Dynamics of a quadratic map in two complex variables

We study the map Psi: C-2 --> C-2 defined by Psi (w,z) = (z, z+w(2)) and the associated collection of sequences {z(j)} satisfying the recurrence z(j+2) = z(j+1) + z(j)(2). Iteration of Psi is equivalent to the study of such sequences. We analyze growth rates of the sequences, positive sequences, periodic and asymptotically periodic sequences and establish the existence of doubly infinite homoclinic sequences, non-zero sequences whose limits as j --> +/-infinity are 0. We investigate some associated functional equations, f(x+2) = f(x+1) + f(x)(2) and L(L(x)) = L(x) + x(2). (C) 2001 Academic Press.