Moscow Mathematical Journal

Volume 20, Issue 1, January–March 2020 pp. 153–183.

Algebraic Curves $A^{\circ l}(x)-U(y)=0$ and Arithmetic of Orbits of Rational Functions

Authors: F. Pakovich (1)
Author institution:(1) Department of Mathematics, Ben-Gurion University of the Negev, P.O.B. 653 Beer Sheva, 8410501 Israel

Summary:

We give a description of pairs of complex rational functions $A$ and $U$ of degree at least two such that for every $d\geq 1$ the algebraic curve $A^{\circ d}(x)-U(y)=0$ 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 $U\circ V=A^{\circ l}$ for some $l\geq 1$ . We also prove a version of the dynamical Mordell–Lang conjecture, concerning intersections of orbits of points from $\mathbb{P}^1(K)$ under iterates of $A$ with the value set $U(\mathbb{P}^1(K))$ , where $A$ and $U$ are rational functions defined over a number field $K$ .

2010 Math. Subj. Class. Primary: 37F10; Secondary: 37P55, 14G05, 14H45 .

Keywords: Semiconjugate rational functions, dynamical Mordell–Lang conjecture, Riemann surface orbifolds, separated variable curves.

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