Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Powers in orbits of rational functions: cases of an arithmetic dynamical
  Mordell-Lang conjecture   Let $K$ be a finitely generated field of characteristic zero. We study, for
fixed $m \geq 2$, the rational functions $\phi$ defined over $K$ that have a
$K$-orbit containing infinitely many distinct $m$th powers. For $m \geq 5$ we
show the only such functions are those of the form $cx^j(\psi(x))^m$ with $\psi
\in K(x)$, and for $m \leq 4$ we show the only additional cases are certain
Latt\`es maps and four families of rational functions whose special properties
appear not to have been studied before.
  With additional analysis, we show that the index set $\{n \geq 0 :
\phi^{n}(a) \in \lambda(\mathbb{P}^1(K))\}$ is a union of finitely many
arithmetic progressions, where $\phi^{n}$ denotes the $n$th iterate of $\phi$
and $\lambda \in K(x)$ is any map M\"obius-conjugate over $K$ to $x^m$. When
the index set is infinite, we give bounds on the number and moduli of the
arithmetic progressions involved. These results are similar in flavor to the
dynamical Mordell-Lang conjecture, and motivate a new conjecture on the
intersection of an orbit with the value set of a morphism. A key ingredient in
our proofs is a study of the curves $y^m = \phi^{n}(x)$. We describe all $\phi$
for which these curves have an irreducible component of genus at most 1, and
show that such $\phi$ must have two distinct iterates that are equal in
$K(x)^*/K(x)^{*m}$.

32
The categories of this paper are: atom-ph
0
The categories of this paper are: hep-ph
12
The categories of this paper are: nucl-th
14
The categories of this paper are: math-ph
20
The categories of this paper are: acc-phys
8
The categories of this paper are: hep-ex
7
The categories of this paper are: hep-th
34
The categories of this paper are: chem-ph
35
The categories of this paper are: mtrl-th
10
The categories of this paper are: q-bio
6
The categories of this paper are: astro-ph
3
The categories of this paper are: physics
19
The categories of this paper are: eess
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The categories of this paper are: alg-geom
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The categories of this paper are: stat
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The categories of this paper are: ao-sci