Nonarchimedean Green functions and dynamics on projective space
Tóm tắt
Let
$${\varphi: \mathbb{P}^N_K\to\mathbb{P}^N_K}$$
be a morphism of degree d ≥ 2 defined over a field K that is algebraically closed field and complete with respect to a nonarchimedean absolute value. We prove that a modified Green function
$${\hat{g}_\varphi}$$
associated to
$${\varphi}$$
is Hölder continuous on
$${\mathbb{P}^N(K)}$$
and that the Fatou set
$${\mathcal{F}(\varphi)}$$
of
$${\varphi}$$
is equal to the set of points at which
$${\hat{g}_\Phi}$$
is locally constant. Further,
$${\hat{g}_\varphi}$$
vanishes precisely on the set of points P such that
$${\varphi}$$
has good reduction at every point in the forward orbit
$${\mathcal{O}_\varphi(P)}$$
of P. We also prove that the iterates of
$${\varphi}$$
are locally uniformly Lipschitz on
$${\mathcal{F}(\varphi)}$$
.
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