Algebraic Curves A ∘ l ( x ) − U ( y ) = 0 and Arithmetic of Orbits of Rational Functions

Abstract

We give a description of pairs of complex rational functions A and U of degree at least two such that for every d ≥ 1 the algebraic curve A •d (x) − U (y) = 0 has a factor of genus zero or one. In particular, we show that if A is not a "generalized Lattès map", then this condition is satisfied if and only if there exists a rational function V such that U • V = A •l for some l ≥ 1. We also prove a version of the dynamical Mordell-Lang conjecture, concerning intersections of orbits of points from P 1 (K) under iterates of A with the value set U (P 1 (K)), where A and U are rational functions defined over a number field K.

Lower bounds for genera of fiber products

“…Remark 3.6. Theorem 3.5 was also proved in the paper [49] by a modification of the method of [47] (see [49], Theorem 3.1). Unfortunately, by the mistake of the author, the formulation of the corresponding result in [49] was partly copied from an earlier version of the paper.…”

“…Third, the low genus problem naturally arises in the new emerging field of arithmetic dynamics. For example, the problem of describing rational functions A and B such that all curves A ˝npxq ´Bpyq " 0, n ě 1, have a factor of genus zero or one is a geometric counterpart of the following problem of the arithmetic nature (see [11], [30], [49]): which rational functions A defined over a number field K have a K-orbit containing infinitely many points from the value set BpP 1 pKqq ? More generally, the problem of describing pairs of rational functions A and B such that all curves (3)…”

Finiteness theorems for commuting and semiconjugate rational functions

“…Proof. The theorem is a particular case of Theorem 2.10 in [21]. For the reader convenience we provide a short independent proof.…”

“…Moreover, deg R > 1 since otherwise equalities (19), (21), and (22) imply that the curve F (x, y) = 0 coincides with (20).…”

“…where A, B, and X are rational functions, playing along with equation 1an important role in complex and arithmetic dynamics (see e.g. [1], [3], [4], [7], [11], [17], [21]). We always will assume that A and B in (3) have degree at least two, while X has degree at least one.…”