Abstract and Applied Analysis

On Boundaries of Parallelizable Regions of Flows of Free Mappings

Zbigniew Leśniak

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Abstract

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.

Article information

Source
Abstr. Appl. Anal., Volume 2007 (2007), Article ID 31693, 8 pages.

Dates
First available in Project Euclid: 27 February 2008

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1204126598

Digital Object Identifier
doi:10.1155/2007/31693

Mathematical Reviews number (MathSciNet)
MR2353781

Zentralblatt MATH identifier
1146.37026

Citation

Leśniak, Zbigniew. On Boundaries of Parallelizable Regions of Flows of Free Mappings. Abstr. Appl. Anal. 2007 (2007), Article ID 31693, 8 pages. doi:10.1155/2007/31693. https://projecteuclid.org/euclid.aaa/1204126598


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References

  • 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.
  • 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.
  • W. Kaplan, ``Regular curve-families filling the plane---I,'' Duke Mathematical Journal, vol. 7, pp. 154--185, 1940.
  • W. Kaplan, ``Regular curve-families filling the plane---II,'' Duke Mathematical Journal, vol. 8, pp. 11--46, 1941.
  • 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.
  • Z. Leśniak, ``On parallelizability of flows of free mappings,'' Aequationes Mathematicae, vol. 71, no. 3, pp. 280--287, 2006.
  • 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.
  • R. C. McCann, ``Planar dynamical systems without critical points,'' Funkcialaj Ekvacioj, vol. 13, pp. 67--95, 1970.
  • Z. Leśniak, ``On parallelizable regions of flows of the plane,'' Grazer Mathematische Berichte, vol. 350, pp. 175--183, 2006.