Potential theory and dynamics on the Berkovich projective line.

*(English)*Zbl 1196.14002
Mathematical Surveys and Monographs 159. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4924-8/hbk). xxxiii, 428 p. (2010).

The book under review deals with non-Archimedean potential theory on the projective line. In time, it has become clear that the right setting for this theory was that of Berkovich non-Archimedean analytic spaces, and this is indeed the point of view the authors have chosen. They derive applications to dynamics of rational functions and equidistribution problems.

The book is self-contained and pleasant to read. The authors recall everything from the basics of Berkovich spaces and potential theory on finite graphs but still manage to end with sophisticated results on rational maps and equidistribtion. The results are stated and proven in great generality (for example, all of them, except in the last chapter, hold not only for \({\mathbb{C}}_{p}\) or in characteristic 0 but for arbitrary algebraically closed complete nontrivially valued fields). The manuscript is at the same time introductory and thorough and is bound to become a reference in the field.

Let us now go into the details of the book. The first two chapters are devoted to the Berkovich projective line \({\mathbb{P}}^1_{\mathrm{Berk}}\) over an algebraically closed field \(K\) which is complete with respect to a nontrivial non-Archimedean absolute value (the basic example being \({\mathbb{C}}_{p}\)). Its construction is given in great detail with emphasis on the \({\mathbb{R}}\)-tree structure and the metric properties. The authors also give an alternative “Proj” construction of \({\mathbb{P}}^1_{\mathrm{Berk}}\) on which the action of the field of rational maps \(K(T)\) becomes clear.

Chapter 3 to 5 are then devoted to defining a Laplacian operator on the Berkovich projective line. First the construction is carried out for finite metrized graphs. Let \(\Gamma\) be such a graph, \(S\) a finite subset of it and \(f\) a real-valued continuous function on \(\Gamma\) which is affine on each edge of \(\Gamma \setminus S\). Then the Laplacian of \(f\) is defined as the sum of Dirac measures \[ \Delta(f) = - \sum_{p \in S} a_{p} \delta_{p}, \] where \(a_{p}\) equals the sum of the derivatives of \(f\) at the point \(p\) along the outward going unit vectors. The Laplacian can actually be defined for a bigger class of maps, called \(BDV(\Gamma\)), which stands for bounded differential variation. Using the fact that \({\mathbb{P}}^1_{\mathrm{Berk}}\) is a projective limit of finite graphs, the class \(BDV\) and the Laplacian are then extended to the whole projective line (and its domains). In some sense, the class \(BDV\) is the largest possible since any finite signed Borel measure of total mass zero can then be obtained as a Laplacian. Let us mention that there have already been other constructions of the Laplacian by Favre, Jonsson and Rivera-Letelier [see C. Favre, M. Jonsson, The valuative tree. Lecture Notes in Mathematics 1853. Berlin: Springer (2004; Zbl 1064.14024) and C. Favre, J. Rivera-Letelier, Math. Ann. 335, No. 2, 311–361 (2006; Zbl 1175.11029)], as well as A. Thuillier (whose construction works for arbitrary Berkovich curves, see [Théorie du potentiel sur les courbes en géométrie analytique non archimédienne. Applications à la théorie d’Arakelov. PhD thesis, University of Rennes, (2005)]). Those constructions give rise to the same object, up to sign.

In chapter 6, the authors define the capacity of a compact subset of \({\mathbb{P}}^1_{\mathrm{Berk}}\) (a measure of the size of the compact set). They use it to prove a Fekete-Szegö theorem. Let \(k\) be a global field. For each finite (resp. infinite) place \(v\) of \(k\), let \(E_{v}\) be a nonempty compact subset of the Berkovich affine line over \({\mathbb{C}}_{v}\) (resp. a compact subset of \({\mathbb{C}}\)) stable under Galois action. Assume that the family \({\mathbb{E}}=(E_{v})_{v}\) is adelic: except a finite number of them, all the \(E_{v}\)’s are the closed unit disc (in particular, they have capacity 1). Let \(\gamma\) be the product of the logarithmic capacities of the \(E_{v}\)’s. If \(\gamma<1\), then there is a Berkovich adelic neighborhood \({\mathbb{U}}\) of \({\mathbb{E}}\) such that the set of points of \({\mathbb{P}}^1(\bar{k})\), all of whose conjugates are contained in \({\mathbb{U}}\), is finite. If \(\gamma>1\), then for every Berkovich adelic neighborhood \({\mathbb{U}}\) of \({\mathbb{E}}\) such that the set of points of \({\mathbb{P}}^{1}(\bar{k})\), all of whose conjugates are contained in \({\mathbb{U}}\), is infinite. This theorem generalizes the one of R. S. Rumely [Capacity theory on algebraic curves. Lecture Notes in Mathematics, 1378. Berlin etc.: Springer-Verlag. (1989; Zbl 0679.14012)] and its proof actually relies heavily on it.

Chapter 7 and 8 are devoted to the study of harmonic and suharmonic functions. Many of the classical results from the complex theory are extended to \({\mathbb{P}}^1_{\mathrm{Berk}}\): let us mention the maximum principle (a subharmonic function on a domain \(U\) that attains its maximum value in \(U\) is constant), the Poisson formula (a harmonic function on a nice enough domain extends to the boundary and its values can be explicitely computed from its values on this boundary) and Harnack’s principle (over a domain, an increasing limit of harmonic function if either harmonic or identically \(+\infty\)). Such results (for arbitrary Berkovich curves) are also to be found in Thuillier’s thesis.

In chapter 9, the authors define and study multiplicities of rational maps. This could be done algebraically using rank of modules, but they choose a different definition through the Laplacian of a proximity function. They prove several results about those multiplicities from which they manage to derive interesting consequences concerning the geometry of nonconstant rational maps: they define open and surjective maps \({\mathbb{P}}^1_{\mathrm{Berk}} \to {\mathbb{P}}^1_{\mathrm{Berk}}\), they send a disc to either a disc or \({\mathbb{P}}^1_{\mathrm{Berk}}\), etc. Those results had already been obtained by J. Rivera-Letelier, although with different proofs [see Astérisque 287, 147–230 (2003; Zbl 1140.37336) and Comment. Math. Helv. 80, No. 3, 593–629 (2005; Zbl 1140.37337)].

The last chapter is devoted to applications of the theory to the dynamics of rational functions. Let \(\varphi\) be a rational function of degree \(d \geq 2\). The authors first show that there exists a probability measure \(\mu_{\varphi}\) on \({\mathbb{P}}^1_{\mathrm{Berk}}\), called the canonical measure, satisfying \(\varphi_{*}(\mu_{\varphi})=\mu_{\varphi}\) and \(\varphi^*(\mu_{\varphi})=d\mu_{\varphi}\). This enables them to state and prove a theorem of adelic equidistribution of small points. Let \(k\) be a number field and \(\varphi\) be a rational function of degree \(\geq 2\). Let \((P_{n})_{n}\) be a sequence of distinct points whose canonical heights relative to \(\varphi\) tend to 0. Let \(v\) be a place of \(k\). For any \(n\), let \(\delta_{n}\) be the discrete probability measure on \({\mathbb{P}}^1_{\mathrm{Berk},{\mathbb{C}}_{v}}\) supported equally on the \(\bar{k}/k\)-conjugates of \(P_{n}\). Then the sequence \((\delta_{n})_{n}\) converges weakly to the canonical measure \(\mu_{\varphi,v}\) on \({\mathbb{P}}^1_{\mathrm{Berk},{\mathbb{C}}_{v}}\) associated to \(\varphi\). This result can also be found in [Ann. Inst. Fourier 56, No. 3, 625–688 (2006; Zbl 1234.11082 ); J. Reine Angew. Math. 595, 215–235 (2006; Zbl 1112.14022)] and [Math. Ann. 335, No. 2, 311–361 (2006; Zbl 1175.11029)].

Then the authors prove an equidistribution result for the preimages by a rational map. Let \(K\) be an algebraically closed complete non-Archimedean field with nontrivial valuation. Assume moreover that \(K\) has characteristic 0. Let \(\varphi\) be a rational map of degree \(\geq 2\). For \(y \in {\mathbb{P}}^1_{\mathrm{Berk}}\), let \(\mu^y_{\varphi^{(n)}}\) be the discrete probability measure supported by the \(n^{th}\) preimages of \(y\) counted with multiplicity. Then, there exists a finite subset \(E_{\varphi}\) of \({\mathbb{P}}^1_{\mathrm{Berk}}\) such that, for every point \(y \notin E_{\varphi}\), the sequence \((\mu^y_{\varphi^{(n)}})_{n}\) converges weakly to the canonical measure \(\mu_{\varphi}\). This result is due to C. Favre and J. Rivera-Letelier [Proc. Lond. Math. Soc. (3) 100, No. 1, 116-154 (2010; Zbl 1254.37064)]) and actually holds in any characteristic.

The last sections of the book deal with Fatou and Julia sets for rational maps on \({\mathbb{P}}^1_{\mathrm{Berk}}\) over an algebraically closed complete nontrivially valued non-Archimedean field of characteristic 0. Like in the complex case, one can give several definitions of those sets. These sets are related to fixed points, which the authors also study. In the case where the field is \({\mathbb{C}}_{p}\), J. Rivera-Letelier has been able to give more precise results (see his articles mentioned above as well as [Compos. Math. 138, No. 2, 199–231 (2003; Zbl 1041.37021); Sur la structure des ensembles de Fatou \(p\)-adiques, Preprint (2004; arXiv:math/0412180)], which are summarized at the end of the book.

The book is self-contained and pleasant to read. The authors recall everything from the basics of Berkovich spaces and potential theory on finite graphs but still manage to end with sophisticated results on rational maps and equidistribtion. The results are stated and proven in great generality (for example, all of them, except in the last chapter, hold not only for \({\mathbb{C}}_{p}\) or in characteristic 0 but for arbitrary algebraically closed complete nontrivially valued fields). The manuscript is at the same time introductory and thorough and is bound to become a reference in the field.

Let us now go into the details of the book. The first two chapters are devoted to the Berkovich projective line \({\mathbb{P}}^1_{\mathrm{Berk}}\) over an algebraically closed field \(K\) which is complete with respect to a nontrivial non-Archimedean absolute value (the basic example being \({\mathbb{C}}_{p}\)). Its construction is given in great detail with emphasis on the \({\mathbb{R}}\)-tree structure and the metric properties. The authors also give an alternative “Proj” construction of \({\mathbb{P}}^1_{\mathrm{Berk}}\) on which the action of the field of rational maps \(K(T)\) becomes clear.

Chapter 3 to 5 are then devoted to defining a Laplacian operator on the Berkovich projective line. First the construction is carried out for finite metrized graphs. Let \(\Gamma\) be such a graph, \(S\) a finite subset of it and \(f\) a real-valued continuous function on \(\Gamma\) which is affine on each edge of \(\Gamma \setminus S\). Then the Laplacian of \(f\) is defined as the sum of Dirac measures \[ \Delta(f) = - \sum_{p \in S} a_{p} \delta_{p}, \] where \(a_{p}\) equals the sum of the derivatives of \(f\) at the point \(p\) along the outward going unit vectors. The Laplacian can actually be defined for a bigger class of maps, called \(BDV(\Gamma\)), which stands for bounded differential variation. Using the fact that \({\mathbb{P}}^1_{\mathrm{Berk}}\) is a projective limit of finite graphs, the class \(BDV\) and the Laplacian are then extended to the whole projective line (and its domains). In some sense, the class \(BDV\) is the largest possible since any finite signed Borel measure of total mass zero can then be obtained as a Laplacian. Let us mention that there have already been other constructions of the Laplacian by Favre, Jonsson and Rivera-Letelier [see C. Favre, M. Jonsson, The valuative tree. Lecture Notes in Mathematics 1853. Berlin: Springer (2004; Zbl 1064.14024) and C. Favre, J. Rivera-Letelier, Math. Ann. 335, No. 2, 311–361 (2006; Zbl 1175.11029)], as well as A. Thuillier (whose construction works for arbitrary Berkovich curves, see [Théorie du potentiel sur les courbes en géométrie analytique non archimédienne. Applications à la théorie d’Arakelov. PhD thesis, University of Rennes, (2005)]). Those constructions give rise to the same object, up to sign.

In chapter 6, the authors define the capacity of a compact subset of \({\mathbb{P}}^1_{\mathrm{Berk}}\) (a measure of the size of the compact set). They use it to prove a Fekete-Szegö theorem. Let \(k\) be a global field. For each finite (resp. infinite) place \(v\) of \(k\), let \(E_{v}\) be a nonempty compact subset of the Berkovich affine line over \({\mathbb{C}}_{v}\) (resp. a compact subset of \({\mathbb{C}}\)) stable under Galois action. Assume that the family \({\mathbb{E}}=(E_{v})_{v}\) is adelic: except a finite number of them, all the \(E_{v}\)’s are the closed unit disc (in particular, they have capacity 1). Let \(\gamma\) be the product of the logarithmic capacities of the \(E_{v}\)’s. If \(\gamma<1\), then there is a Berkovich adelic neighborhood \({\mathbb{U}}\) of \({\mathbb{E}}\) such that the set of points of \({\mathbb{P}}^1(\bar{k})\), all of whose conjugates are contained in \({\mathbb{U}}\), is finite. If \(\gamma>1\), then for every Berkovich adelic neighborhood \({\mathbb{U}}\) of \({\mathbb{E}}\) such that the set of points of \({\mathbb{P}}^{1}(\bar{k})\), all of whose conjugates are contained in \({\mathbb{U}}\), is infinite. This theorem generalizes the one of R. S. Rumely [Capacity theory on algebraic curves. Lecture Notes in Mathematics, 1378. Berlin etc.: Springer-Verlag. (1989; Zbl 0679.14012)] and its proof actually relies heavily on it.

Chapter 7 and 8 are devoted to the study of harmonic and suharmonic functions. Many of the classical results from the complex theory are extended to \({\mathbb{P}}^1_{\mathrm{Berk}}\): let us mention the maximum principle (a subharmonic function on a domain \(U\) that attains its maximum value in \(U\) is constant), the Poisson formula (a harmonic function on a nice enough domain extends to the boundary and its values can be explicitely computed from its values on this boundary) and Harnack’s principle (over a domain, an increasing limit of harmonic function if either harmonic or identically \(+\infty\)). Such results (for arbitrary Berkovich curves) are also to be found in Thuillier’s thesis.

In chapter 9, the authors define and study multiplicities of rational maps. This could be done algebraically using rank of modules, but they choose a different definition through the Laplacian of a proximity function. They prove several results about those multiplicities from which they manage to derive interesting consequences concerning the geometry of nonconstant rational maps: they define open and surjective maps \({\mathbb{P}}^1_{\mathrm{Berk}} \to {\mathbb{P}}^1_{\mathrm{Berk}}\), they send a disc to either a disc or \({\mathbb{P}}^1_{\mathrm{Berk}}\), etc. Those results had already been obtained by J. Rivera-Letelier, although with different proofs [see Astérisque 287, 147–230 (2003; Zbl 1140.37336) and Comment. Math. Helv. 80, No. 3, 593–629 (2005; Zbl 1140.37337)].

The last chapter is devoted to applications of the theory to the dynamics of rational functions. Let \(\varphi\) be a rational function of degree \(d \geq 2\). The authors first show that there exists a probability measure \(\mu_{\varphi}\) on \({\mathbb{P}}^1_{\mathrm{Berk}}\), called the canonical measure, satisfying \(\varphi_{*}(\mu_{\varphi})=\mu_{\varphi}\) and \(\varphi^*(\mu_{\varphi})=d\mu_{\varphi}\). This enables them to state and prove a theorem of adelic equidistribution of small points. Let \(k\) be a number field and \(\varphi\) be a rational function of degree \(\geq 2\). Let \((P_{n})_{n}\) be a sequence of distinct points whose canonical heights relative to \(\varphi\) tend to 0. Let \(v\) be a place of \(k\). For any \(n\), let \(\delta_{n}\) be the discrete probability measure on \({\mathbb{P}}^1_{\mathrm{Berk},{\mathbb{C}}_{v}}\) supported equally on the \(\bar{k}/k\)-conjugates of \(P_{n}\). Then the sequence \((\delta_{n})_{n}\) converges weakly to the canonical measure \(\mu_{\varphi,v}\) on \({\mathbb{P}}^1_{\mathrm{Berk},{\mathbb{C}}_{v}}\) associated to \(\varphi\). This result can also be found in [Ann. Inst. Fourier 56, No. 3, 625–688 (2006; Zbl 1234.11082 ); J. Reine Angew. Math. 595, 215–235 (2006; Zbl 1112.14022)] and [Math. Ann. 335, No. 2, 311–361 (2006; Zbl 1175.11029)].

Then the authors prove an equidistribution result for the preimages by a rational map. Let \(K\) be an algebraically closed complete non-Archimedean field with nontrivial valuation. Assume moreover that \(K\) has characteristic 0. Let \(\varphi\) be a rational map of degree \(\geq 2\). For \(y \in {\mathbb{P}}^1_{\mathrm{Berk}}\), let \(\mu^y_{\varphi^{(n)}}\) be the discrete probability measure supported by the \(n^{th}\) preimages of \(y\) counted with multiplicity. Then, there exists a finite subset \(E_{\varphi}\) of \({\mathbb{P}}^1_{\mathrm{Berk}}\) such that, for every point \(y \notin E_{\varphi}\), the sequence \((\mu^y_{\varphi^{(n)}})_{n}\) converges weakly to the canonical measure \(\mu_{\varphi}\). This result is due to C. Favre and J. Rivera-Letelier [Proc. Lond. Math. Soc. (3) 100, No. 1, 116-154 (2010; Zbl 1254.37064)]) and actually holds in any characteristic.

The last sections of the book deal with Fatou and Julia sets for rational maps on \({\mathbb{P}}^1_{\mathrm{Berk}}\) over an algebraically closed complete nontrivially valued non-Archimedean field of characteristic 0. Like in the complex case, one can give several definitions of those sets. These sets are related to fixed points, which the authors also study. In the case where the field is \({\mathbb{C}}_{p}\), J. Rivera-Letelier has been able to give more precise results (see his articles mentioned above as well as [Compos. Math. 138, No. 2, 199–231 (2003; Zbl 1041.37021); Sur la structure des ensembles de Fatou \(p\)-adiques, Preprint (2004; arXiv:math/0412180)], which are summarized at the end of the book.

Reviewer: Jérôme Poineau (Strasbourg)

##### MSC:

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

37-02 | Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory |

14G20 | Local ground fields in algebraic geometry |

14G22 | Rigid analytic geometry |

14G40 | Arithmetic varieties and schemes; Arakelov theory; heights |

37P50 | Dynamical systems on Berkovich spaces |

37P40 | Non-Archimedean Fatou and Julia sets |

31C15 | Potentials and capacities on other spaces |

31C45 | Other generalizations (nonlinear potential theory, etc.) |