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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.
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