## Institute for Mathematical Sciences

## Preprint ims06-01

** A. Bonifant, M. Dabija, J. Milnor**
*Elliptic curves as attractors in P^2, Part 1: dynamics*

Abstract: 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.

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