Dynamics of a Quadratic Map in Two Complex Variables

We study the map Ψ: 2→ 2 defined by Ψ(w, z)=(z, z+w2) and the associated collection of sequences {zj} satisfying the recurrence zj+2=zj+1+z2j. Iteration of Ψ 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→±∞ are 0. We investigate some associated functional equations, f (x+2)=f(x+1)+f(x)2 and L(L(x))=L(x)+x2.