  We 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.
