34 research outputs found
Invariant curves for endomorphisms of
Let be rational functions of degree at least two
that are neither Latt\`es maps nor conjugate to or We
describe invariant, periodic, and preperiodic algebraic curves for
endomorphisms of of the form In particular, we show that if is not
a ``generalized Latt\`es map'', then any -invariant curve has genus zero
and can be parametrized by rational functions commuting with . As an
application, for defined over a subfield of we give a
criterion for a point of to have a Zariski dense -orbit in terms of canonical heights, and deduce from this criterion a
version of a conjecture of Zhang on the existence of rational points with
Zariski dense forward orbits. We also prove a result about functional
decompositions of iterates of rational functions, which implies in particular
that there exist at most finitely many -invariant curves of any
given bi-degree Comment: A polished and extended version, containing a proof of the Zhang
conjecture for endomorphisms of $\mathbb P^1\times \mathbb P^1.
On algebraic curves A(x)-B(y)=0 of genus zero
Using a geometric approach involving Riemann surface orbifolds, we provide
lower bounds for the genus of an irreducible algebraic curve of the form
, where . We also investigate
"series" of curves of genus zero, where by a series we mean a family
with the "same" . We show that for a given rational function a sequence
of rational functions , such that and
all the curves are irreducible and have genus zero, exists if
and only if the Galois closure of the field extension has genus zero or one.Comment: published versio
Polynomial semiconjugacies, decompositions of iterations, and invariant curves
We study the functional equation , where , and
are polynomials over . Using previous results of the author about
polynomials sharing preimages of compact sets, we show that for given its
solutions may be described in terms of the filled-in Julia set of . On this
base, we prove a number of results describing a general structure of solutions.
The results obtained imply in particular the result of Medvedev and Scanlon
about invariant curves of maps of the
form , where is a polynomial, and a version
of the result of Zieve and M\"uller about decompositions of iterations of a
polynomial.Comment: The final version accepted by Ann. Sc. Norm. Super. Pisa Cl. Sc
On polynomials orthogonal to all powers of a given polynomial on a segment
In this paper we investigate the following "polynomial moment problem": for a
complex polynomial and distinct complex numbers to describe
polynomials orthogonal to all integer non-negative powers of on
the segment Comment: 27 pages, 9 figure