Abstract
Suppose ϕ is a flow in \( {mathbb{S}^2} \), and \( S{\rm{(}}\varphi {\rm{)}}\;{\rm{ = }}\;square. \) □. As we shall show later, the behaviour of ϕ is very simple. In fact, if Ω ia any component of \( G(\varphi ) \), then Ω is an annulus, and if x ∈ Ω, then either
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(i)
\( {O_\varphi }(x) \) is a Jordan curve which separates ∂(Ω), or
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(ii)
pϕ (x) = ∞, and there exist z1 and z2 in Ω, such that \( {O_\varphi }({z_1}) \) and \( {O_\varphi }({z_2}) \) are Jordan curves, x belongs to the annulus A between \( {O_\varphi }({z_1}) \) and \( {O_\varphi }({z_2}) \), and each point x′ ∈ A satisfies that \( {O_\varphi }(x') \) α-spirals to \( {O_\varphi }({z_1}) \) and ω-spirals to \( {O_\varphi }({z_2}) \).
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© 1974 Springer-Verlag Berlin · Heidelberg
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Beck, A. (1974). Regular and Singular Points. In: Continuous Flows in the Plane. Die Grundlehren der mathematischen Wissenschaften, vol 201. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-65548-7_4
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DOI: https://doi.org/10.1007/978-3-642-65548-7_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-65550-0
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