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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Dynamical systems with cross-sections


Author: Dean A. Neumann
Journal: Proc. Amer. Math. Soc. 56 (1976), 339-344
MSC: Primary 58F99; Secondary 57D50
DOI: https://doi.org/10.1090/S0002-9939-1976-0407903-9
MathSciNet review: 0407903
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Abstract: The problem of classifying dynamical systems (flows) with global cross-sections in terms of the associated diffeomorphisms of the cross-sections is considered. Suppose that, for $ i = 1,2,{\phi _i}$ is a $ {C^r}$ flow $ (r \geqslant 0)$ on the $ {C^r}$ manifold $ {M_i}$ that admits a global cross-section $ {S_i} \subseteq {M_i}$ with associated diffeomorphism ('first return map') $ {d_i}$. If rank $ ({H_1}({M_1};{\mathbf{Z}})) = 1$, then $ ({M_1},{\phi _1})$ is $ {C^s}$ equivalent $ (s \leqslant r)$ to $ ({M_2},{\phi _2})$ if and only if $ {d_1}$ is $ {C^s}$ conjugate to $ {d_2}$. If rank $ ({H_1}({M_1};{\mathbf{Z}})) \ne 1$ and $ {\phi _1}$ has a periodic orbit, then there are infinitely many global cross-sections $ {T_i} \subseteq {M_1}$ of $ {\phi _1}$, such that the associated diffeomorphisms are pairwise nonconjugate.


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DOI: https://doi.org/10.1090/S0002-9939-1976-0407903-9
Article copyright: © Copyright 1976 American Mathematical Society