Index 1 fixed points of orientation reversing
planar homeomorphisms
| dc.contributor.author | Romero Ruiz del Portal, Francisco | |
| dc.contributor.author | Salazar, J. M. | |
| dc.date.accessioned | 2023-06-19T14:58:00Z | |
| dc.date.available | 2023-06-19T14:58:00Z | |
| dc.date.issued | 2015 | |
| dc.description.abstract | Let U subset of R-2 be an open subset, f : U -> f (U) subset of R-2 be an orientation reversing homeomorphism and let 0 is an element of U be an isolated, as a periodic orbit, fixed point. The main theorem of this paper says that if the fixed point indices i(R2)(f, 0) = i(R2)(f(2), 0) = 1 then there exists an orientation preserving dissipative homeomorphism phi: R-2 -> R-2 such that f(2) = phi in a small neighbourhood of 0 and {0} is a global attractor for phi. As a corollary we have that for orientation reversing planar homeomorphisms a fixed point, which is an isolated fixed point for f(2), is asymptotically stable if and only if it is stable. We also present an application to periodic differential equations with symmetries where orientation reversing homeomorphisms appear naturally | |
| dc.description.department | Depto. de Álgebra, Geometría y Topología | |
| dc.description.faculty | Fac. de Ciencias Matemáticas | |
| dc.description.refereed | TRUE | |
| dc.description.sponsorship | Ministerio de Ciencia e Innovación (MICINN) | |
| dc.description.status | pub | |
| dc.eprint.id | https://eprints.ucm.es/id/eprint/34835 | |
| dc.identifier.doi | 10.12775/TMNA.2015.044 | |
| dc.identifier.issn | 1230-3429 | |
| dc.identifier.officialurl | http://apcz.pl/czasopisma/index.php/TMNA/article/view/TMNA.2015.044 | |
| dc.identifier.uri | https://hdl.handle.net/20.500.14352/34979 | |
| dc.issue.number | 1 | |
| dc.journal.title | Topological Methods in Nonlinear Analysis | |
| dc.language.iso | eng | |
| dc.page.final | 226 | |
| dc.page.initial | 223 | |
| dc.publisher | Juliusz Schauder Center | |
| dc.relation.projectID | MTM 2012-30719 | |
| dc.rights.accessRights | restricted access | |
| dc.subject.cdu | 514 | |
| dc.subject.keyword | Fixed point index | |
| dc.subject.keyword | Conley index | |
| dc.subject.keyword | Orientation reversing homeomorphisms | |
| dc.subject.keyword | Attractors | |
| dc.subject.keyword | Stability | |
| dc.subject.ucm | Geometría | |
| dc.subject.unesco | 1204 Geometría | |
| dc.title | Index 1 fixed points of orientation reversing planar homeomorphisms | |
| dc.type | journal article | |
| dc.volume.number | 46 | |
| dcterms.references | [1] J.M. Alonso, J. Campos, The index and the asymptotic stability of fixed points in two dimensions: a counterexample. Nonlinear Anal. 32 (1998) 719-725. [2] M. Barge, R.M. Gillette, A fixed point theorem for plane separating continua, Topology and its applications 43 (1992) 203-212. [3] C. Bonatti, B. Kolev, Surface homeomorphisms with zero-dimensional singular set, Topology Appl. 90 (1998) 69-95. [4] L.E. Brouwer, Beweis des ebenen translationssatzes, Math. Annalen, 72 (1912) 37-54. [5] R.F. Brown, The Lefschetz fixed point theorem, Scott Foreman Co. Glenview Illinois, London (1971). [6] C. Carathéodory, Über die Begrenzung einfach zusammenhängender Gebiete, Math. Annalen, 73 (1913) 323-370. [7] M.L.Cartwright, J.E. Littlewood, Some fixed point theorems, Annals of Mathematics, Vol 54, 1, (1951), 1-37. [8] C.O. Christenson, W.L. Voxman, Aspects of Topology, BCS Associates,Moscow, Idaho, (1998). [9] E.N. Dancer, On the indices of fixed point of mappings in cones and applications, J. Math. Anal. Appl. 91 (1983) 131-151. [10] E.N. Dancer, R. Ortega, The index or Lyapunov stable fixed points, Journal Dynamics and Diff. Equations, 6 (1994) 631-637. [11] A. Dold Fixed point index and fixed point theorem for Euclidean neighborhood retracts, Topology, 4 (1965),1-8. [12] J. Franks, D. Richeson, Shift equivalence and the Conley index, Trans. Amer. Math. Soc. 352, 7 (2000), 3305-3322. [13] G. Graff, P. Nowak-Przygodzki, Fixed point indices of the iterations of planar homeomorphisms, Topol. Methods Nonlinear Anal. 22 (2003) 159-166. [14] J.Jezierski, W. Marzantowicz, Homotopy methods in topological fixed and periodic point theory, Springer (2005). [15] B. Kerékjártó, Voresungen Uber Topologie (I), Springer, Berlin (1923). [16] B. Kerékjártó, Sur le caractèr topologique des représentations conformes, C.R. Acad. Sci. 198 (1934) 317-320. [17] K. Kuperberg, Fixed points of orientation reversing homeomorphisms of the plane, Proc. Amer. Math. Soc. 112, (1991), 1, 223-229. [18] P. Le Calvez, Une propriété dynamique des homéomorphismes du plan au voisinage d’un point fixe d’indice > 1, Topology, 38, 1 (1999) 23-35. [19] P. Le Calvez, Dynamique des homéomorphismes du plan au voisinage d’un point fixe. Ann. Sci. Ecole Norm. Sup. (4) 36 (2003), no. 1, 139-171. [20] P. Le Calvez, J.-C. Yoccoz, Un théoréme d’indice pour les homéomorphismes du plan au voisinage d’un point fixe. Annals of Math. 146 (1997) 241-293. [21] F. Le Roux, Homéomorphismes de surfaces - Théore`mes de la fleur de Leau-Fatou et de la variété stable. astérisque 292 (2004). [22] J.N. Mather, Topological proofs of some purely topological consequences of Carathéodory’s theory of prime ends. Selected studies: physicsastrophysics, mathematics, history of science, pp. 225-255, NorthHolland, Amsterdam-New York, 1982. [23] J.R. Munkres, Topology, Prentice Hall, 2000. [24] R.D. Nussbaum, The fixed point index and some applications, Séminaire de Mathématiques supérieures, Les Presses de L’Université de Montréal, 1985. [25] R. Ortega, A criterion for asymptotic stability based on topological degree. In Proceedings of the First World Congress of Nonlinear Analysts, Tampa (1992) 383-394. [26] R. Ortega, F.R. Ruiz del Portal, Attractors with vanishing rotation number, Journal Eur. Math. Soc. 13 (2011) 1567-1588. [27] Ch. Pommerenke, Boundary Behavior of Conformal Maps, Grundlehren der mathematischen Wissenschaften 299, 1992, Springer-Verlag, Berlin, Heidelberg, New York. [28] S. Pelikan, E.E. Slaminka, A bound for the fixed point index of areapreserving homeomorphisms of two-manifolds, Ergodic Th. and Dynamical systems, 7 (1987) 463-479. [29] F.R. Ruiz del Portal Planar isolated and stable fixed points have index=1, Journal of Diff. Equations, 199 (2004) 179-188. [30] F.R. Ruiz del Portal, J.M. Salazar, Fixed point index of iterations of local homeomorphisms of the plane: a Conley-index approach, Topology 41 (2002) 1199-1212. | |
| dspace.entity.type | Publication |
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