×

zbMATH — the first resource for mathematics

On the structure of Brouwer homeomorphisms embeddable in a flow. (English) Zbl 1288.54027
Summary: We present two theorems describing the structure of the set of all regular points and the set of all irregular points for a Brouwer homeomorphism which is embeddable in a flow. The theorems are counterparts of structure theorems proved by T. Homma and H. Terasaka [Osaka Math. J. 5, 233–266 (1953; Zbl 0051.14701)]. To obtain our results, we use properties of the codivergence relation.

MSC:
54H20 Topological dynamics (MSC2010)
37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] H. Nakayama, “Limit sets and square roots of homeomorphisms,” Hiroshima Mathematical Journal, vol. 26, no. 2, pp. 405-413, 1996. · Zbl 0914.54035
[2] L. E. J. Brouwer, “Beweis des ebenen Translationssatzes,” Mathematische Annalen, vol. 72, no. 1, pp. 37-54, 1912. · JFM 43.0569.02
[3] M. Brown, E. E. Slaminka, and W. Transue, “An orientation preserving fixed point free homeomorphism of the plane which admits no closed invariant line,” Topology and its Applications, vol. 29, no. 3, pp. 213-217, 1988. · Zbl 0668.54024
[4] E. W. Daw, “A maximally pathological Brouwer homeomorphism,” Transactions of the American Mathematical Society, vol. 343, no. 2, pp. 559-573, 1994. · Zbl 0871.54041
[5] P. Le Calvez and A. Sauzet, “Une démonstration dynamique du théorème de translation de Brouwer,” Expositiones Mathematicae, vol. 14, no. 3, pp. 277-287, 1996. · Zbl 0859.54029
[6] T. Homma and H. Terasaka, “On the structure of the plane translation of Brouwer,” Osaka Journal of Mathematics, vol. 5, pp. 233-266, 1953. · Zbl 0051.14701
[7] S. A. Andrea, “On homoeomorphisms 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
[8] 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
[9] Z. Leśniak, “On parallelizability of flows of free mappings,” Aequationes Mathematicae, vol. 71, no. 3, pp. 280-287, 2006. · Zbl 1097.39005
[10] H. Nakayama, “On dimensions of non-Hausdorff sets for plane homeomorphisms,” Journal of the Mathematical Society of Japan, vol. 47, no. 4, pp. 789-793, 1995. · Zbl 0847.54011
[11] N. P. Bhatia and G. P. Szegö, Stability Theory of Dynamical Systems, Springer, New York, NY, USA, 1970. · Zbl 0213.10904
[12] 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
[13] Z. Leśniak, “On a decomposition of the plane for a flow of free mappings,” Publicationes Mathematicae Debrecen, vol. 75, no. 1-2, pp. 191-202, 2009. · Zbl 1194.37037
[14] Z. Leśniak, “On boundaries of parallelizable regions of flows of free mappings,” Abstract and Applied Analysis, vol. 2007, Article ID 31693, 8 pages, 2007. · Zbl 1146.37026
[15] F. Béguin and F. Le Roux, “Ensemble oscillant d’un homéomorphisme de Brouwer, homéomorphismes de Reeb,” Bulletin de la Société Mathématique de France, vol. 131, no. 2, pp. 149-210, 2003. · Zbl 1026.37033
[16] P. Hartman, Ordinary Differential Equations, vol. 38, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA, 2002. · Zbl 1009.34001
[17] S. N. Elaydi, Discrete Chaos, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2nd edition, 2008.
[18] F. Le Roux, “Il n’y a pas de classification borélienne des homéomorphismes de Brouwer,” Ergodic Theory and Dynamical Systems, vol. 21, no. 1, pp. 233-247, 2001. · Zbl 0992.37034
[19] F. Le Roux, “Classes de conjugaison des flots du plan topologiquement équivalents au flot de Reeb,” Comptes Rendus de l’Académie des Sciences. Série I. Mathématique, vol. 328, no. 1, pp. 45-50, 1999. · Zbl 0922.58069
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.