Let A1, A2∈ C(z) be rational functions of degree at least two that are neither Lattès maps nor conjugate to z±n or ± Tn. We describe invariant, periodic, and preperiodic algebraic curves for endomorphisms of (P1(C))2 of the form (z1, z2) → (A1(z1) , A2(z2)). In particular, we show that if A∈ C(z) is not a “generalized Lattès map”, then any (A, A)-invariant curve has genus zero and can be parametrized by rational functions commuting with A. As an application, for A defined over a subfield K of C we give a criterion for a point of (P1(K))2 to have a Zariski dense (A, A)-orbit in terms of canonical heights, and deduce from this criterion a version of a conjecture of Zhang on the existence of rational points with Zariski dense forward orbits. We also prove a result about functional decompositions of iterates of rational functions, which implies in particular that there exist at most finitely many (A1, A2) -invariant curves of any given bi-degree (d1, d2).
ASJC Scopus subject areas
- Mathematics (all)