Global linearization of periodic difference equations

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May 2012, 32(5): 1575-1595. doi: 10.3934/dcds.2012.32.1575

Global linearization of periodic difference equations

Anna Cima ^1, , Armengol Gasull ^2, and Francesc Mañosas ^1,

1.	Dept. de Matemàtiques, Universitat Autònoma de Barcelona, Ediﬁci C, 08193 Bellaterra, Barcelona, Spain, Spain
2.	Dept. de Matemàtiques, Universitat Autònoma de Barcelona, Edifici C, 08193 Bellaterra, Barcelona

Received November 2010 Revised March 2011 Published January 2012

Abstract
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We deal with $m$-periodic, $n$-th order difference equations and study whether they can be globally linearized. We give an affirmative answer when $m=n+1$ and for most of the known examples appearing in the literature. Our main tool is a refinement of the Montgomery-Bochner Theorem.

Keywords: recurrences, periodic difference equations, Periodic maps, linearization..

Mathematics Subject Classification: Primary: 39A05; Secondary: 39A20, 39B1.

Citation: Anna Cima, Armengol Gasull, Francesc Mañosas. Global linearization of periodic difference equations. Discrete and Continuous Dynamical Systems, 2012, 32 (5) : 1575-1595. doi: 10.3934/dcds.2012.32.1575

References:

[1]	R. M. Abu-Saris and Q. M. Al-Hassan, On global periodicity of difference equations, J. Math. Anal. Appl., 283 (2003), 468-477. doi: 10.1016/S0022-247X(03)00272-5.
[2]	K. I. T. Al-Dosary, Global periodicity: An inverse problem, Appl. Math. Lett., 18 (2005), 1041-1045. doi: 10.1016/j.aml.2003.12.010.
[3]	F. Balibrea and A. Linero, Some new results and open problems on periodicity of difference equations, in "Iteration Theory" (ECIT '04), Grazer Math. Ber., 350, Karl-Franzens-Univ. Graz, Graz, (2006), 15-38.
[4]	F. Balibrea and A. Linero, On the global periodicity of some difference equations of third order, J. Difference Equ. Appl., 13 (2007), 1011-1027. doi: 10.1080/10236190701388518.
[5]	F. Balibrea, A. Linero Bas, G. Soler López and S. Stević, Global periodicity of $x_{n+k+1}=f_k(x_{n+k})...f_2(x_ {n+2})f_1(x_{x+1})$, J. Difference Equ. Appl., 13 (2007), 901-910. doi: 10.1080/10236190701351144.
[6]	R. H. Bing, A homeomorphism between the $3$-sphere and the sum of two solid horned spheres, Ann. of Math. (2), 56 (1952), 354-362. doi: 10.2307/1969804.
[7]	R. H. Bing, Inequivalent families of periodic homeomorphisms of $E_3$, Ann. of Math. (2), 80 (1964), 78-93. doi: 10.2307/1970492.
[8]	J. S. Cánovas, A. Linero and G. Soler, A characterization of $k$-cycles, Nonlinear Anal., 72 (2010), 364-372. doi: 10.1016/j.na.2009.06.070.
[9]	A. Cima, A. Gasull and V. Mañosa, Global periodicity and complete integrability of discrete dynamical systems, J. Difference Equ. Appl., 12 (2006), 697-716. doi: 10.1080/10236190600703031.
[10]	A. Cima, A. Gasull and F. Mañosas, On periodic rational difference equations of order $k$, J. Difference Equ. and Appl., 10 (2004), 549-559. doi: 10.1080/10236190410001667977.
[11]	A. Cima, A. Gasull and F. Mañosas, On Coxeter recurrences, J. Difference Equ. Appl., to appear.
[12]	P. E. Conner and E. E. Floyd, On the construction of periodic maps without fixed points, Proc. Amer. Math. Soc., 10 (1959), 354-360. doi: 10.1090/S0002-9939-1959-0105115-X.
[13]	A. Constantin and B. Kolev, The theorem of Kerékjártó on periodic homeomorphisms of the disc and the sphere, Enseign. Math. (2), 40 (1994), 193-204.
[14]	H. S. M. Coxeter, Frieze patterns, Acta Arith., 18 (1971), 297-310.
[15]	M. Csörnyei and M. Laczkovich, Some periodic and non-periodic recursions, Monatshefte für Mathematik, 132 (2001), 215-236. doi: 10.1007/s006050170042.
[16]	R. Haynes, S. Kwasik, J. Mast and R. Schultz, Periodic maps on $\R^7$ without fixed points, Math. Proc. Cambridge Philos. Soc., 132 (2002), 131-136.
[17]	J. M. Kister, Differentiable periodic actions on $E^8$ without fixed points, Amer. J. Math., 85 (1963), 316-319. doi: 10.2307/2373217.
[18]	M. Kuczma, B. Choczewski and R. Ger, "Iterative Functional Equations," Encyclopedia of Mathematics and its Applications, 32, Cambridge University Press, Cambridge, 1990.
[19]	R. P. Kurshan and B. Gopinath, Recursively generated periodic sequences, Canad. J. Math., 26 (1974), 1356-1371. doi: 10.4153/CJM-1974-129-6.
[20]	B. D. Mestel, On globally periodic solutions of the difference equation $x_{n+1}=f(x_n)$/$x_{n-1}$, J. Difference Equations and Appl., 9 (2003), 201-209. doi: 10.1080/1023619031000061061.
[21]	D. Montgomery and L. Zippin, "Topological Transformation Groups,'' Interscience Publishers, New-York-London, 1955.
[22]	R. Plastock, Homeomorphisms between Banach spaces, Transactions of the American Mathematical Society, 200 (1974), 169-183. doi: 10.1090/S0002-9947-1974-0356122-6.

show all references

References:

[1]	R. M. Abu-Saris and Q. M. Al-Hassan, On global periodicity of difference equations, J. Math. Anal. Appl., 283 (2003), 468-477. doi: 10.1016/S0022-247X(03)00272-5.
[2]	K. I. T. Al-Dosary, Global periodicity: An inverse problem, Appl. Math. Lett., 18 (2005), 1041-1045. doi: 10.1016/j.aml.2003.12.010.
[3]	F. Balibrea and A. Linero, Some new results and open problems on periodicity of difference equations, in "Iteration Theory" (ECIT '04), Grazer Math. Ber., 350, Karl-Franzens-Univ. Graz, Graz, (2006), 15-38.
[4]	F. Balibrea and A. Linero, On the global periodicity of some difference equations of third order, J. Difference Equ. Appl., 13 (2007), 1011-1027. doi: 10.1080/10236190701388518.
[5]	F. Balibrea, A. Linero Bas, G. Soler López and S. Stević, Global periodicity of $x_{n+k+1}=f_k(x_{n+k})...f_2(x_ {n+2})f_1(x_{x+1})$, J. Difference Equ. Appl., 13 (2007), 901-910. doi: 10.1080/10236190701351144.
[6]	R. H. Bing, A homeomorphism between the $3$-sphere and the sum of two solid horned spheres, Ann. of Math. (2), 56 (1952), 354-362. doi: 10.2307/1969804.
[7]	R. H. Bing, Inequivalent families of periodic homeomorphisms of $E_3$, Ann. of Math. (2), 80 (1964), 78-93. doi: 10.2307/1970492.
[8]	J. S. Cánovas, A. Linero and G. Soler, A characterization of $k$-cycles, Nonlinear Anal., 72 (2010), 364-372. doi: 10.1016/j.na.2009.06.070.
[9]	A. Cima, A. Gasull and V. Mañosa, Global periodicity and complete integrability of discrete dynamical systems, J. Difference Equ. Appl., 12 (2006), 697-716. doi: 10.1080/10236190600703031.
[10]	A. Cima, A. Gasull and F. Mañosas, On periodic rational difference equations of order $k$, J. Difference Equ. and Appl., 10 (2004), 549-559. doi: 10.1080/10236190410001667977.
[11]	A. Cima, A. Gasull and F. Mañosas, On Coxeter recurrences, J. Difference Equ. Appl., to appear.
[12]	P. E. Conner and E. E. Floyd, On the construction of periodic maps without fixed points, Proc. Amer. Math. Soc., 10 (1959), 354-360. doi: 10.1090/S0002-9939-1959-0105115-X.
[13]	A. Constantin and B. Kolev, The theorem of Kerékjártó on periodic homeomorphisms of the disc and the sphere, Enseign. Math. (2), 40 (1994), 193-204.
[14]	H. S. M. Coxeter, Frieze patterns, Acta Arith., 18 (1971), 297-310.
[15]	M. Csörnyei and M. Laczkovich, Some periodic and non-periodic recursions, Monatshefte für Mathematik, 132 (2001), 215-236. doi: 10.1007/s006050170042.
[16]	R. Haynes, S. Kwasik, J. Mast and R. Schultz, Periodic maps on $\R^7$ without fixed points, Math. Proc. Cambridge Philos. Soc., 132 (2002), 131-136.
[17]	J. M. Kister, Differentiable periodic actions on $E^8$ without fixed points, Amer. J. Math., 85 (1963), 316-319. doi: 10.2307/2373217.
[18]	M. Kuczma, B. Choczewski and R. Ger, "Iterative Functional Equations," Encyclopedia of Mathematics and its Applications, 32, Cambridge University Press, Cambridge, 1990.
[19]	R. P. Kurshan and B. Gopinath, Recursively generated periodic sequences, Canad. J. Math., 26 (1974), 1356-1371. doi: 10.4153/CJM-1974-129-6.
[20]	B. D. Mestel, On globally periodic solutions of the difference equation $x_{n+1}=f(x_n)$/$x_{n-1}$, J. Difference Equations and Appl., 9 (2003), 201-209. doi: 10.1080/1023619031000061061.
[21]	D. Montgomery and L. Zippin, "Topological Transformation Groups,'' Interscience Publishers, New-York-London, 1955.
[22]	R. Plastock, Homeomorphisms between Banach spaces, Transactions of the American Mathematical Society, 200 (1974), 169-183. doi: 10.1090/S0002-9947-1974-0356122-6.

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