# American Institute of Mathematical Sciences

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

## Global linearization of periodic difference equations

 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

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.
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., (). [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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##### 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., (). [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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