Contemporary Mathematics Volume 311, 2002 SELF-MAPS OF JP2 WITH INVARIANT ELLIPTIC CURVES ARACELI M. BONIFANT AND MARIUS DABIJA ABSTRACT. We discuss the geometric and dynamical properties of the holo- morphic self-maps of JP2 that leave invariant an elliptic plane curve. 1. INTRODUCTION. Given an elliptic plane curve Q, we consider the problem of constructing holo- morphic self-maps f of JP2 that leave Q invariant. In section 2, we state the criterion for a self-map of Q to extend to JP2 . We look in section 3 at the singular points of Q. In contrast with the smooth case, most singular elliptic curves do not admit non- trivial self-maps. The obstructions given by the singular points of Q are discussed in section 3. We define two invariants, in terms of Weierstrass' a and ( functions, and state an invariance criterion for the elliptic plane curves with ordinary singularities. In section 4, we prove that do not exist self-maps of JP2 , for which Q is critical and invariant. We prove in section 5 that the backward orbit of any point of Q is dense in the Julia set of f. In section 6 we discuss the case when Q is a smooth cubic. The classic tangent process on Q provides examples of self-maps that leave Q invariant. If we require f to leave invariant a line of lines, Q must be isomorphic to the Fermat cubic. We also discuss in this section the case when f has minimal degree 2. When an elliptic plane curve has enough symmetries, the invariants associated to its singular points can be calculated easily. The simplest case is the dual of a smooth cubic (section 7). In section 8, we consider special families of elliptic quartics with two singular points. Computer-generated pictures illustrate tangent processes on such curves. 2. PRELIMINARIES. Let Cd denote the space of homogeneous polynomials of degree d in three vari- ables. A rational self-map f of JP2 , of algebraic degree d(f) = d is given by three polynomials po, Pl, P2 in Cd with no common divisors, according to the formula f[xo,xl.x2] = [po(xo,xl,x2),pl(xo,xl,x2),p2(xo,xl,x2)]. Let I(!) denote the set of indeterminacy of J, formed by the common zeroes in JP2 of the polynomials Pi. 1991 Mathematics Subject Classification. 32H50 14H52, 14N05. The first author was partially supported by Proyecto Conacyt-Mexico, Sistemas Dinamicos 27958E. © 2002 American Mathematical Society http://dx.doi.org/10.1090/conm/311/05444
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