×

zbMATH — the first resource for mathematics

On families of invariant lines of a Brouwer homeomorphism. (English) Zbl 1427.37034
Summary: We present properties of equivalence classes of the codivergency relation defined for a Brouwer homeomorphism for which there exists a family of invariant pairwise disjoint lines covering the plane. In particular, using the codivergency relation we describe the sets of regular and irregular points for such Brouwer homeomorphisms. Moreover, we show that under this assumption the interior of each equivalence class of this relation is invariant and simply connected.
MSC:
37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
39B12 Iteration theory, iterative and composite equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Andrea, S. A., On homeomorphisms of the plane which have no fixed points, Abh. Math. Sem. Hamburg, 30, 61-74 (1967) · Zbl 0156.43704
[2] Béguin, F.; Leroux, F., Ensemble oscillant d’un homéomorphisme de Brouwer, homéomorphismes de Reeb, Bull. Soc. Math. France, 131, 2, 149-210 (2003) · Zbl 1026.37033
[3] Brouwer, L. E.J., Beweis des ebenen Translationssatzes, Math. Ann., 72, 37-54 (1912) · JFM 43.0569.02
[4] Brown, M., Fundamental regions of planar homeomorphisms, Contemp. Math., 117, 49-56 (1991)
[5] Daw, E. W., A maximally pathological Brouwer homeomorphism, Trans. Amer. Math. Soc., 343, 559-573 (1994) · Zbl 0871.54041
[6] Homma, T.; Terasaka, H., On the structure of the plane translation of Brouwer, Osaka. Math. J., 5, 233-266 (1953) · Zbl 0051.14701
[7] Leroux, F.; O’Farrell, A. G.; Roginskaya, M.; Short, I., Flowability of plane homeomorphisms, Ann. Inst. Fourier, 62, 2, 619-639 (2012) · Zbl 1296.37032
[8] Leśniak, Z., On an equivalence relation for free mappings embeddable in a flow, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13, 7, 1911-1915 (2003) · Zbl 1056.37057
[9] Leśniak, Z., On parallelizability of flows of free mappings, Aequationes Math., 71, 3, 280-287 (2006) · Zbl 1097.39005
[10] Leśniak, Z., On fractional iterates of a Brouwer homeomorphism embeddable in a flow, J. Math. Anal. Appl., 366, 1, 310-318 (2010) · Zbl 1211.37050
[11] Leśniak, Z., On the structure of Brouwer homeomorphisms embeddable in a flow, Abstr. Appl. Anal. 2012 (2012), 8 pp. Article ID 248413.
[12] Leśniak, Z., On properties of the set of invariant lines of a Brouwer homeomorphism, J. Differ. Equ. Appl., 24, 746-752 (2018) · Zbl 1390.39069
[13] Nakayama, H., A non flowable plane homeomorphism whose non Hausdorff set consists of two disjoint lines, Houston J. Math., 21, 3, 569-572 (1995) · Zbl 0857.54040
[14] Nakayama, H., Limit sets and square roots of homeomorphisms, Hiroshima Math. J., 26, 405-413 (1996) · Zbl 0914.54035
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.