  A study of rational maps of the real or complex projective plane of degree
two or more, concentrating on those which map an elliptic curve onto itself,
necessarily by an expanding map. We describe relatively simple examples with a
rich variety of exotic dynamical behaviors which are perhaps familiar to the
applied dynamics community but not to specialists in several complex variables.
For example, we describe smooth attractors with riddled or intermingled
attracting basins, and we observe ``blowout'' bifurcations when the transverse
Lyapunov exponent for the invariant curve changes sign. In the complex case,
the elliptic curve (a topological torus) can never have a trapping
neighborhood, yet it can have an attracting basin of large measure (perhaps
even of full measure). We also describe examples where there appear to be
Herman rings (that is topological cylinders mapped to themselves with
irrational rotation number) with open attracting basin. In some cases we
provide proofs, but in other cases the discussion is empirical, based on
numerical computation.
