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

This work deals with non-autonomous Lyness type recurrences of the formxn+2=an+xn+1xn, where {an}n{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∘Fak−1∘⋯∘Fa1Fak∘Fak−1∘⋯∘Fa1 where FaFa is defined by Fa(x,y)=(y,a+yx). For the cases k∈{1,2,3,6}k∈{1,2,3,6} the corresponding mappings have a rational first integral. By calculating the dynamical degree we show that for k=4k=4 and for k=5k=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}k∈{1,2,3,6}.