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#### Algebraic curves A^{ol}(x) - U(y) = 0 and arithmetic of orbits of rational functions

##### External Resource

https://doi.org/10.17323/1609-4514-2020-20-1-153-183

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1801.01985.pdf

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##### Citation

Pakovich, F. (2020). Algebraic curves A^{ol}(x) - U(y) = 0 and arithmetic of
orbits of rational functions.* Moscow Mathematical Journal,* *20*(1),
153-183. doi:10.17323/1609-4514-2020-20-1-153-183.

Cite as: https://hdl.handle.net/21.11116/0000-0006-0C2D-C

##### Abstract

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 Latt\`es 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 $K.$