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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: [email protected]; [email protected]; [email protected]
A. GASULL
Affiliation:
Departament de Matemàtiques, Edifici C Universitat Autònoma de Barcelona, Bellaterra (Barcelona), Spain. e-mails: [email protected]; [email protected]; [email protected]
F. MAÑOSAS
Affiliation:
Departament de Matemàtiques, Edifici C Universitat Autònoma de Barcelona, Bellaterra (Barcelona), Spain. e-mails: [email protected]; [email protected]; [email protected]
R. ORTEGA
Affiliation:
Departamento de Matemática Aplicada, Universidad de Granada, Granada, Spain. e-mail: [email protected]
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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 

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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