Abstract
These are expanded lecture notes for the summer school on Berkovich spaces that took place at the Institut de Mathématiques de Jussieu, Paris, during June 28–July 9, 2010. They serve to illustrate some techniques and results from the dynamics on low-dimensional Berkovich spaces and to exhibit the structure of these spaces.
Keywords
- Berkovich Spaces
- Semivaluations
- Exceptional Prime
- Berkovich Projective Line
- Integral Affine Structure
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptions















Notes
- 1.
For a precise definition of what we mean by “tree”, see Sect. 2.
- 2.
Its one-point compactification is the Berkovich projective line \(\mathop{\mathbf{P}_{\mathrm{Berk}}^{1}}\nolimits =\mathop{ \mathbf{A}_{\mathrm{Berk}}^{1}}\nolimits \cup \{\infty \}\).
- 3.
- 4.
The apex of the cone does not define an element in \(\mathop{\mathbf{A}_{\mathrm{Berk}}^{2}}\nolimits\).
- 5.
- 6.
In [BR10], the local degree is called multiplicity.
- 7.
- 8.
Except for the missing condition (RT3), see Remark 2.5.
- 9.
- 10.
They are sometimes called rigid points as they are the points that show up rigid analytic geometry [BGR84].
- 11.
This differs from the “algebraic degree” used by Trucco, see [Tru09, Definition 5.1].
- 12.
The terminology “tame” follows Trucco [Tru09].
- 13.
- 14.
The analytification of a general variety or scheme over K is defined by gluing the analytifications of open affine subsets, see [Ber90, §3.5].
- 15.
The unit polydisc is denoted by E(0, 1) in [Ber90, §1.5.2].
- 16.
- 17.
This is a divisorial valuation given by the order of vanishing along the exceptional divisor of the blowup of Y, see Sect. 6.10.
- 18.
A local branch is a preimage of a point of C under the normalization map.
- 19.
The acronym “nef” is due to M. Reid who meant it to stand for “numerically eventually free” although many authors refer to it as “numerically effective”.
- 20.
- 21.
- 22.
We shall not, however, actually consider the toric variety defined by the fan \(\hat{\Delta }(\pi )\).
- 23.
The increasing parametrization −α is denoted by α and called skewness in [FJ04]. The increasing parametrization A is called thinness in loc. cit..
- 24.
The notation reflects the fact that \(\vert \cdot \vert:= e^{-v}\) is a seminorm on R, see (30).
- 25.
If \(\varphi \in \mathop{ SH}\nolimits (\mathcal{V}_{0})\), then \(-\varphi\) is a positive tree potential in the sense of [FJ04].
- 26.
The existence of such a relation has been established by W. Gignac and M. Ruggiero in arXiv:1209.3450.
- 27.
- 28.
In [FJ04] the parametrization A is called thinness whereas −α is called skewness.
- 29.
As in Sect. 7.8 the notation reflects the fact that \(\vert \cdot \vert:= e^{-v}\) is a seminorm on R.
- 30.
P. Mondal has given examples in arXiv:1301.3172 showing that equality does not always hold.
References
M. Abate, Discrete holomorphic local dynamical systems, in Holomorphic Dynamical Systems, Lecture Notes in Mathematics, vol 1998 (Springer, 2010), pp. 1–55
S.S. Abhyankar, T.T. Moh, Newton-Puiseux expansion and generalized Tschirnhausen transformation. I, II. J. Reine Angew. Math. 260, 47–83 (1973); 261, 29–53 (1973)
M. Artin, On isolated rational singularities of surfaces. Am. J. Math. 88, 129136 (1966)
M. Baker, L. DeMarco, Preperiodic points and unlikely intersections. Duke Math. J. 159, 1–29 (2011)
I.N. Baker, Fixpoints of polynomials and rational functions. J. Lond. Math. Soc. 39, 615–622 (1964)
M. Baker, A lower bound for average values of dynamical Green’s functions. Math. Res. Lett. 13, 245–257 (2006)
M. Baker, An introduction to Berkovich analytic spaces and non-Archimedean potential theory on curves. In p-adic Geometry, 123–174. Univ. Lecture Ser., vol. 45 (American Mathematical Society, Providence, RI, 2008)
M. Baker, A finiteness theorem for canonical heights attached to rational maps over function fields. J. Reine Angew. Math. 626, 205–233 (2009)
M. Baker, L.-C. Hsia, Canonical heights, transfinite diameters, and polynomial dynamics. J. Reine Angew. Math. 585, 61–92 (2005)
M. Baker, R. Rumely, Equidistribution of small points, rational dynamics, and potential theory. Ann. Inst. Fourier 56, 625–688 (2006)
M. Baker, R. Rumely, Potential Theory on the Berkovich Projective Line. Mathematical Surveys and Monographs, vol. 159 (American Mathematical Society, Providence, RI, 2010)
A.F. Beardon, Iteration of Rational Functions. Graduate Texts in Mathematics, vol. 132 (Springer, New York, 1991)
E. Bedford, B.A. Taylor, A new capacity for plurisubharmonic functions. Acta Math. 149, 1–40 (1982)
E. Bedford, B.A. Taylor, Fine topology, Shilov boundary and \((dd^{c})^{n}\). J. Funct. Anal. 72, 225–251 (1987)
R.L. Benedetto, Fatou components in p-adic dynamics. Ph.D. Thesis. Brown University, 1998. Available at www.cs.amherst.edu/~rlb/papers/
R.L. Benedetto, p-adic dynamics and Sullivan’s no wandering domains theorem. Compositio Math. 122, 281–298 (2000)
R.L. Benedetto, Reduction, dynamics, and Julia sets of rational functions. J. Number Theory 86, 175–195 (2001)
R.L. Benedetto, Hyperbolic maps in p-adic dynamics. Ergodic Theory Dynam. Syst. 21, 1–11 (2001)
R.L. Benedetto, Components and periodic points in non-Archimedean dynamics. Proc. Lond. Math. Soc. 84, 231–256 (2002)
R.L. Benedetto, Examples of wandering domains in p-adic polynomial dynamics. C. R. Math. Acad. Sci. Paris 335, 615–620 (2002)
R.L. Benedetto, Wandering domains and nontrivial reduction in non-Archimedean dynamics. Ill. J. Math. 49, 167–193 (2005)
R.L. Benedetto, Heights and preperiodic points of polynomials over function fields. Int. Math. Res. Not. 62, 3855–3866 (2005)
R.L. Benedetto, Wandering domains in non-Archimedean polynomial dynamics. Bull. Lond. Math. Soc. 38, 937–950 (2006)
R.L. Benedetto, Non-Archimedean dynamics in dimension one. Lecture notes from the 2010 Arizona Winter School, http://math.arizona.edu/~swc/aws/2010/
V.G. Berkovich, Spectral Theory and Analytic Geometry Over Non-Archimedean Fields. Mathematical Surveys and Monographs, vol. 33 (American Mathematical Society, Providence, RI, 1990)
R.L. Benedetto, Étale cohomology for non-Archimedean analytic spaces. Publ. Math. Inst. Hautes Études Sci. 78, 5–161 (1993)
R.L. Benedetto, Vanishing cycles for formal schemes. Invent. Math. 115, 539–571 (1994)
R.L. Benedetto, Smooth p-adic analytic spaces are locally contractible. I. Invent. Math. 137, 1–84 (1999)
R.L. Benedetto, Smooth p-adic analytic spaces are locally contractible. II. Geometric Aspects of Dwork Theory. (Walter de Gruyter and Co. KG, Berlin, 2004), pp. 293–370
R.L. Benedetto, A non-Archimedean interpretation of the weight zero subspaces of limit mixed Hodge structures. Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Progr. Math., vol. 269 (Birkhäuser, Boston, MA, 2009), pp. 49–67
S. Boucksom, T. de Fernex, C. Favre, The volume of an isolated singularity. Duke Math. J. 161(8), 1455–1520 (2012)
S. Boucksom, C. Favre, M. Jonsson, Degree growth of meromorphic surface maps. Duke Math. J. 141, 519–538 (2008)
S. Boucksom, C. Favre, M. Jonsson, Valuations and plurisubharmonic singularities. Publ. Res. Inst. Math. Sci. 44, 449–494 (2008)
S. Boucksom, C. Favre, M. Jonsson, Singular semipositive metrics in non-Archimedean geometry. arXiv:1201.0187. To appear in J. Algebraic Geom.
S. Boucksom, C. Favre, M. Jonsson, Solution to a non-Archimedean Monge-Ampère equation. J. Amer. Math. Soc., electronically published on May 22, 2014
S. Bosch, U. Güntzer, R. Remmert, Non-Archimedean Analysis (Springer, Berlin, Heidelberg, 1994)
J.-Y. Briend, J. Duval, Deux caractérisations de la mesure d’équilibre d’un endomorphisme de \(\mathrm{P}^{k}(\mathbf{C})\). Publ. Math. Inst. Hautes Études Sci. 93, 145–159 (2001)
H. Brolin, Invariant sets under iteration of rational functions. Ark. Mat. 6, 103–144 (1965)
A. Campillo, O. Piltant, A. Reguera, Cones of curves and of line bundles on surfaces associated with curves having one place at infinity. Proc. Lond. Math. Soc. 84, 559–580 (2002)
A. Campillo, O. Piltant, A. Reguera, Cones of curves and of line bundles at infinity. J. Algebra 293, 513–542 (2005)
L. Carleson, T. Gamelin, Complex Dynamics (Springer, New York, 1993)
A. Chambert-Loir, Mesures et équidistribution sur les espaces de Berkovich. J. Reine Angew. Math. 595, 215–235 (2006)
B. Conrad, Several approaches to non-archimedean geometry. In p-adic Geometry (Lectures from the 2007 Arizona Winter School). AMS University Lecture Series, vol. 45 (Amer. Math. Soc., Providence, RI, 2008)
T. Coulbois, A. Hilion, M. Lustig, Non-unique ergodicity, observers’ topology and the dual algebraic lamination for \(\mathbb{R}\)-trees. Ill. J. Math. 51, 897–911 (2007)
J. Diller, R. Dujardin, V. Guedj, Dynamics of meromorphic maps with small topological degree I: from cohomology to currents. Indiana Univ. Math. J. 59, 521–562 (2010)
J. Diller, R. Dujardin, V. Guedj, Dynamics of meromorphic maps with small topological degree II: Energy and invariant measure. Comment. Math. Helv. 86, 277–316 (2011)
J. Diller, R. Dujardin, V. Guedj, Dynamics of meromorphic maps with small topological degree III: geometric currents and ergodic theory. Ann. Sci. École Norm. Sup. 43, 235–278 (2010)
T.-C. Dinh, N. Sibony, Equidistribution towards the Green current for holomorphic maps. Ann. Sci. École Norm. Sup. 41, 307–336 (2008)
L. Ein, R. Lazarsfeld, K.E. Smith, Uniform approximation of Abhyankar valuations in smooth function fields. Am. J. Math. 125, 409–440 (2003)
X. Faber, Equidistribution of dynamically small subvarieties over the function field of a curve. Acta Arith. 137, 345–389 (2009)
X. Faber, Topology and geometry of the Berkovich ramification locus for rational functions. I. Manuscripta Math. 142, 439–474 (2013)
X. Faber, Topology and geometry of the Berkovich ramification locus for rational functions, II. Math. Ann. 356, 819–844 (2013)
X. Faber, Rational functions with a unique critical point. Int. Math. Res. Not. IMRN no. 3, 681–699 (2014)
C. Favre, Arbres réels et espaces de valuations, Thèse d’habilitation, 2005
C. Favre, M. Jonsson, Brolin’s theorem for curves in two complex dimensions. Ann. Inst. Fourier 53, 1461–1501 (2003)
C. Favre, M. Jonsson, The Valuative Tree. Lecture Notes in Mathematics, vol. 1853 (Springer, New York, 2004)
C. Favre, M. Jonsson, Valuative analysis of planar plurisubharmonic functions. Invent. Math. 162(2), 271–311 (2005)
C. Favre, M. Jonsson, Valuations and multiplier ideals. J. Am. Math. Soc. 18, 655–684 (2005)
C. Favre, M. Jonsson, Eigenvaluations. Ann. Sci. École Norm. Sup. 40, 309–349 (2007)
C. Favre, M. Jonsson, Dynamical compactifications of C 2. Ann. Math. 173, 211–249 (2011)
C. Favre, J. Kiwi, E. Trucco, A non-archimedean Montel’s theorem. Compositio 148, 966–990 (2012)
C. Favre, J. Rivera-Letelier, Théorème d’équidistribution de Brolin en dynamique p-adique. C. R. Math. Acad. Sci. Paris 339, 271–276 (2004)
C. Favre, J. Rivera-Letelier, Équidistribution quantitative des points de petite hauteur sur la droite projective. Math. Ann. 335, 311–361 (2006)
C. Favre, J. Rivera-Letelier, Théorie ergodique des fractions rationnelles sur un corps ultramétrique. Proc. Lond. Math. Soc. 100, 116–154 (2010)
A. Freire, A. Lopez, R. Mañé, An invariant measure for rational maps. Bol. Soc. Bras. Mat. 14, 45–62 (1983)
G.B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd edn. Pure and Applied Mathematics (New York). A Wiley-Interscience Publication (Wiley, New York, 1999)
D.H. Fremlin, Real-valued measurable cardinals. In Set Theory of the Reals (Ramat Gan, 1991). Israel Math. Conf. Proc., vol. 6 (Bar-Ilan Univ, Ramat Gan, 1993), pp. 151–304. See also www.essex.ac.uk/maths/people/fremlin/papers.htm
W. Fulton, Introduction to Toric Varieties. Annals of Mathematics Studies, vol. 131 (Princeton University Press, Princeton, NJ, 1993)
C. Galindo, F. Monserrat, On the cone of curves and of line bundles of a rational surface. Int. J. Math. 15, 393–407 (2004)
C. Galindo, F. Monserrat, The cone of curves associated to a plane configuration. Comment. Math. Helv. 80, 75–93 (2005)
D. Ghioca, T.J. Tucker, M.E. Zieve, Linear relations between polynomial orbits. Duke Math. J. 161, 1379–1410 (2012)
É. Ghys, P. de la Harpe, Sur les groupes hyperboliques d’après Mikhael Gromov. Progress in Mathematics, vol. 83 (Birkhäuser, Boston, 1990)
A. Granja, The valuative tree of a two-dimensional regular local ring. Math. Res. Lett. 14, 19–34 (2007)
W. Gubler, Equidistribution over function fields. Manuscripta Math. 127, 485–510 (2008)
R. Hartshorne, Algebraic Geometry. Graduate Texts in Mathematics, vol. 52 (Springer, New York-Heidelberg, 1977)
L.-C. Hsia, Closure of periodic points over a non-Archimedean field. J. Lond. Math. Soc. 62, 685–700 (2000)
R. Hübl, I. Swanson, Discrete valuations centered on local domains. J. Pure Appl. Algebra 161, 145–166 (2001)
S. Izumi, A measure of integrity for local analytic algebras. Publ. RIMS Kyoto Univ. 21, 719–735 (1985)
T. Jech, Set Theory. The third millenium edition, revised and expanded. Springer Monographs in Mathematics. (Springer, Berlin, 2003)
M. Jonsson, M. Mustaţă, Valuations and asymptotic invariants for sequences of ideals. Ann. Inst. Fourier 62, 2145–2209 (2012)
K. Kedlaya, Good formal structures for flat meromorphic connections, I: surfaces. Duke Math. J. 154, 343–418 (2010)
K. Kedlaya, Good formal structures for flat meromorphic connections, II: excellent schemes. J. Am. Math. Soc. 24, 183–229 (2011)
K. Kedlaya, Semistable reduction for overconvergent F-isocrystals, IV: Local semistable reduction at nonmonomial valuations. Compos. Math. 147, 467–523 (2011)
G. Kempf, F.F. Knudsen, D. Mumford, B. Saint-Donat, Toroidal Embeddings. I. Lecture Notes in Mathematics, vol. 339 (Springer, Berlin, 1973)
J. Kiwi, Puiseux series polynomial dynamics and iteration of complex cubic polynomials. Ann. Inst. Fourier 56, 1337–1404 (2006)
J. Kiwi, Puiseux series dynamics of quadratic rational maps. Israel J. Math. 201, 631–700 (2014)
T. Kishimoto, A new proof of a theorem of Ramanujan-Morrow. J. Math. Kyoto 42, 117–139 (2002)
J. Kollár, Singularities of Pairs. Proc. Symp. Pure Math., vol. 62, Part 1 (AMS, Providence, RI, 1997)
J. Kollár, S. Mori, Birational Geometry of Algebraic Varieties. Cambridge Tracts in Mathematics, vol. 134 (Cambridge University Press, Cambridge, 1998)
S. Lang, Algebra. Revised third edition. Graduate Texts in Mathematics, vol. 211 (Springer, New York, 2002)
J. Lipman, Rational singularities with applications to algebraic surfaces and unique factorization. Publ. Math. Inst. Hautes Études Sci. 36, 195–279 (1969)
M. Lyubich, Entropy properties of rational endomorphisms of the Riemann sphere. Ergodic Theory Dynam. Syst. 3, 351–385 (1983)
S. MacLane, A construction for prime ideals as absolute values of an algebraic field. Duke M. J. 2, 363–395 (1936)
H. Matsumura, Commutative Ring Theory. Cambridge Studies in Advanced Mathematics, vol. 8 (Cambridge University Press, Cambridge, 1989)
J. Milnor, Dynamics in One Complex Variable. Annals of Mathematics Studies, vol. 160 (Princeton University Press, Princeton, NJ, 2006)
F. Monserrat, Curves having one place at infinity and linear systems on rational surfaces. J. Pure Appl. Algebra 211, 685–701 (2007)
J.A. Morrow, Minimal normal compactifications of C 2. In Complex Analysis, 1972 (Proc. Conf., Rice Univ. Houston, TX, 1972) Rice Univ. Studies 59, 97–112 (1973)
P. Morton, J.H. Silverman, Periodic points, multiplicities, and dynamical units. J. Reine Angew. Math. 461, 81–122 (1995)
T. Oda, Convex Bodies and Algebraic Geometry. An Introduction to the Theory of Toric Varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 15 (Springer, Berlin, 1988)
Y. Okuyama, Repelling periodic points and logarithmic equidistribution in non-archimedean dynamics. Acta Arith. 152(3), 267–277 (2012)
Y. Okuyama, Feketeness, equidistribution and critical orbits in non-archimedean dynamics. Math. Z (2012). DOI:10.1007/s00209-012-1032-x
M.R. Parra, The Jacobian cocycle and equidistribution towards the Green current. arXiv:1103.4633
P. Popescu-Pampu, Le cerf-volant d’une constellation. Enseign. Math. 57, 303–347 (2011)
C. Petsche, L Szpiro, M. Tepper, Isotriviality is equivalent to potential good reduction for endomorphisms of \(\mathbb{P}^{N}\) over function fields. J. Algebra 322, 3345–3365 (2009)
D. Rees, Izumi’s theorem. In Commutative Algebra (Berkeley, CA, 1987). Math. Sci. Res. Inst. Publ., vol. 15 (Springer, New York 1989), pp. 407–416
J. Rivera-Letelier, Dynamique des fonctions rationnelles sur des corps locaux. Astérisque 287, 147–230 (2003)
J. Rivera-Letelier, Espace hyperbolique p-adique et dynamique des fonctions rationnelles. Compositio Math. 138, 199–231 (2003)
J. Rivera-Letelier, Sur la structure des ensembles de Fatou p-adiques. Available at arXiv:math/0412180
J. Rivera-Letelier, Points périodiques des fonctions rationnelles dans l’espace hyperbolique p-adique. Comment. Math. Helv. 80, 593–629 (2005)
A. Robert, A Course in p-Adic Analysis. Graduate Texts in Mathematics, vol. 198 (Springer, New York, 2000)
M. Ruggiero, Rigidification of holomorphic germs with non-invertible differential. Michigan Math. J. 61, 161–185 (2012)
I.P. Shestakov, U.U. Umirbaev, The tame and the wild automorphisms of polynomial rings in three variables. J. Am. Math. Soc, 17, 197–227 (2004)
N. Sibony, Dynamique des applications rationnelles de P k. In Dynamique et géométrie complexes (Lyon, 1997). Panor. Synthèses, vol. 8 (Soc. Math. France, Paris, 1999), pp. 97–185
J.H. Silverman, The Arithmetic of Dynamical Systems. Graduate Texts in Mathematics, vol. 241 (Springer, New York, 2007)
J.H. Silverman, Lecture notes on arithmetic dynamics. Lecture notes from the 2010 Arizona Winter School. math.arizona.edu/ ∼ swc/aws/10/
M. Spivakovsky, Valuations in function fields of surfaces. Am. J. Math. 112, 107–156 (1990)
M. Suzuki, Propriétés topologiques des polynômes de deux variables complexes, et automorphismes algébriques de l’espace C 2. J. Math. Soc. Jpn. 26, 241–257 (1974)
M. Temkin, Stable modification of relative curves. J. Algebraic Geometry 19, 603–677 (2010)
M. Temkin, Introduction to Berkovich Analytic spaces. In this volume
A. Thuillier, Théorie du potentiel sur les courbes en géométrie analytique non archimédienne. Applications à la théorie d’Arakelov, Ph.D. thesis, University of Rennes, 2005. tel.archives-ouvertes.fr/tel-00010990
A. Thuillier, Géométrie toroïdale et géométrie analytique non archimédienne. Application au type d’homotopie de certains schémas formels. Manuscripta Math. 123(4), 381–451 (2007)
J.-C. Tougeron, Idéaux de fonctions différentiables. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 71 (Springer, Berlin-New York, 1972)
E. Trucco, Wandering Fatou components and algebraic Julia sets. To appear in Bull. de la Soc. Math. de France
M. Vaquié, Valuations. In Resolution of Singularities (Obergurgl, 1997). Progr. Math., vol. 181 (Birkhaüser, Basel, 2000), pp. 539–590
M. Vaquié, Extension d’une valuation. Trans. Am. Math. Soc. 359, 3439–3481 (2007)
X. Yuan, Big line bundles over arithmetic varieties. Invent. Math. 173, 603–649 (2008)
X. Yuan, S.-W. Zhang, Calabi-Yau theorem and algebraic dynamics. Preprint www.math.columbia.edu/~szhang/papers/Preprints.htm
X. Yuan, S.-W. Zhang, Small points and Berkovich metrics. Preprint, 2009, available at www.math.columbia.edu/~szhang/papers/Preprints.htm
O. Zariski, P. Samuel, Commutative Algebra, vol 2. Graduate Texts in Mathematics, vol. 29 (Springer, New York, 1975)
Acknowledgements
I would like to express my gratitude to many people, first and foremost to Charles Favre for a long and fruitful collaboration and without whom these notes would not exist. Likewise, I have benefitted enormously from working with Sébastien Boucksom. I thank Matt Baker for many interesting discussions; the book by Matt Baker and Robert Rumely has also served as an extremely useful reference for dynamics on the Berkovich projective line. I am grateful to Michael Temkin and Antoine Ducros for answering various questions about Berkovich spaces and to Andreas Blass for help with Remark 2.6 and Example 2.7. Conversations with Dale Cutkosky, William Gignac, Olivier Piltant and Matteo Ruggiero have also been very helpful, as have comments by Yûsuke Okuyama. Vladimir Berkovich of course deserves a special acknowledgment as neither these notes nor the summer school itself would have been possible without his work. Finally I am grateful to the organizers and the sponsors of the summer school. My research has been partially funded by grants DMS-0449465 and DMS-1001740 from the NSF.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Jonsson, M. (2015). Dynamics on Berkovich Spaces in Low Dimensions. In: Ducros, A., Favre, C., Nicaise, J. (eds) Berkovich Spaces and Applications. Lecture Notes in Mathematics, vol 2119. Springer, Cham. https://doi.org/10.1007/978-3-319-11029-5_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-11029-5_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-11028-8
Online ISBN: 978-3-319-11029-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)
