2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/222747We consider the family $f_{a,b}(x,y)=(y,(y+a)/(x+b))$ of birational maps of the plane and the parameter values $(a,b)$ for which $f_{a,b}$ gives an automorphism of a rational surface. In particular, we find values for which $f_{a,b}$ is an automorphism of positive entropy but no invariant curve. The Main Theorem: If $f_{a,b}$ is an automorphism with an invariant curve and positive entropy, then either (1) $(a,b)$ is real, and the restriction of $f$ to the real points has maximal entropy, or (2) $f_{a,b}$ has a rotation (Siegel) domain.24 pages, 7 figures, A companion Mathematica notebook is available at: http://www.math.fsu.edu/~kim/Dynamical Systems37F99;32M99;32H50text