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

This work deals with non-autonomous Lyness type recurrences of the form x n + 2 = a n + x n + 1 x n , where { a n } 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 F a k ∘ F a k − 1 ∘ ⋯ ∘ F a 1 where F a is defined by F a ( x , y ) = ( y , a + y x ) . 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 } .