2.4. Polarized dynamical systems [01JJ]

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2.4. Polarized dynamical systems

We now explain another example of metrized line bundles : the canonical metrics associated to dynamical system.

Lemma 2.4.1.

Let $\mathrm{X}$ be the analytic space associated to a proper $K$ -scheme and let $f\colon\mathrm{X}\rightarrow\mathrm{X}$ be a finite morphism. Let $L$ be a line bundle on $\mathrm{X}$ , $d$ an integer such that $d\geq 2$ and an isomorphism $\varepsilon\colon f^{*}L\simeq L^{d}$ . The line bundle $L$ possesses a unique continuous metric such that the isomorphism $\varepsilon$ is an isometry. If $L$ is ample, then this metric is semi-positive.

In essence, this result, or at least its proof, goes back to Tate’s construction of the “Néron–Tate” canonical height for abelian varieties. In the slightly different language of local heights and Néron functions, it has been proved by Call–Silverman [16]. In the asserted form, it is due to Zhang [59].

Démonstration.

Let us first prove uniqueness. If $\overline{L}$ and $\overline{L}^{\prime}$ are two metrics on $L$ , let $\varphi$ be the continuous function such that $\left\|{\cdot}\right\|^{\prime}=e^{-\varphi}\left\|{\cdot}\right\|$ . Assuming that $\varepsilon$ is an isometry for these two metrics, one obtains the following equation

\varphi(f(x))=d\varphi(x),

for any $x\in\mathrm{X}$ . Since $\mathrm{X}$ is compact, $\varphi$ is bounded and this equation implies that $\left\|{\varphi}\right\|_{\infty}\leq\frac{1}{d}\left\|{\varphi}\right\|_{\infty}$ . Since $d\geq 2$ , one concludes that $\varphi\equiv 0$ .

For the existence, one begins with any continuous metric $\overline{L}_{0}$ on $L$ . Let us then consider the sequence of metrics $(\overline{L}_{n})$ on $L$ induced by the pull-backs on $L^{d}=\varepsilon f^{*}L$ , $L^{d^{2}}=(\varepsilon f^{*})^{2}L$ , etc., hence on $L$ . Since $d\geq 2$ , a similar contraction argument as the one used for uniqueness shows that this is a Cauchy sequence of metrics on $L$ ; consequently, it converges to a continuous metric on $L$ . If $\overline{L}_{0}$ is chosen to be semi-positive, which we may if $L$ is ample, then all al of the metrized line bundles $\overline{L}_{n}$ are semi-positive, hence the canonical metric is semi-positive.

Concretely, in the non-archimedean case, one begins with a model $(\mathfrak{X}_{0},\mathfrak{L}_{0},e)$ such that $\mathfrak{L}_{0}$ is numerically effective. Then one considers the map $f\colon X\rightarrow\mathfrak{X}_{0}$ and the normalization $\mathfrak{X}_{1}$ of $\mathfrak{X}_{0}$ in $X$ ; this is a projective model $\mathfrak{X}_{1}$ , equiped with a finite morphism $f_{1}\colon\mathfrak{X}_{1}\rightarrow\mathfrak{X}_{0}$ extending $f$ . Moreover, $\mathfrak{L}_{1}=f_{1}^{*}\mathfrak{L}_{0}$ is a model of $f^{*}L^{e}$ which is identified with $L^{ed}$ via the fixed isomorphism $\varepsilon$ . Iterating this construction defines a sequence $(\mathfrak{X}_{n},\mathfrak{L}_{n},ed^{n})$ of models of $(X,L)$ , with finite morphisms $f_{n}\colon\mathfrak{X}_{n}\rightarrow\mathfrak{X}_{n-1}$ such that $f_{n}^{*}\mathfrak{L}_{n-1}=\mathfrak{L}_{n}$ . The metric on $L$ defined by any of these models is semi-positive, hence so is their uniform limit. ∎

The canonical measure

The measure $c_{1}(\overline{L})^{n}$ on $\mathrm{X}$ defined by the metrized line bundle $\overline{L}$ is a very important invariant of the dynamical system. It satisfies the functional equations

f^{*}c_{1}(\overline{L})^{n}=d^{n}c_{1}(\overline{L})^{n}\quad\text{and}\quad f_{*}c_{1}(\overline{L})^{n}=c_{1}(\overline{L})^{n}.

The first follows by a general functorial property proved in [18] ; it implies the second. The support of the canonical measure is therefore totally invariant under $f$ .

The Fatou set

Generalizing results of Kawaguchi–Silverman in [42] and Baker–Rumely [6], we want to show here that the canonical measure vanishes on any open set $\mathrm{U}$ of $\mathrm{X}$ where the sequence $(f^{n}|_{\mathrm{U}})$ of iterates of $f$ is equicontinuous.

Let $\mathrm{U}$ be an open set in $\mathrm{X}$ and $\mathscr{F}$ be a family of continuous maps from $\mathrm{U}$ to $\mathrm{X}$ . One says that this family is equicontinuous if for any $x\in\mathrm{U}$ and any finite covering $(\mathrm{V}_{j})$ of $\mathrm{X}$ by affinoid spaces, there exists a neighbourhood $\mathrm{U}_{x}$ of $x$ in $\mathrm{U}$ such that for any $\varphi\in\mathscr{F}$ , there exists an index $j$ such that $\varphi(\mathrm{U}_{x})\subset\mathrm{V}_{j}$ . (This definition is adapted from Definition 10.63 in [6] ; it is the definition of equicontinuity associated to the canonical uniform structure of the compact space $\mathrm{X}$ .)

We define the equicontinuous locus of $f$ as the largest open subset $\mathrm{E}_{f}$ of $\mathrm{X}$ over which the sequence of iterates of $f$ is equicontinuous.

Proposition 2.4.2.

If $L$ is ample, then the metric $\overline{L}$ is strongly pluriharmonic on $\mathrm{E}_{f}$ .³³ 3 The ampleness assumption should not be necessary for the result to hold.

Démonstration.

The proof is inspired from the above-mentioned sources, which in turns is an adaptation of the complex case [41] (see also [52]).

We may replace $L$ by a positive power of itself and assume that it is very ample, induced by a closed embedding of $\mathrm{X}$ in $\mathrm{P}^{n}$ , and that the natural map $\Gamma(\mathrm{{\mathbf{P}}}^{n},\mathscr{O}(d))\rightarrow\Gamma(\mathrm{X},\mathscr{O}(d))$ is surjective. Then, there are homogeneous polynomials $(F_{0},\dots,F_{n})$ , of degree $d$ , with coefficients in $K$ , and without common zeroes on $\mathrm{X}$ , such that $f([x_{0}:\dots:x_{n}])=[F_{0}(x):\dots:F_{n}(x)]$ for any $x=[x_{0}:\dots:x_{n}]\in\mathrm{P}^{n}$ . One considers the polynomial map $F\colon\mathrm{A}^{n+1}\rightarrow\mathrm{A}^{n+1}$ ; it lifts a rational map on $\mathrm{P}^{n}$ which extends the morphism $f$ .

For $(x_{0},\dots,x_{n})\in\mathrm{A}^{n+1}$ , define $\left\|{x}\right\|=\max(\left|{x_{0}}\right|,\dots,\left|{x_{n}}\right|)$ . The Weil metric on $\mathscr{O}(1)$ is given by

\log\left\|{s_{P}(x)}\right\|^{-1}=\log\left|{P(x)}\right|^{-1}+\deg(P)\log\left\|{x}\right\|,

where $P$ is an homogeneous polynomial, $s_{P}$ the corresponding global section of $\mathscr{O}(\deg(P))$ , and $x$ is a point of $\mathrm{A}^{n+1}$ such that $P(x)\neq 0$ . The restriction to $\mathrm{X}$ of this metric is a semi-positive metric $\left\|{\cdot}\right\|_{0}$ on $L$ . The construction of the canonical metric on $L$ introduces a sequence of semi-positive metrics $\left\|{\cdot}\right\|_{n}$ on $L$ ; these metrics are given by the following explicit formula

\log\left\|{s_{P}(x)}\right\|_{k}^{-1}=\log\left|{P(x)}\right|^{-1}+\deg(P)d^{-k}\log\left\|{F^{(k)}(x)}\right\|,

where $F^{(k)}:\mathrm{A}^{n+1}\rightarrow\mathrm{A}^{n+1}$ is the $k$ th iterate of $F$ .

The convergence of this sequence is therefore equivalent to the convergence of the sequence $(d^{-k}\log\left\|{F^{(k)}}\right\|)_{k}$ towards a continuous fonction on the preimage of $\mathrm{X}$ under the projection map $\mathrm{A}^{n+1}\setminus\{0\}\rightarrow\mathrm{P}^{n}$ . The limit is usually called the homogeneous Green function.

For $0\leq i\leq n$ , let $\mathrm{V}_{i}$ be the open set of points $x=[x_{0}:\dots:x_{n}]\in\mathrm{P}^{n}$ such that $\left|{x_{i}}\right|>\frac{1}{2}\left\|{x}\right\|$ . They form an open covering of $\mathrm{P}^{n}$ ; their intersections with $\mathrm{X}$ form an open covering of $\mathrm{X}$ .

Fix $x\in\mathrm{E}_{f}$ and let $\mathrm{U}$ be an open neighbourhood of $x$ such that for any positive integer $k$ , there exists $i\in\{0,\dots,n\}$ such that $f^{k}(\mathrm{U})\subset\mathrm{V}_{i}$ . For any $i$ , let $N_{i}$ be the set of integers $k$ such that $f^{k}(\mathrm{U})\subset\mathrm{V}_{i}$ . Let us consider any index $i$ such that $N_{i}$ is infinite ; to fix ideas, let us assume that $i=0$ . The canonical norm of a section $s_{P}$ at a point $y\in\mathrm{U}$ is given by

	$\displaystyle\log\left\\|{s_{P}(y)}\right\\|^{-1}$	$\displaystyle=\log\left\|{P(y)}\right\|^{-1}+\deg(P)\lim_{\begin{subarray}{c}k\rightarrow\infty\\ k\in N_{0}\end{subarray}}d^{-k}\log\left\\|{F^{(k)}(y)}\right\\|$
		$\displaystyle=\log\left\|{P(y)}\right\|^{-1}$
		$\displaystyle\quad+\deg(P)\lim_{\begin{subarray}{c}k\rightarrow\infty\\ k\in N_{0}\end{subarray}}d^{-k}\left(\log\left\|{F_{0}^{(k)}(y)}\right\|+\log\max_{0\leq i\leq m}\left\|{F_{i}^{(k)}(y)/F_{0}^{(k)}(y)}\right\|\right).$

Observe that $[F_{0}^{(k)}(y):\dots:F^{(k)}_{m}(y)]$ are the homogeneous coordinates of the point $f^{k}(y)$ . Since $y\in\mathrm{U}$ and $f^{k}(\mathrm{U})\subset\mathrm{V}_{0}$ , one has $\left|{F^{(k)}_{i}(y)}\right|\leq 2\left|{F^{(k)}_{0}(y)}\right|$ , so that the last term is bounded by $d^{-k}\log 2$ and uniformly converges to $0$ on $\mathrm{U}$ . Finally, uniformly on $\mathrm{U}$ ,

\log\left\|{s_{P}(y)}\right\|^{-1}=\log\left|{P(y)}\right|^{-1}+\deg(P)\lim_{\begin{subarray}{c}k\rightarrow\infty\\ k\in N_{0}\end{subarray}}d^{-k}\log\left|{F_{0}^{(k)}(y)}\right|.

This shows that $\log\left\|{s_{P}}\right\|^{-1}$ is strongly harmonic on $\mathrm{U}$ , as claimed. ∎

Corollary 2.4.3.

The canonical measure $c_{1}(\overline{L})^{n}$ vanishes on $\mathrm{E}_{f}$ .

Démonstration.

It suffices to apply Prop. 2.3.3. ∎

Remarks

1) The particular case $\mathrm{X}=\mathrm{P}^{n}$ generalizes Theorem 6 in [42] according to which canonical metrics are locally constant on the classical Fatou set (meaning that the norm of a non-vanishing local section is locally constant). Indeed, the restriction to the set of smooth rigid points of a strongly harmonic function is locally constant. This follows from the fact that any such point has an affinoid neighbourhood $\mathrm{U}$ which is a polydisk, so that the absolute value of any invertible function on $\mathrm{U}$ , hence any harmonic function on $\mathrm{U}$ is constant.

2) In the case $\mathrm{X}=\mathrm{P}^{1}$ , Fatou and Julia sets in the Berkovich framework have been studied by Rivera-Letelier [48] and Benedetto [9] ; see also [6] for a detailed exposition of the theory and further references. An example of Rivera-Letelier on the projective line (Example 10.70 of [6]) shows that the equicontinuity locus $\mathrm{E}_{f}$ may be smaller than the complement of the support of the measure $c_{1}(\overline{L})$ .

Anyway, this proposition suggests the interest of a general study of Fatou sets and of pluripotential theory on Berkovich spaces. For example, is there an interesting theory of pseudoconvexity for Berkovich spaces ? Is it related to Stein spaces ? By analogy to the complex case (see [52]), are Berkovich Fatou components pseudoconvex ? Stein ?

	$\displaystyle\log\left\\|{s_{P}(y)}\right\\|^{-1}$	$\displaystyle=\log\left\|{P(y)}\right\|^{-1}+\deg(P)\lim_{\begin{subarray}{c}k\rightarrow\infty\\ k\in N_{0}\end{subarray}}d^{-k}\log\left\\|{F^{(k)}(y)}\right\\|$
		$\displaystyle=\log\left\|{P(y)}\right\|^{-1}$
		$\displaystyle\quad+\deg(P)\lim_{\begin{subarray}{c}k\rightarrow\infty\\ k\in N_{0}\end{subarray}}d^{-k}\left(\log\left\|{F_{0}^{(k)}(y)}\right\|+\log\max_{0\leq i\leq m}\left\|{F_{i}^{(k)}(y)/F_{0}^{(k)}(y)}\right\|\right).$