In that situation, the reduction graph $R(\mathfrak{X})$ is a metrized graph defined as follows. It has for vertices the irreducible components of the special fiber, with as many edges of length $\log\left|{\pi}\right|^{-1}$ between two vertices as the number of intersection points of the corresponding components. In an neighbourghood of a double point, $\mathfrak{X}$ looks like (i.e., has an étale map to) the scheme with equation $xy=\pi$ in the affine plane $\mathbf{A}^{2}_{K^{\circ}}$.

If one replaces the field $K$ by a finite extension $K^{\prime}$, the base change $\mathfrak{X}\otimes_{K^{\circ}}(K^{\prime})^{\circ}$ may no more be regular. Indeed, $\mathfrak{X}\otimes_{K^{\circ}}(K^{\prime})^{\circ}$ is étale locally isomorphic to $xy=(\pi^{\prime})^{e}$, where $\pi^{\prime}$ is a uniformizing element of $K^{\prime}$, and $e$ is the ramification index. When $e>1$, the origin is a singular point of that scheme and one needs to blow it up repeatedly in order to obtain a regular scheme, which is a semi-stable model of $\mathrm{X}_{K^{\prime}}$ over $(K^{\prime})^{\circ}$. The two initial components are replaced by a chain of $e+1$ components, the $e-1$ intermediate ones being projective lines. In other words, $e-1$ vertices have been added, regularly spaced along each edge. One concludes that the reduction graph has not changed, as a topological space. Its metric has not changed neither, since the $e$ edges that partition an original edge (of length $\log\left|{\pi}\right|^{-1}$) have length $\log\left|{\pi^{\prime}}\right|^{-1}=\frac{1}{e}\log\left|{\pi}\right|^{-1}$.

We say that a function on $R(\mathfrak{X})$ is piecewise linear if, up to passing to a finite extension (which replaces each edge by $e$ edges of length equal to $1/e$th of the initial one), it is linear on each edge.

Let us analyse the situation from the Berkovich viewpoint. As we have seen, the generic points of the special fiber are the reductions of canonical points of $\mathrm{X}$ : the vertices of the graph $R(\mathfrak{X})$ naturally live in $\mathrm{X}$. The same holds for the edges, but is a bit more subtle. As we have seen, blowing-up intersection points of components in the special fiber gives rise to new components, hence to new points of $\mathrm{X}$. Would we enlarge the ground field and blow-up indefinitely, the constellation of points in $\mathrm{X}$ that we draw converges to a graph which is isomorphic to $R(\mathfrak{X})$.

According to Berkovich [12], a far more precise result holds. Let us consider a neighborhood $\mathfrak{U}$ of a singular point of the special fiber, pretending it is isomorphic to the locus defined by the equation $xy-\pi$ in $\mathbf{A}^{2}$ ; so $\mathfrak{U}=\operatorname{Spec}(K^{\circ}[x,y]/(xy-\pi))$. Its generic fibre is the affinoid space $\mathrm{U}$ defined by the inequality $\left|{xy}\right|=\left|{\pi}\right|$ in the unit polydisk $\mathrm{B}^{2}=\mathscr{M}(K\langle x,y\rangle$. The affinoid algebra of $\mathrm{U}$ is the quotient

whose elements $f$ are (non-uniquely) represented by a series

with $a_{m,n}\rightarrow 0$ when $m+n\rightarrow\infty$. However, observing that $x$ is invertible in this algebra, with inverse $\pi^{-1}y$, so that $y=\pi x^{-1}$, we can replace each product $x^{m}y^{n}$ by $\pi^{n}x^{m-n}$, leading to an expression of the form

where $\left|{a_{n}}\right|\rightarrow 0$ when $n\rightarrow+\infty$ and $\left|{a_{n}}\right|\pi^{-n}\rightarrow 0$ when $n\rightarrow-\infty$. Such an expression is now unique, and is called the Laurent expansion of $f$.

It leads to a natural family $(\gamma_{r})_{r\in[0,\log\left|{\pi}\right|^{-1}]}$ of multiplicative seminorms on the algebra $\mathscr{O}(\mathrm{U})$, parametrized by the unit interval in ${\mathbf{R}}$. Namely, for each real number $r\in[0,\log\left|{\pi}\right|^{-1}]$, we can set

Obviously, $\gamma_{r}$ is a norm on $\mathscr{O}(\mathrm{U})$ which extends the absolute value of $K$ ; its multiplicativity is proved analogously that of the Gauß norm. It is easy to check that the map $[0,\log\left|{\pi}\right|^{-1}]\rightarrow\mathrm{U}$ defined by $r\mapsto\gamma_{r}$ is continuous (this amounts to the fact that the maps $r\mapsto\gamma_{r}(f)$ are continuous), hence defines an parametrized path in the topological space $\mathrm{U}$.

Let $S(\mathfrak{U})$ be its image (with the induced distance) ; Berkovich calls it the skeleton of the formal scheme obtained by completing $\mathfrak{U}$ along its special fibre. A point $u$ in $\mathrm{U}$ has two coordinates $(x(u),y(u))$ in the completed residue field $\mathscr{H}(u)$ which are elements of absolute value $\leq 1$ satisfying $x(u)y(u)=\pi$. In particular,

The map $\rho\colon u\mapsto\gamma_{r(u)}$ is a continuous from $\mathrm{U}$ to $S(\mathfrak{U})$.

Let us compute the image of $\gamma_{r}$ by this map. By definition of $\gamma_{r}$, one has

hence $r(\gamma_{r})=r$ and $\rho(\gamma_{r})=\gamma_{r}$. In other words, the map $\rho$ is a retraction of $\mathrm{U}$ onto the skeleton $S(\mathfrak{U})$.

The special fiber of $\mathfrak{U}$ is defined by the equation $xy=0$ in $\mathbf{A}^{2}_{\tilde{K}}$, hence has two components. One can check that the point $\gamma_{0}$ reduces to the generic point of the component with equation $y=0$, while $\gamma_{\log\left|{\pi}\right|^{-1}}$ reduces to the generic point of the component with equation $x=0$.

These constructions have to be done around each singular point of the special fiber of $\mathfrak{X}$, locally for the étale topology of $\mathfrak{X}$. Berkovich proves that they can be glued, so that the graph $R(\mathfrak{X})$ is again canonically interpreted as an actual metrized graph drawn on the analytic space $\mathrm{X}$ ; we write $\iota\colon R(\mathfrak{X})\hookrightarrow\mathrm{X}$ for the canonical embedding. The map $\iota$ admits a continuous retraction $\rho\colon\mathrm{X}\rightarrow R(\mathfrak{X})$.

Although we will not use this fact, we must mention that the retraction $\rho$ is a deformation retraction. (For any $t\in[0,1]$ and any $x\in\mathrm{X}$, $\rho_{t}(x)$ is the semi-norm $x^{1-t}\rho(x)^{t}$.)

A construction of S. Zhang [57], building on prior results of Chinburg–Rumely [20], furnishes continuous metrics on divisors from continuous functions on the reduction graph $R(\mathfrak{X})$. It works as follows. First of all, if $P\in\mathrm{X}(K)$ is a rational point, there is a unique morphism $\varepsilon_{P}\colon\operatorname{Spec}K^{\circ}\rightarrow\mathfrak{X}$ which extends the point $P$ viewed as a morphism from $\operatorname{Spec}K$ to $X$. The image of this section is a divisor $D_{P}$ on $\mathfrak{X}$ and the line bundle $\mathscr{O}(D_{P})$ on $\mathfrak{X}$ defines a smooth metric on $\mathscr{O}(P)$ ; we write $\overline{\mathscr{O}(P)}_{\mathfrak{X}}$ for the corresponding metrized line bundle. We also define $\mu_{P}$ as the Dirac measure at the vertex of the graph corresponding to the (unique) irreducible component of the special fiber by which $D_{P}$ passes through. The construction and the notation is extended by additivity for divisors which are sums of rational points. More generally, if $P$ is only a closed point of $X$, we do this construction after the finite extension $K(P)/K$, so that $P$ becomes a sum of rational points, using for model the minimal resolution of $\mathfrak{X}\otimes K(P)^{\circ}$ described earlier.

If $f$ is any continuous function on $R(\mathfrak{X})$ and $D$ a divisor on $\mathrm{X}$, the metrized line bundle $\mathscr{O}(D+f)_{\mathfrak{X}}$ is deduced from $\overline{\mathscr{O}(D)}_{\mathfrak{X}}$ by multiplying the metric by $e^{-f}$. When $f$ is piecewise linear, this metrized line bundle is smooth. To prove that, we may extend the scalars and assume that $D$ is a sum of rational points $\sum n_{j}P_{j}$ and that $f$ is linear on each edge corresponding to an intersection point of components of the special fiber. Letting $(V_{i})$ being the family of these components, and writing $v_{i}$ for the vertex of $R(\mathfrak{X})$ corresponding to $V_{i}$, the divisor

defines the metrized line bundle $\mathscr{O}(D+f)_{\mathfrak{X}}$.

In this context, Zhang has defined a curvature operator, which associates to a metrized line bundle a distribution on the graph $R(\mathfrak{X})$, defined in such a way that

• —

  for any divisor $D$ on $\mathrm{X}$, ${\operatorname{curv}}(\overline{\mathscr{O}(D)}_{\mathfrak{X}})=\mu_{D}$ ;

• —

  for any continuous function $f$, ${\operatorname{curv}}(\overline{O}(f)_{\mathfrak{X}})=-\Delta f$, where $\Delta$ is the Laplacian operator of the graph $R(\mathfrak{X})$,

and depending linearly on the metrized line bundle. The following lemma compares this construction with the general one on Berkovich spaces.

In other words, the measure $c_{1}(\overline{L})$ is supported by the graph $R(\mathfrak{X})$ where it coincides essentially with Zhang’s curvature.

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

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 $f$.

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.

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 ?

Let $L$ be a line bundle on $\mathrm{X}$. Let $0$ be the neutral element of $\mathrm{X}$ and let us fix a trivialization $L_{0}$ of $L$ at $0$.

The line bundle $L^{\otimes 2}$ is canonically decomposed as the tensor product of an even and an odd line bundle :

By the theorem of the cube, an even line bundle $L$ satisfies $[m]^{*}L\simeq L^{\otimes m^{2}}$, while for an odd line bundle $L$, one has $[m]^{*}L\simeq L^{\otimes m}$ ; moreover, there are in each case a unique isomorphism compatible with the trivialization at the origin. By Lemma 2.4.1, an even (resp. an odd) line bundle possesses a canonical continuous metric making this isomorphism an isometry. This furnishes a canonical metric on $L^{\otimes 2}$, hence on $L$. According to this lemma, this metric is semi-positive if $L$ is ample and even. Using a Lemma of Künnemann, ([17], Lemme 2.3), one proves that this also holds if $L$ is algebraically equivalent to $0$. In any case, the canonical metrics are admissible.

When the variety $\mathrm{X}$ has good reduction, the canonical metrics and the associated measures are fairly easy to describe. Indeed, let $\mathfrak{X}$ be the Néron model of $\mathrm{X}$ over $K^{\circ}$, an Abelian scheme. For any line bundle $L$ on $\mathrm{X}$ there is a unique line bundle $\mathfrak{L}$ on $\mathfrak{X}$ which extends $L$ and which admits a trivialization at the $0$ section extending the given one over $K$. By the theorem of the cube for the Abelian scheme $\mathfrak{X}$, the isomorphism $[m]^{*}L\simeq L^{\otimes m^{a}}$ (with $a=1$ or $2$, according to whether $L$ is odd or even) extends uniquely to an isomorphism $[m]^{*}\mathfrak{L}\simeq\mathfrak{L}^{\otimes m^{a}}$. This implies that the canonical metrics are algebraic, induced by these models.

The description of the canonical measures on $\mathfrak{X}$ follows at once. Let $\xi$ be the point of $\mathrm{X}$ whose reduction is the generic point of the special fiber of $\mathfrak{X}$. Then, for any family $(L_{1},\dots,L_{n})$ of line bundles on $\mathrm{X}$, one has

We see in particular that they only depend on the classes of the line bundles $L_{j}$ modulo numerical equivalence.

In the case of bad reduction, the description of the canonical measures has been established by W. Gubler [39].

Up to replacing $K$ by a finite extension, we assume that $\mathrm{X}$ has split semi-stable reduction. Raynaud’s uniformization involves an analytic group $\mathrm{E}$ which is an extension of an abelian variety with good reduction $\mathrm{Y}$ by a split torus $\mathrm{T}\simeq{\mathbf{G}_{\mathrm{m}}}^{t}$, where $t\in\{1,\dots,n\}$ — the so-called Raynaud extension of $\mathrm{X}$. One has $t\geq 1$ since we assume bad reduction ; moreover, $\operatorname{dim}\mathrm{Y}=\operatorname{dim}\mathrm{E}-t=n-t$. There is a morphism $p\colon\mathrm{E}\rightarrow\mathrm{X}$, whose kernel is a discrete subgroup $M$ of $\mathrm{E}(K)$, so that the induced map $\mathrm{E}/\Lambda\rightarrow\mathrm{X}$ is an isomorphism. When $t=n$, one says that $\mathrm{X}$ has totally degenerate reduction, and the morphism $p$ is the rigid analytic uniformization of the abelian variety $\mathrm{X}$.

Moreover, $\mathrm{E}$ is constructed as a contracted product $(\mathrm{E}_{1}\times\mathrm{T})/\mathrm{T}_{1}$ from an extension $\mathrm{E}_{1}$ of $\mathrm{Y}$ by the “unit subtorus” $\mathrm{T}_{1}$ of $\mathrm{T}$ (defined by the equalities $\left|{T_{j}(x)}\right|=1$ for $j\in\{1,\dots,t\}$ and $x\in\mathrm{T}$). The natural map $\lambda_{\mathrm{T}}\colon\mathrm{T}\rightarrow{\mathbf{R}}^{t}$ defined by

is continuous and surjective ; it admits a canonical section $\iota_{\mathrm{T}}$ which maps a point $(u_{1},\dots,u_{t})\in{\mathbf{R}}^{t}$ to the semi-norm

The map $\lambda_{\mathrm{T}}$ extends uniquely to a morphism $\lambda\colon\mathrm{E}\rightarrow{\mathbf{R}}^{t}$ whose kernel contains $\mathrm{E}_{1}$. The image $\Lambda=\lambda(M)$ is a lattice of ${\mathbf{R}}^{t}$, and the morphism $p$ induces a continuous proper morphism $\rho\colon\mathrm{X}\rightarrow{\mathbf{R}}^{t}/\Lambda$. Composing the section $\iota_{\mathrm{T}}$ with the projection $p$ furnishes a section $\iota\colon{\mathbf{R}}^{t}/\Lambda\rightarrow\mathrm{X}$ of $\rho$. Its image is the skeleton of $\mathrm{X}$. Gubler’s theorem ([39], Cor. 7.3) is the following :