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On boundaries of parallelizable regions of flows of free mappings. (English) Zbl 1146.37026

Summary: We are interested in the first prolongational limit set of the boundary of parallelizable regions of a given flow of the plane which has no fixed points. We prove that for every point from the boundary of a maximal parallelizable region, there exists exactly one orbit contained in this region which is a subset of the first prolongational limit set of the point. Using these uniquely determined orbits, we study the structure of maximal parallelizable regions.

MSC:

37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
37B99 Topological dynamics
39B12 Iteration theory, iterative and composite equations
54H20 Topological dynamics (MSC2010)
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References:

[1] Z. Leśniak, “On an equivalence relation for free mappings embeddable in a flow,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 13, no. 7, pp. 1911-1915, 2003. · Zbl 1056.37057 · doi:10.1142/S0218127403007746
[2] S. A. Andrea, “On homeomorphisms of the plane which have no fixed points,” Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. 30, pp. 61-74, 1967. · Zbl 0156.43704 · doi:10.1007/BF02993992
[3] W. Kaplan, “Regular curve-families filling the plane-I,” Duke Mathematical Journal, vol. 7, pp. 154-185, 1940. · Zbl 0024.19001 · doi:10.1215/S0012-7094-40-00710-4
[4] W. Kaplan, “Regular curve-families filling the plane-II,” Duke Mathematical Journal, vol. 8, pp. 11-46, 1941. · Zbl 0025.09301 · doi:10.1215/S0012-7094-41-00802-5
[5] Z. Leśniak, “On maximal parallelizable regions of flows of the plane,” International Journal of Pure and Applied Mathematics, vol. 30, no. 2, pp. 151-156, 2006. · Zbl 1106.39023
[6] Z. Leśniak, “On parallelizability of flows of free mappings,” Aequationes Mathematicae, vol. 71, no. 3, pp. 280-287, 2006. · Zbl 1097.39005 · doi:10.1007/s00010-005-2808-4
[7] N. P. Bhatia and G. P. Szegö, Stability Theory of Dynamical Systems, vol. 161 of Die Grundlehren der mathematischen Wissenschaften, Springer, New York, NY, USA, 1970. · Zbl 0213.10904
[8] R. C. McCann, “Planar dynamical systems without critical points,” Funkcialaj Ekvacioj, vol. 13, pp. 67-95, 1970. · Zbl 0228.54041
[9] Z. Leśniak, “On parallelizable regions of flows of the plane,” Grazer Mathematische Berichte, vol. 350, pp. 175-183, 2006. · Zbl 1132.37302
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