Integrability and algebraic entropy of k-periodic non-autonomous Lyness recurrences

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Abstract

This work deals with non-autonomous Lyness type recurrences of the formxn+2=an+xn+1xn, where {an}n is a k-periodic sequence of complex numbers with minimal period k. We treat such non-autonomous recurrences via the autonomous dynamical system generated by the birational mapping Fak{ring operator}Fak-1{ring operator}⋯{ring operator}Fa1 where Fa is defined by Fa(x,y)=(y,a+yx). For the cases k∈{1, 2, 3, 6} the corresponding mappings have a rational first integral. By calculating the dynamical degree we show that for k=4 and for k=5 generically the dynamical system is no longer rationally integrable. We also prove that the only values of k for which the corresponding dynamical system is rationally integrable for all the values of the involved parameters, are k∈{1, 2, 3, 6}. © 2013.
Original languageEnglish
Pages (from-to)20-34
JournalJournal of Mathematical Analysis and Applications
Volume413
Issue number1
DOIs
Publication statusPublished - 1 May 2014

Keywords

  • Algebraic entropy
  • Blow-ups
  • Dynamical degree
  • Integrability
  • Lyness equation
  • Picard group
  • Recurrences

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