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Variation of periods modulo $$p$$ in arithmetic dynamics. (English) Zbl 1153.11028
Let $$V$$ be a quasi-projective variety over a number field $$K$$, and let $$\varphi: V \to V$$ be a $$K$$-morphism. We fix a $$K$$-rational point $$P$$ on $$V$$, and assume that $$O_{\varphi}(P) := \{ \varphi^n(P)\mid n \in \mathbb{N} \}$$ is an infinite set. If $$\varphi$$ has good reduction at a finite place $$\mathfrak{p}$$ of $$K$$, then we set $$m_{\mathfrak{p}}(\varphi, P) := \# \{ \widetilde{Q} \mid Q \in O_{\varphi}(P) \}$$, where $$\widetilde{Q}$$ is the reduction of $$P$$ at $$\mathfrak{p}$$. Otherwise, we set $$m_{\mathfrak{p}}(\varphi, P) := \infty$$. The author proves that, roughly speaking, $$m_{\mathfrak{p}}\varphi, P)$$ is almost as large as $$\log N_{K/\mathbb{Q}} \mathfrak{p}$$ for most $$\mathfrak{p}$$. To state precise statements, given a set $$\mathcal{P}$$ of finite places of $$K$$, we define $$\delta(\mathcal{P})$$ (resp. $$\underline{\delta}(\mathcal{P})$$) to be the limit (resp. liminf) of $$(d \log \zeta_K(\mathcal{P}, s))(d \log \zeta_K(s))^{-1}$$ as $$s \to 1+$$, where $$\zeta_K(s)$$ is the Dedekind zeta function of $$K$$ and $$\zeta_K(\mathcal{P}, s) = \prod_{\mathfrak{p} \in \mathcal{P}} (1 - (N_{K/\mathbb{Q}} \mathfrak{p})^{-s})^{-1}$$ is the partial zeta function for $$\mathcal{P}$$. The main results of the paper under review are the following:
(1) For any $$\gamma < 1$$, we have
$\delta\{ \mathfrak{p}\mid m_{\mathfrak{p}}(\varphi, P) \geq (\log N_{K/\mathbb{Q}} \mathfrak{p})^{\gamma} \} = 1.$ (2) There exists $$C=C(K, V,\varphi, P)$$ so that for all $$\varepsilon >0$$
$\underline{\delta}\{ \mathfrak{p}\mid m_{\mathfrak{p}}(\varphi, P) \geq \varepsilon \log N_{K/\mathbb{Q}} \mathfrak{p} \} \geq 1- C \varepsilon.$ The author expects this lower bound is far from strict. Based on experimental and heuristic arguments, he made a conjecture that for any $$\varepsilon >0$$, one should have $\delta\{ \mathfrak{p} ~| ~ m_{\mathfrak{p}}(\varphi, P) \leq N_{K/\mathbb{Q}} \mathfrak{p}^{\frac{N}{2} - \varepsilon} \} =0$ where $$\varphi: \mathbb{P}^N \to \mathbb{P}^N$$ is a morphism of degree $$\geq 2$$ and if $$O_{\varphi}(P)$$ is Zariski dense in $$\mathbb{P}^N$$.

##### MSC:
 11G35 Varieties over global fields 11B37 Recurrences 14G40 Arithmetic varieties and schemes; Arakelov theory; heights 37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
##### Keywords:
Arithmetic dynamical systems; orbit modulo $$p$$
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