A pathological flow in the torus
Proc. Amer. Math. Soc. 82 (1981), 303-306
Primary 54H20; Secondary 28A75
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Abstract: In a recent book (Continuous flows in the plane, Springer Grundlehren, vol. 201, 1974) the author of this article comments that there exist flows in the plane which have the property that some of their orbits have positive planar measure. Indeed, there are flows with the property that the complement of a countable set of orbits has measure 0. This fact does not appear in the literature, and the author has been asked to provide a proof. He proves instead the existence of a continuous flow in the torus in which the complement of a single orbit has measure 0. Then, lifting the torus to the plane gives the other example.
Beck, Continuous flows in the plane, Springer-Verlag, New
York-Heidelberg, 1974. With the assistance of Jonathan Lewin and Mirit
Lewin; Die Grundlehren der mathematischen Wissenschaften, Band 201. MR 0500869
Anatole Beck, M. N. Bleicher and D. W. Crowe, Excursions into mathematics, Worth, New York, 1969.
F. Osgood, A Jordan curve of positive
area, Trans. Amer. Math. Soc.
4 (1903), no. 1,
- Anatole Beck, Continuous flows in the plane, Die Grundlehren der Math. Wissenschaften, Band 201, Springer-Verlag, Berlin and New York, 1974. MR 0500869 (58:18379)
- Anatole Beck, M. N. Bleicher and D. W. Crowe, Excursions into mathematics, Worth, New York, 1969.
- W. F. Osgood, A Jordan curve of positive area [sic], Trans. Amer. Math. Soc. 4 (1903), 107-112. MR 1500628
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