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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

R. H. Bing, Inequivalent families of periodic homeomorphisms of $E_3$, Ann. of Math. (2), 80 (1964), 78-93. doi: 10.2307/1970492.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

A. Cima, A. Gasull and F. Mañosas, On Coxeter recurrences,, J. Difference Equ. Appl., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

H. S. M. Coxeter, Frieze patterns, Acta Arith., 18 (1971), 297-310.  Google Scholar

[15]

M. Csörnyei and M. Laczkovich, Some periodic and non-periodic recursions, Monatshefte für Mathematik, 132 (2001), 215-236. doi: 10.1007/s006050170042.  Google Scholar

[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.  Google Scholar

[17]

J. M. Kister, Differentiable periodic actions on $E^8$ without fixed points, Amer. J. Math., 85 (1963), 316-319. doi: 10.2307/2373217.  Google Scholar

[18]

M. Kuczma, B. Choczewski and R. Ger, "Iterative Functional Equations," Encyclopedia of Mathematics and its Applications, 32, Cambridge University Press, Cambridge, 1990.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

D. Montgomery and L. Zippin, "Topological Transformation Groups,'' Interscience Publishers, New-York-London, 1955.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

R. H. Bing, Inequivalent families of periodic homeomorphisms of $E_3$, Ann. of Math. (2), 80 (1964), 78-93. doi: 10.2307/1970492.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

A. Cima, A. Gasull and F. Mañosas, On Coxeter recurrences,, J. Difference Equ. Appl., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

H. S. M. Coxeter, Frieze patterns, Acta Arith., 18 (1971), 297-310.  Google Scholar

[15]

M. Csörnyei and M. Laczkovich, Some periodic and non-periodic recursions, Monatshefte für Mathematik, 132 (2001), 215-236. doi: 10.1007/s006050170042.  Google Scholar

[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.  Google Scholar

[17]

J. M. Kister, Differentiable periodic actions on $E^8$ without fixed points, Amer. J. Math., 85 (1963), 316-319. doi: 10.2307/2373217.  Google Scholar

[18]

M. Kuczma, B. Choczewski and R. Ger, "Iterative Functional Equations," Encyclopedia of Mathematics and its Applications, 32, Cambridge University Press, Cambridge, 1990.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

D. Montgomery and L. Zippin, "Topological Transformation Groups,'' Interscience Publishers, New-York-London, 1955.  Google Scholar

[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.  Google Scholar

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