Flowbox manifolds

Aarts, J. M.; Oversteegen, L. G.

doi:10.1090/S0002-9947-1991-1042286-8

Flowbox manifolds
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by J. M. Aarts and L. G. Oversteegen PDF

Trans. Amer. Math. Soc. 327 (1991), 449-463 Request permission

Abstract:

A separable and metrizable space $X$ is called a flowbox manifold if there exists a base for the open sets each of whose elements has a product structure with the reals $\operatorname {Re}$ as a factor such that a natural consistency condition is met. We show how flowbox manifolds can be divided into orientable and nonorientable ones. We prove that a space $X$ is an orientable flowbox manifold if and only if $X$ can be endowed with the structure of a flow without restpoints. In this way we generalize Whitney’s theory of regular families of curves so as to include self-entwined curves in general separable metric spaces. All spaces under consideration are separable and metrizable.

References

J. M. Aarts, The structure of orbits in dynamical systems, Fund. Math. 129 (1988), no. 1, 39–58. MR 954894, DOI 10.4064/fm-129-1-39-58
J. M. Aarts, Orientation of orbits in flows, Papers on general topology and related category theory and topological algebra (New York, 1985/1987) Ann. New York Acad. Sci., vol. 552, New York Acad. Sci., New York, 1989, pp. 1–7. MR 1020768, DOI 10.1111/j.1749-6632.1989.tb22380.x
J. M. Aarts and M. Martens, Flows on one-dimensional spaces, Fund. Math. 131 (1988), no. 1, 53–67. MR 970914, DOI 10.4064/fm-131-1-53-67
J. M. Aarts, C. L. Hagopian, and L. G. Oversteegen, The orientability of matchbox manifolds, Pacific J. Math. 150 (1991), no. 1, 1–12. MR 1120708
David H. Carlson, A generalization of Vinograd’s theorem for dynamical systems, J. Differential Equations 11 (1972), 193–201. MR 289891, DOI 10.1016/0022-0396(72)90089-7
W. C. Chewning, A dynamical system on $E^{4}$ neither isomorphic nor equivalent to a differential system, Bull. Amer. Math. Soc. 80 (1974), 150–153. MR 326781, DOI 10.1090/S0002-9904-1974-13396-3

Geometric theory of foliations

Andrzej Gutek, On compact spaces which are locally Cantor bundles, Fund. Math. 108 (1980), no. 1, 27–31. MR 585557, DOI 10.4064/fm-108-1-27-31
Otomar Hájek, Dynamical systems in the plane, Academic Press, London-New York, 1968. MR 0240418
Otomar Hájek, Local characterisation of local semi-dynamical systems, Math. Systems Theory 2 (1968), 17–25. MR 239575, DOI 10.1007/BF01691343
Michael W. Mislove and James T. Rogers Jr., Local product structures on homogeneous continua, Topology Appl. 31 (1989), no. 3, 259–267. MR 997493, DOI 10.1016/0166-8641(89)90022-9
Sam B. Nadler Jr., Hyperspaces of sets, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 49, Marcel Dekker, Inc., New York-Basel, 1978. A text with research questions. MR 0500811
V. V. Nemytskii and V. V. Stepanov, Qualitative theory of differential equations, Princeton Mathematical Series, No. 22, Princeton University Press, Princeton, N.J., 1960. MR 0121520
Taro Ura, Isomorphism and local characterization of local dynamical systems, Funkcial. Ekvac. 12 (1969), 99–122. MR 259886
Hassler Whitney, Regular families of curves, Ann. of Math. (2) 34 (1933), no. 2, 244–270. MR 1503106, DOI 10.2307/1968202