Powers in Orbits of Rational Functions: Cases of an Arithmetic Dynamical Mordell–Lang Conjecture
Let $K$ be a finitely generated field of characteristic zero. For fixed $m\geqslant 2$ , we study the rational functions $\unicode[STIX]{x1D719}$ defined over $K$ that have a $K$ -orbit containing infinitely many distinct $m$ -th powers. For $m\geqslant 5$ we show that the only such functions are th...
Uloženo v:
| Vydáno v: | Canadian journal of mathematics Ročník 71; číslo 4; s. 773 - 817 |
|---|---|
| Hlavní autoři: | , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Canada
Canadian Mathematical Society
01.08.2019
Cambridge University Press |
| Témata: | |
| ISSN: | 0008-414X, 1496-4279 |
| On-line přístup: | Získat plný text |
| Tagy: |
Přidat tag
Žádné tagy, Buďte první, kdo vytvoří štítek k tomuto záznamu!
|
| Shrnutí: | Let
$K$
be a finitely generated field of characteristic zero. For fixed
$m\geqslant 2$
, we study the rational functions
$\unicode[STIX]{x1D719}$
defined over
$K$
that have a
$K$
-orbit containing infinitely many distinct
$m$
-th powers. For
$m\geqslant 5$
we show that the only such functions are those of the form
$cx^{j}(\unicode[STIX]{x1D713}(x))^{m}$
with
$\unicode[STIX]{x1D713}\in K(x)$
, and for
$m\leqslant 4$
we show that the only additional cases are certain Lattès 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
$\{n\geqslant 0:\unicode[STIX]{x1D719}^{n}(a)\in \unicode[STIX]{x1D706}(\mathbb{P}^{1}(K))\}$
is a union of finitely many arithmetic progressions, where
$\unicode[STIX]{x1D719}^{n}$
denotes the
$n$
-th iterate of
$\unicode[STIX]{x1D719}$
and
$\unicode[STIX]{x1D706}\in K(x)$
is any map Möbius-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
$y^{m}=\unicode[STIX]{x1D719}^{n}(x)$
. We describe all
$\unicode[STIX]{x1D719}$
for which these curves have an irreducible component of genus at most 1, and show that such
$\unicode[STIX]{x1D719}$
must have two distinct iterates that are equal in
$K(x)^{\ast }/K(x)^{\ast m}$
. |
|---|---|
| Bibliografie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0008-414X 1496-4279 |
| DOI: | 10.4153/CJM-2018-026-x |