zbl 0618.58010 · doi:10.1002/cpa.3160380603
[6] Cappell, Nonlinear similarity begins in dimension six, Amer. J. Math. 111 pp 717– (1989) · Zbl 0686.57025 · doi:10.2307/2374878
[7] Colonius, The dynamics of control (2000) · doi:10.1007/978-1-4612-1350-5
[8] Cruz, Linear and Lipschitz similarity, Linear Algebra Appl. 151 pp 17– (1991) · Zbl 0724.15006 · doi:10.1016/0024-3795(91)90352-W
[9] De Rham, Sur les nouveaux invariants topologiques de M. Reidemeister, Rec. Math. [Mat. Sbornik] N.S. 1 ((43)) pp 737– (1936)
[10] De Rham, Reidemeister’s torsion invariant and rotations of Sn, in: Differential analysis pp 27– (1964)
[11] Ferraiol, Jordan decomposition and dynamics on flag manifolds, Discrete Contin. Dyn. Syst. 26 pp 923– (2010) · Zbl 1183.37028 · doi:10.3934/dcds.2010.26.923
[12] Hambleton, Non-linear similarity revisited, in: Prospects in topology (Princeton, NJ, 1994) pp 157– (1995)
[13] Hatcher, Algebraic topology (2002)
[14] Hsiang, When are topologically equivalent orthogonal transformations linearly equivalent?, Invent. Math. 68 pp 275– (1982) · Zbl 0505.57016 · doi:10.1007/BF01394060
[15] Kawan, Lipschitz conjugacy of linear flows, J. London Math. Soc. 80 ((2)) pp 699– (2009) · Zbl 1191.34060 · doi:10.1112/jlms/jdp034
[16] Kuiper, The topology of the solutions of a linear differential equation on Rn, in: Manifolds-Tokyo 1973 (Proc. Internat. Conf., Tokyo, 1973) pp 195– (1975)
[17] Kuiper, Topological conjugacy of real projective transformations, Topology 15 pp 13– (1976) · Zbl 0317.58015 · doi:10.1016/0040-9383(76)90046-X
[18] Kuiper, Topological classification of linear endomorphisms, Invent. Math. 19 pp 83– (1973) · Zbl 0251.58008 · doi:10.1007/BF01418922
[19] Ladis, The topological equivalence of linear flows, Differ. Equ. 9 pp 938– (1975) · Zbl 0305.34014
[20] McSwiggen, Conjugate phase portraits of linear systems, Amer. Math. Monthly 115 pp 596– (2008) · Zbl 1167.34013
[21] Poincaré, Sur les courbes definies par les equations differentielles, in: Oeuvres de H. Poincaré (1928)
[22] Robinson, Dynamical systems. Stability, symbolic dynamics, and chaos, in: Studies in Advanced Mathematics (1999) · Zbl 0914.58021
[23] Schultz, On the topological classification of linear representations, Topology 16 pp 263– (1977) · Zbl 0367.55018 · doi:10.1016/0040-9383(77)90007-6
[24] Strelcyn, On topological conjugation in linear groups, Studia Math. 35 pp 261– (1970) · Zbl 0197.11005
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