Invariant curves for endomorphisms of P1×P1\mathbb P^1\times \mathbb P^1

Let $A_1, A_2\in \mathbb C(z)$ be rational functions of degree at least two that are neither Latt\`es maps nor conjugate to $z^{\pm n}$ or $\pm T_n.$ We describe invariant, periodic, and preperiodic algebraic curves for endomorphisms of $(\mathbb P^1(\mathbb C))^2$ of the form $(z_1,z_2)\rightarrow (A_1(z_1),A_2(z_2)).$ In particular, we show that if $A\in \mathbb C(z)$ is not a ``generalized Latt\`es 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 $\mathbb C$ we give a criterion for a point of $(\mathbb P^1(K))^2$ 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 $(A_1, A_2)$-invariant curves of any given bi-degree $(d_1,d_2).$Comment: A polished and extended version, containing a proof of the Zhang conjecture for endomorphisms of $\mathbb P^1\times \mathbb P^1.

On algebraic curves A(x)-B(y)=0 of genus zero

Using a geometric approach involving Riemann surface orbifolds, we provide lower bounds for the genus of an irreducible algebraic curve of the form $E_{A,B}:\, A(x)-B(y)=0$, where $A, B\in\mathbb C(z)$. We also investigate "series" of curves $E_{A,B}$ of genus zero, where by a series we mean a family with the "same" $A$. We show that for a given rational function $A$ a sequence of rational functions $B_i$, such that ${\rm deg}\, B_i \rightarrow \infty$ and all the curves $A(x)-B_i(y)=0$ are irreducible and have genus zero, exists if and only if the Galois closure of the field extension $\mathbb C(z)/\mathbb C(A)$ has genus zero or one.Comment: published versio

Polynomial semiconjugacies, decompositions of iterations, and invariant curves

We study the functional equation $A\circ X=X\circ B$, where $A,$ $B$, and $X$ are polynomials over $\mathbb C$. Using previous results of the author about polynomials sharing preimages of compact sets, we show that for given $B$ its solutions may be described in terms of the filled-in Julia set of $B$. On this base, we prove a number of results describing a general structure of solutions. The results obtained imply in particular the result of Medvedev and Scanlon about invariant curves of maps $F:\,\mathbb C^2 \rightarrow \mathbb C^2$ of the form $(x,y)\rightarrow (f(x),f(y))$, where $f$ is a polynomial, and a version of the result of Zieve and M\"uller about decompositions of iterations of a polynomial.Comment: The final version accepted by Ann. Sc. Norm. Super. Pisa Cl. Sc

On polynomials orthogonal to all powers of a given polynomial on a segment

In this paper we investigate the following "polynomial moment problem": for a complex polynomial $P(z)$ and distinct complex numbers $a,b$ to describe polynomials $q(z)$ orthogonal to all integer non-negative powers of $P(z)$ on the segment $[a,b].$Comment: 27 pages, 9 figure