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GLOBAL PERIODICITY CONDITIONS FOR MAPS AND RECURRENCES VIA NORMAL FORMS

Departament de Matemàtiques, Facultat de Ciències, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain

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ARMENGOL GASULL

Departament de Matemàtiques, Facultat de Ciències, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain

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VÍCTOR MAÑOSA

Departament de Matemàtica Aplicada III (MA3), Control, Dynamics and Applications Group (CoDALab), Universitat Politècnica de Catalunya (UPC), Colom 1, 08222 Terrassa, Spain

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https://doi.org/10.1142/S0218127413501824Cited by:5 (Source: Crossref)

Abstract

We face the problem of characterizing the periodic cases in parametric families of rational diffeomorphisms of 𝕂^k, where 𝕂 is ℝ or ℂ, having a fixed point. Our approach relies on the Normal Form Theory, to obtain necessary conditions for the existence of a formal linearization of the map, and on the introduction of a suitable rational parametrization of the parameters of the family. Using these tools we can find a finite set of values p for which the map can be p-periodic, reducing the problem of finding the parameters for which the periodic cases appear to simple computations. We apply our results to several two- and three-dimensional classes of polynomial or rational maps. In particular, we find the global periodic cases for several Lyness-type recurrences.

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References

R. M. Abu-Saris and Q. M. Al-Hassan, J. Math. Anal. Appl. 283, 468 (2003), DOI: 10.1016/S0022-247X(03)00272-5. Crossref, Web of Science, Google Scholar
R. M. Abu-Saris, Appl. Math. Lett. 19, 1078 (2006), DOI: 10.1016/j.aml.2005.11.022. Crossref, Web of Science, Google Scholar
D. K. Arrowsmith and C. M. Place , An Introduction to Dynamical Systems ( Cambridge University Press , Cambridge , 1990 ) . Google Scholar
I. Bajo and E. Liz, J. Diff. Eq. Appl. 12, 1201 (2006), DOI: 10.1080/10236190600949782. Crossref, Web of Science, Google Scholar
F. Balibrea and A. Linero, Grazer Math. Ber. 350, 15 (2006). Google Scholar
F. Balibrea and A. Linero, J. Diff. Eq. Appl. 13, 1011 (2007), DOI: 10.1080/10236190701388518. Crossref, Web of Science, Google Scholar
L. Berg and S. Stević, J. Diff. Eq. Appl. 12, 827 (2006), DOI: 10.1080/10236190600761575. Crossref, Web of Science, Google Scholar
R. H. Bing, Ann. Math. 56, 354 (1952), DOI: 10.2307/1969804. Crossref, Web of Science, Google Scholar
A. Caro and A. Linero, J. Diff. Eq. Appl. 15, 489 (2009), DOI: 10.1080/10236190802162739. Crossref, Web of Science, Google Scholar
A. Caro and A. Linero, Int. J. Bifurcation and Chaos 20, 2735 (2010), DOI: 10.1142/S0218127410027325. Link, Web of Science, Google Scholar
A. Cima, A. Gasull and F. Mañosas, J. Diff. Eq. Appl. 10, 549 (2004), DOI: 10.1080/10236190410001667977. Crossref, Web of Science, Google Scholar
A. Cima, A. Gasull and F. Mañosas, Discr. Contin. Dyn. Syst. A 32, 1575 (2012), DOI: 10.3934/dcds.2012.32.1575. Crossref, Web of Science, Google Scholar
M. Csörnyei and M. Laczkovich, Monatsh. Math. 132, 215 (2001). Web of Science, Google Scholar
J. J. Duistermaat , Discrete Integrable Systems: QRT Maps and Elliptic Surfaces , Springer Monographs in Mathematics ( Springer , NY , 2010 ) . Crossref, Google Scholar
E. A. Grove and G. Ladas , Periodicities in Nonlinear Difference Equations , Advances in Discrete Math. and Appl 4 ( Chapman & Hall , Boca Raton FL , 2005 ) . Google Scholar
I. Gumovski and Ch. Mira , Recurrences and Discrete Dynamic Systems , Lecture Notes in Mathematics 809 ( Springer-Verlag , Berlin , 1980 ) . Crossref, Google Scholar
R. Hayneset al., Math. Proc. Cambr. Philos. Soc. 132, 131 (2002), DOI: 10.1017/S0305004101005345. Crossref, Web of Science, Google Scholar
D. Jogia, J. A. G. Roberts and F. Vivaldi, J. Phys. A: Math. Gen. 39, 1133 (2006), DOI: 10.1088/0305-4470/39/5/008. Crossref, Google Scholar
J. M. Kister, Amer. J. Math. 85, 316 (1963), DOI: 10.2307/2373217. Crossref, Web of Science, Google Scholar
Maubach, S. [2003] Polynomial Endomorphisms and Kernels of Derivations, PhD thesis, University of Nijemegen, Nijmegen . Google Scholar
B. D. Mestel, J. Diff. Eq. Appl. 9, 201 (2003), DOI: 10.1080/1023619031000061061. Crossref, Web of Science, Google Scholar
D. Montgomery and L. Zippin , Topological Transformation Groups ( Interscience , NY , 1955 ) . Google Scholar
I. Niven, H. S. Zukerman and H. L. Montgomery, An Introduction to the Theory of Numbers, 5th edn. (John Wiley & Sons, NY, 1991). Google Scholar
J. Rubió-Massegú and V. Mañosa, J. Math. Anal. Appl. 332, 896 (2007). Crossref, Web of Science, Google Scholar
J. Rubió-Massegú, J. Math. Anal. Appl. 343, 182 (2008). Crossref, Web of Science, Google Scholar
T. Tsuda, J. Phys. A: Math. Gen. 37, 2721 (2004), DOI: 10.1088/0305-4470/37/7/014. Crossref, Google Scholar
A. van den Essen , Polynomial Automorphisms and the Jacobian Conjecture , Progress in Mathematics 190 ( Birkhäuser-Verlag , Basel , 2000 ) . Crossref, Google Scholar