Nonarchimedean Green functions and dynamics on projective space. (English) Zbl 1161.32009
Summary: Let \({\varphi: \mathbb{P}^N_K\to\mathbb{P}^N_K}\) be a morphism of degree \(d \geq 2\) defined over a field \(K\) that is algebraically closed 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 \({\mathcal{F}(\varphi)}\).
MSC:
| 32P05 | Non-Archimedean analysis |
| 11G25 | Varieties over finite and local fields |
| 14G20 | Local ground fields in algebraic geometry |
| 32U35 | Plurisubharmonic extremal functions, pluricomplex Green functions |
| 37F10 | Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets |
| 37F50 | Small divisors, rotation domains and linearization in holomorphic dynamics |
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