Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-19T15:49:40.628Z Has data issue: false hasContentIssue false
  • English
  • Français

Smooth linearisation of planar periodic maps

Published online by Cambridge University Press:  21 May 2018

A. CIMA ,
A. GASULL ,
F. MAÑOSAS  and
R. ORTEGA
A. CIMA
Affiliation:
Departament de Matemàtiques, Edifici C Universitat Autònoma de Barcelona, Bellaterra (Barcelona), Spain. e-mails: cima@mat.uab.cat; gasull@mat.uab.cat; manyosas@mat.uab.cat
A. GASULL
Affiliation:
Departament de Matemàtiques, Edifici C Universitat Autònoma de Barcelona, Bellaterra (Barcelona), Spain. e-mails: cima@mat.uab.cat; gasull@mat.uab.cat; manyosas@mat.uab.cat
F. MAÑOSAS
Affiliation:
Departament de Matemàtiques, Edifici C Universitat Autònoma de Barcelona, Bellaterra (Barcelona), Spain. e-mails: cima@mat.uab.cat; gasull@mat.uab.cat; manyosas@mat.uab.cat
R. ORTEGA
Affiliation:
Departamento de Matemática Aplicada, Universidad de Granada, Granada, Spain. e-mail: rortega@ugr.es
Get access Rights & Permissions [Opens in a new window]

Abstract

The celebrated Kerékjártó theorem asserts that planar continuous periodic maps can be continuously linearised. We prove that for each k ∈ {1, 2,. . ., ∞}, ${\Mathcal {C}}$k-planar periodic maps can be ${\Mathcal {C}}$k-linearised.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 167 , Issue 2 , September 2019 , pp. 295 - 320
Copyright
Copyright © Cambridge Philosophical Society 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Bing, R. H. A homeomorphism between the 3-sphere and the sum of two solid horned spheres. Ann. of Math. 56 (1952), 354–362.Google Scholar
[2] Bing, R. H. Inequivalent families of periodic homeomorphisms of E 3. Ann. of Math. 80 (1964), 78–93.Google Scholar
[3] Cima, A., Gasull, A. and Mañosas, F. Simple examples of planar involutions with non-global Montgomery–Bochner linearisations. Appl. Math. Lett. 25 (2012), 2086–2088.Google Scholar
[4] Cima, A., Gasull, A., Mañosas, F. and Ortega, R.. Linearisation of planar involutions in ${\Mathcal {C}}$1. Ann. Mat. Pur. Appl. 194 (2015), 1349–1357.Google Scholar
[5] Conner, P. E. and Floyd, E. E. On the construction of periodic maps without fixed points. Proc. Amer. Math. Soc. 10 (1959), 354–360.Google Scholar
[6] Constantin, A. and Kolev, B. The theorem of Kerékjártó on periodic homeomorphisms of the disc and the sphere. Enseign. Math. 40 (1994), 193–204.Google Scholar
[7] Epstein, D. B. A., Pointwise periodic homeomorphisms. Proc. London Math. Soc. 42 (1981), 415–460.Google Scholar
[8] Haynes, R., Kwasik, S., Mast, J. and Schultz, R.. Periodic maps on R7 without fixed points. Math. Proc. Camb. Phil. Soc. 132 (2002), 131–136.Google Scholar
[9] Hirsch, M. W. Differential Topology (Springer-Verlag, New York 1976).Google Scholar
[10] Kister, J. M. Differentiable periodic actions on E 8 without fixed points. Amer. J. Math. 85 (1963), 316–319.Google Scholar
[11] Kuczma, M., Choczewski, B. and Ger, R. Iterative functional equations. Encyclopedia of Mathematics and its Applications 32 (Cambridge University Press, Cambridge, 1990).Google Scholar
[12] Meyer, K. R. Counterexamples in dynamical systems via normal form theory. SIAM Rev. 28 (1986), 41–51.Google Scholar
[13] Meyer, K. R., Hall, G. R. and Offin, D. Introduction to Hamiltonian dynamical systems and the N-body problem (Springer, 2009).Google Scholar
[14] Montgomery, D. Pointwise periodic homeomorphisms. Amer. J. Math. 59 (1937), 118–120.Google Scholar
[15] Montgomery, D. and Zippin, L. Topological transformation groups (Interscience, New York 1955).Google Scholar
[16] Morse, M. Differentiable mappings in the Shoenflies theorem. Compositio Math. 14 (1959), 83–151.Google Scholar
[17] Munkres, J. R. Elementary differential topology (Princeton University Press, Princeton 1966).Google Scholar
[18] Ortega, R. Retracts, fixed points and differential equations. Rev. R. Acad. Cien. Serie A. Mat. 102 (2008), 89–100.Google Scholar
[19] Ortega, R. A dynamical characterization of planar symmetries. Qual. Theory Dyn. Syst. 10 (2011), 197–201.Google Scholar
[20] Palais, R. S. Natural operations on differential forms. Trans. Amer. Math. Soc. 92 (1959), 125–141.Google Scholar
[21] Palais, R. S. Extending diffeomorphisms. Proc. Amer. Math. Soc. 11 (1960), 274–277.Google Scholar