Dynamics in One Non-Archimedean Variable

Benedetto, Robert

doi:10.1090/gsm/198

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Dynamics in One Non-Archimedean Variable

About this Title

Robert L. Benedetto, Amherst College, Amherst, MA

Publication: Graduate Studies in Mathematics
Publication Year: 2019; Volume 198
ISBNs: 978-1-4704-4688-8 (print); 978-1-4704-5106-6 (online)
DOI: https://doi.org/10.1090/gsm/198
MathSciNet review: MR3890051
MSC: Primary 37-01; Secondary 11S82, 37Pxx

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Sergio Albeverio, Andrei Khrennikov, and Brunello Tirozzi, $p$-adic dynamical systems and neural networks, Math. Models Methods Appl. Sci. 9 (1999), no. 9, 1417–1437. MR 1725825, DOI 10.1142/S0218202599000634
Yvette Amice, Les nombres $p$-adiques, Collection SUP: “Le Mathématicien”, vol. 14, Presses Universitaires de France, Paris, 1975 (French). Préface de Ch. Pisot. MR 0447195
David K. Arrowsmith and Franco Vivaldi, Geometry of $p$-adic Siegel discs, Phys. D 71 (1994), no. 1-2, 222–236. MR 1264116, DOI 10.1016/0167-2789(94)90191-0
Dvij Bajpai, Robert L. Benedetto, Ruqian Chen, Edward Kim, Owen Marschall, Darius Onul, and Yang Xiao, Non-archimedean connected Julia sets with branching, Ergodic Theory Dynam. Systems 37 (2017), no. 1, 59–78. MR 3590494, DOI 10.1017/etds.2015.42
M. Baker, personal communication, 2017.
Matthew Baker and Laura DeMarco, Preperiodic points and unlikely intersections, Duke Math. J. 159 (2011), no. 1, 1–29. MR 2817647, DOI 10.1215/00127094-1384773
Matthew H. Baker and Robert Rumely, Equidistribution of small points, rational dynamics, and potential theory, Ann. Inst. Fourier (Grenoble) 56 (2006), no. 3, 625–688 (English, with English and French summaries). MR 2244226
Matthew Baker and Robert Rumely, Potential theory and dynamics on the Berkovich projective line, Mathematical Surveys and Monographs, vol. 159, American Mathematical Society, Providence, RI, 2010. MR 2599526, DOI 10.1090/surv/159
Alan F. Beardon, Iteration of rational functions, Graduate Texts in Mathematics, vol. 132, Springer-Verlag, New York, 1991. Complex analytic dynamical systems. MR 1128089, DOI 10.1007/978-1-4612-4422-6
S. Ben-Menahem, $p$-adic iterations, preprint, TAUP 1627–88, Tel-Aviv University (1988).
John J. Benedetto and Wojciech Czaja, Integration and modern analysis, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Boston, Ltd., Boston, MA, 2009. MR 2559803, DOI 10.1007/978-0-8176-4656-1
Robert L. Benedetto, Fatou components inp-adic dynamics, ProQuest LLC, Ann Arbor, MI, 1998. Thesis (Ph.D.)–Brown University. MR 2697379
Robert L. Benedetto, $p$-adic dynamics and Sullivan’s no wandering domains theorem, Compositio Math. 122 (2000), no. 3, 281–298. MR 1781331, DOI 10.1023/A:1002067315057
Robert L. Benedetto, Hyperbolic maps in $p$-adic dynamics, Ergodic Theory Dynam. Systems 21 (2001), no. 1, 1–11. MR 1826658, DOI 10.1017/S0143385701001043
Robert L. Benedetto, Reduction, dynamics, and Julia sets of rational functions, J. Number Theory 86 (2001), no. 2, 175–195. MR 1813109, DOI 10.1006/jnth.2000.2577
Robert L. Benedetto, Components and periodic points in non-Archimedean dynamics, Proc. London Math. Soc. (3) 84 (2002), no. 1, 231–256. MR 1863402, DOI 10.1112/plms/84.1.231
Robert L. Benedetto, Examples of wandering domains in $p$-adic polynomial dynamics, C. R. Math. Acad. Sci. Paris 335 (2002), no. 7, 615–620 (English, with English and French summaries). MR 1941304
Robert L. Benedetto, Wandering domains and nontrivial reduction in non-Archimedean dynamics, Illinois J. Math. 49 (2005), no. 1, 167–193. MR 2157374
Robert L. Benedetto, Wandering domains in non-Archimedean polynomial dynamics, Bull. London Math. Soc. 38 (2006), no. 6, 937–950. MR 2285248, DOI 10.1112/S0024609306019126
Robert L. Benedetto, An Ahlfors islands theorem for non-Archimedean meromorphic functions, Trans. Amer. Math. Soc. 360 (2008), no. 8, 4099–4124. MR 2395165, DOI 10.1090/S0002-9947-08-04546-7
Robert L. Benedetto, A criterion for potentially good reduction in nonarchimedean dynamics, Acta Arith. 165 (2014), no. 3, 251–256. MR 3263950, DOI 10.4064/aa165-3-4
Robert L. Benedetto, Attaining potentially good reduction in arithmetic dynamics, Int. Math. Res. Not. IMRN 22 (2015), 11828–11846. MR 3456704, DOI 10.1093/imrn/rnv039
Robert Benedetto, Jean-Yves Briend, and Hervé Perdry, Dynamique des polynômes quadratiques sur les corps locaux, J. Théor. Nombres Bordeaux 19 (2007), no. 2, 325–336 (French, with English and French summaries). MR 2394889
Robert L. Benedetto, Dragos Ghioca, Pär Kurlberg, and Thomas J. Tucker, A case of the dynamical Mordell-Lang conjecture, Math. Ann. 352 (2012), no. 1, 1–26. With an appendix by Umberto Zannier. MR 2885573, DOI 10.1007/s00208-010-0621-4
Robert Benedetto, Patrick Ingram, Rafe Jones, and Alon Levy, Attracting cycles in $p$-adic dynamics and height bounds for postcritically finite maps, Duke Math. J. 163 (2014), no. 13, 2325–2356. MR 3265554, DOI 10.1215/00127094-2804674
Vladimir G. Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, vol. 33, American Mathematical Society, Providence, RI, 1990. MR 1070709, DOI 10.1090/surv/033
Jean-Paul Bézivin, Sur les points périodiques des applications rationnelles en dynamique ultramétrique, Acta Arith. 100 (2001), no. 1, 63–74 (French). MR 1864626, DOI 10.4064/aa100-1-5
Jean-Paul Bézivin, Sur la compacité des ensembles de Julia des polynômes $p$-adiques, Math. Z. 246 (2004), no. 1-2, 273–289 (French, with English summary). MR 2031456, DOI 10.1007/s00209-003-0599-7
J.-P. Bézivin, Dynamique des Fractions Rationnelles $p$-adiques, Lecture notes, 2003. Available at http://jp.bezivin.pagesperso-orange.fr/dealatex.pdf
S. Bosch, U. Güntzer, and R. Remmert, Non-Archimedean analysis, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 261, Springer-Verlag, Berlin, 1984. A systematic approach to rigid analytic geometry. MR 746961, DOI 10.1007/978-3-642-52229-1
Patrick Billingsley, Convergence of probability measures, 2nd ed., Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Inc., New York, 1999. A Wiley-Interscience Publication. MR 1700749, DOI 10.1002/9780470316962
L. E. Böttcher, The principal laws of convergence of iterates and their applications to analysis (Russian), Izv. Kazan. Fiz.-Mat. Obshch. 14 (1904), 155-234.
N. Bourbaki, Éléments de mathématique. Fasc. XIII. Livre VI: Intégration. Chapitres 1, 2, 3 et 4: Inégalités de convexité, Espaces de Riesz, Mesures sur les espaces localement compacts, Prolongement d’une mesure, Espaces $L^{p}$, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1175, Hermann, Paris, 1965 (French). Deuxième édition revue et augmentée. MR 0219684
Hans Brolin, Invariant sets under iteration of rational functions, Ark. Mat. 6 (1965), 103–144 (1965). MR 194595, DOI 10.1007/BF02591353
John Bryk and Cesar E. Silva, Measurable dynamics of simple $p$-adic polynomials, Amer. Math. Monthly 112 (2005), no. 3, 212–232. MR 2125384, DOI 10.2307/30037439
A. D. Bryuno, Analytical form of differential equations, Trans. Moscow Math. Soc. 25 (1971), 131–288.
Gregory S. Call and Joseph H. Silverman, Canonical heights on varieties with morphisms, Compositio Math. 89 (1993), no. 2, 163–205. MR 1255693
Serge Cantat and Antoine Chambert-Loir, Dynamique $p$-adique (d’après les exposés de Jean-Christophe Yoccoz), Quelques aspects des systèmes dynamiques polynomiaux, Panor. Synthèses, vol. 30, Soc. Math. France, Paris, 2010, pp. 295–341 (French, with English and French summaries). MR 2932436
Lennart Carleson and Theodore W. Gamelin, Complex dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. MR 1230383, DOI 10.1007/978-1-4612-4364-9
Antoine Chambert-Loir, Mesures et équidistribution sur les espaces de Berkovich, J. Reine Angew. Math. 595 (2006), 215–235 (French). MR 2244803, DOI 10.1515/CRELLE.2006.049
Robert F. Coleman, $p$-adic Banach spaces and families of modular forms, Invent. Math. 127 (1997), no. 3, 417–479. MR 1431135, DOI 10.1007/s002220050127
John B. Conway, Functions of one complex variable, 2nd ed., Graduate Texts in Mathematics, vol. 11, Springer-Verlag, New York-Berlin, 1978. MR 503901
Laura De Marco and Xander Faber, Degenerations of complex dynamical systems, Forum Math. Sigma 2 (2014), Paper No. e6, 36. MR 3264250, DOI 10.1017/fms.2014.8
Laura DeMarco and Xander Faber, Degenerations of complex dynamical systems II: analytic and algebraic stability, Math. Ann. 365 (2016), no. 3-4, 1669–1699. With an appendix by Jan Kiwi. MR 3521102, DOI 10.1007/s00208-015-1331-8
Adrien Douady, Disques de Siegel et anneaux de Herman, Astérisque 152-153 (1987), 4, 151–172 (1988) (French). Séminaire Bourbaki, Vol. 1986/87. MR 936853
John R. Doyle, Kenneth Jacobs, and Robert Rumely, Configuration of the crucial set for a quadratic rational map, Res. Number Theory 2 (2016), Paper No. 11, 16. MR 3518909, DOI 10.1007/s40993-016-0041-y
V. A. Dremov, On a $p$-adic Julia set, Uspekhi Mat. Nauk 58 (2003), no. 6(354), 151–152 (Russian); English transl., Russian Math. Surveys 58 (2003), no. 6, 1194–1195. MR 2054097, DOI 10.1070/RM2003v058n06ABEH000682
V. Dremov, G. Shabat, and P. Vytnova, On the chaotic properties of quadratic maps over non-Archimedean fields, $p$-adic mathematical physics, AIP Conf. Proc., vol. 826, Amer. Inst. Phys., Melville, NY, 2006, pp. 43–54. MR 2258672, DOI 10.1063/1.2193109
J. Écalle, Théorie itérative: introduction à la théorie des invariants holomorphes, J. Math. Pures Appl. (9) 54 (1975), 183–258 (French). MR 499882
Manfred Einsiedler and Thomas Ward, Ergodic theory with a view towards number theory, Graduate Texts in Mathematics, vol. 259, Springer-Verlag London, Ltd., London, 2011. MR 2723325, DOI 10.1007/978-0-85729-021-2
A. Escassut, The ultrametric spectral theory, Period. Math. Hungar. 11 (1980), no. 1, 7–60. MR 571136, DOI 10.1007/BF02019983
Xander Faber, Topology and geometry of the Berkovich ramification locus for rational functions, I, Manuscripta Math. 142 (2013), no. 3-4, 439–474. MR 3117171, DOI 10.1007/s00229-013-0611-4
Xander Faber, Topology and geometry of the Berkovich ramification locus for rational functions, II, Math. Ann. 356 (2013), no. 3, 819–844. MR 3063898, DOI 10.1007/s00208-012-0872-3
P. Fatou, Sur les équations fonctionnelles, Bull. Soc. Math. France 48 (1920), 208–314 (French). MR 1504797
C. Favre and T. Gauthier, Distribution of postcritically finite polynomials, Israel J. Math. 209 (2015), no. 1, 235–292. MR 3430241, DOI 10.1007/s11856-015-1218-0
Charles Favre and Thomas Gauthier, Classification of special curves in the space of cubic polynomials, Int. Math. Res. Not. IMRN 2 (2018), 362–411. MR 3801434, DOI 10.1093/imrn/rnw245
Charles Favre, Jan Kiwi, and Eugenio Trucco, A non-Archimedean Montel’s theorem, Compos. Math. 148 (2012), no. 3, 966–990. MR 2925406, DOI 10.1112/S0010437X11007470
Charles Favre and Mattias Jonsson, The valuative tree, Lecture Notes in Mathematics, vol. 1853, Springer-Verlag, Berlin, 2004. MR 2097722, DOI 10.1007/b100262
Charles Favre and Mattias Jonsson, Eigenvaluations, Ann. Sci. École Norm. Sup. (4) 40 (2007), no. 2, 309–349 (English, with English and French summaries). MR 2339287, DOI 10.1016/j.ansens.2007.01.002
Charles Favre and Juan Rivera-Letelier, Théorème d’équidistribution de Brolin en dynamique $p$-adique, C. R. Math. Acad. Sci. Paris 339 (2004), no. 4, 271–276 (French, with English and French summaries). MR 2092012, DOI 10.1016/j.crma.2004.06.023
Charles Favre and Juan Rivera-Letelier, Équidistribution quantitative des points de petite hauteur sur la droite projective, Math. Ann. 335 (2006), no. 2, 311–361 (French, with English and French summaries). MR 2221116, DOI 10.1007/s00208-006-0751-x
Charles Favre and Juan Rivera-Letelier, Corrigendum à l’article “Équidistribution quantitative des points de petite hauteur sur la droite projective”, Math. Ann. 339 (2007), 799–801.
Charles Favre and Juan Rivera-Letelier, Théorie ergodique des fractions rationnelles sur un corps ultramétrique, Proc. Lond. Math. Soc. (3) 100 (2010), no. 1, 116–154 (French, with English and French summaries). MR 2578470, DOI 10.1112/plms/pdp022
Gerald B. Folland, Real analysis, 2nd ed., Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1999. Modern techniques and their applications; A Wiley-Interscience Publication. MR 1681462
Alexandre Freire, Artur Lopes, and Ricardo Mañé, An invariant measure for rational maps, Bol. Soc. Brasil. Mat. 14 (1983), no. 1, 45–62. MR 736568, DOI 10.1007/BF02584744
Jean Fresnel and Marius van der Put, Rigid analytic geometry and its applications, Progress in Mathematics, vol. 218, Birkhäuser Boston, Inc., Boston, MA, 2004. MR 2014891, DOI 10.1007/978-1-4612-0041-3
T. Gauthier, Y. Okuyama, and G. Vigny, Approximation of non-archimedean Lyapunov exponents and applications over global fields, arXiv:1803.06859 (2018).
D. Ghioca, H. Krieger, K. D. Nguyen, and H. Ye, The dynamical André-Oort conjecture: unicritical polynomials, Duke Math. J. 166 (2017), no. 1, 1–25. MR 3592687, DOI 10.1215/00127094-3673996
David Goss, A short introduction to rigid analytic spaces, The arithmetic of function fields (Columbus, OH, 1991) Ohio State Univ. Math. Res. Inst. Publ., vol. 2, de Gruyter, Berlin, 1992, pp. 131–141. MR 1196516
Fernando Q. Gouvêa, $p$-adic numbers, Universitext, Springer-Verlag, Berlin, 1993. An introduction. MR 1251959, DOI 10.1007/978-3-662-22278-2
Matthias Gundlach, Andrei Khrennikov, and Karl-Olof Lindahl, On ergodic behavior of $p$-adic dynamical systems, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 4 (2001), no. 4, 569–577. MR 1876486, DOI 10.1142/S0219025701000632
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
Michael-R. Herman, Exemples de fractions rationnelles ayant une orbite dense sur la sphère de Riemann, Bull. Soc. Math. France 112 (1984), no. 1, 93–142 (French, with English summary). MR 771920
M. Herman and J.-C. Yoccoz, Generalizations of some theorems of small divisors to non-Archimedean fields, Geometric dynamics (Rio de Janeiro, 1981) Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983, pp. 408–447. MR 730280, DOI 10.1007/BFb0061427
Marc Hindry and Joseph H. Silverman, Diophantine geometry, Graduate Texts in Mathematics, vol. 201, Springer-Verlag, New York, 2000. An introduction. MR 1745599, DOI 10.1007/978-1-4612-1210-2
Liang-Chung Hsia, A weak Néron model with applications to $p$-adic dynamical systems, Compositio Math. 100 (1996), no. 3, 277–304. MR 1387667
Liang-Chung Hsia, Closure of periodic points over a non-Archimedean field, J. London Math. Soc. (2) 62 (2000), no. 3, 685–700. MR 1794277, DOI 10.1112/S0024610700001447
Patrick Ingram, The critical height is a moduli height, Duke Math. J. 167 (2018), no. 7, 1311–1346. MR 3799700, DOI 10.1215/00127094-2017-0053
K. Jacobs, Lower bounds for non-Archimedean Lyapunov exponents, arXiv:1510.02440 (2015).
Nathan Jacobson, Basic algebra. II, 2nd ed., W. H. Freeman and Company, New York, 1989. MR 1009787
Mattias Jonsson, Dynamics of Berkovich spaces in low dimensions, Berkovich spaces and applications, Lecture Notes in Math., vol. 2119, Springer, Cham, 2015, pp. 205–366. MR 3330767, DOI 10.1007/978-3-319-11029-5_{6}
G. Julia, Mémoire sur l’itération des fonctions rationelles, J. Math. Pures Appl. (7) 4 (1918), 47–245.
Jan Kiwi, Puiseux series polynomial dynamics and iteration of complex cubic polynomials, Ann. Inst. Fourier (Grenoble) 56 (2006), no. 5, 1337–1404 (English, with English and French summaries). MR 2273859
Jan Kiwi, Puiseux series dynamics of quadratic rational maps, Israel J. Math. 201 (2014), no. 2, 631–700. MR 3265299, DOI 10.1007/s11856-014-1047-6
Jan Kiwi, Rescaling limits of complex rational maps, Duke Math. J. 164 (2015), no. 7, 1437–1470. MR 3347319, DOI 10.1215/00127094-2916431
Neal Koblitz, $p$-adic numbers, $p$-adic analysis, and zeta-functions, 2nd ed., Graduate Texts in Mathematics, vol. 58, Springer-Verlag, New York, 1984. MR 754003, DOI 10.1007/978-1-4612-1112-9
G. Koenigs, Recherches sur les intégrales de certaines équations fonctionnelles, Ann. Sci. École Norm. Sup. (3) 1 (1884), 3–41 (French). MR 1508749
Serge Lang, Fundamentals of Diophantine geometry, Springer-Verlag, New York, 1983. MR 715605, DOI 10.1007/978-1-4757-1810-2
Serge Lang, Algebra, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. MR 0197234
J. Lee, $J$-stability of expanding maps in non-Archimedean dynamics, Ergodic Theory Dynam. Systems (to appear). https://doi.org/10.1017/etds.2017.53
Hua-Chieh Li, $p$-adic periodic points and Sen’s theorem, J. Number Theory 56 (1996), no. 2, 309–318. MR 1373554, DOI 10.1006/jnth.1996.0020
Hua-Chieh Li, Counting periodic points of $p$-adic power series, Compositio Math. 100 (1996), no. 3, 351–364. MR 1387670
Hua-Chieh Li, $p$-adic dynamical systems and formal groups, Compositio Math. 104 (1996), no. 1, 41–54. MR 1420709
D. A. Lind and T. Ward, Automorphisms of solenoids and $p$-adic entropy, Ergodic Theory Dynam. Systems 8 (1988), no. 3, 411–419. MR 961739, DOI 10.1017/S0143385700004545
Karl-Olof Lindahl, On Siegel’s linearization theorem for fields of prime characteristic, Nonlinearity 17 (2004), no. 3, 745–763. MR 2057125, DOI 10.1088/0951-7715/17/3/001
Karl-Olof Lindahl, Divergence and convergence of conjugacies in non-Archimedean dynamics, Advances in $p$-adic and non-Archimedean analysis, Contemp. Math., vol. 508, Amer. Math. Soc., Providence, RI, 2010, pp. 89–109. MR 2597687, DOI 10.1090/conm/508/09993
Karl-Olof Lindahl, The size of quadratic $p$-adic linearization disks, Adv. Math. 248 (2013), 872–894. MR 3107530, DOI 10.1016/j.aim.2013.07.020
Karl-Olof Lindahl and Juan Rivera-Letelier, Generic parabolic points are isolated in positive characteristic, Nonlinearity 29 (2016), no. 5, 1596–1621. MR 3481344, DOI 10.1088/0951-7715/29/5/1596
Jonathan Lubin, Non-Archimedean dynamical systems, Compositio Math. 94 (1994), no. 3, 321–346. MR 1310863
M. Ju. Ljubich, Entropy properties of rational endomorphisms of the Riemann sphere, Ergodic Theory Dynam. Systems 3 (1983), no. 3, 351–385. MR 741393, DOI 10.1017/S0143385700002030
R. Mañé, P. Sad, and D. Sullivan, On the dynamics of rational maps, Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 2, 193–217. MR 732343
Michelle Manes and Yu Yasufuku, Explicit descriptions of quadratic maps on $\Bbb P^1$ defined over a field $K$, Acta Arith. 148 (2011), no. 3, 257–267. MR 2794930, DOI 10.4064/aa148-3-3
Curtis T. McMullen, The Mandelbrot set is universal, The Mandelbrot set, theme and variations, London Math. Soc. Lecture Note Ser., vol. 274, Cambridge Univ. Press, Cambridge, 2000, pp. 1–17. MR 1765082
John Milnor, Dynamics in one complex variable, Friedr. Vieweg & Sohn, Braunschweig, 1999. Introductory lectures. MR 1721240
John Milnor, Geometry and dynamics of quadratic rational maps, Experiment. Math. 2 (1993), no. 1, 37–83. With an appendix by the author and Lei Tan. MR 1246482
Patrick Morton and Joseph H. Silverman, Rational periodic points of rational functions, Internat. Math. Res. Notices 2 (1994), 97–110. MR 1264933, DOI 10.1155/S1073792894000127
Patrick Morton and Joseph H. Silverman, Periodic points, multiplicities, and dynamical units, J. Reine Angew. Math. 461 (1995), 81–122. MR 1324210, DOI 10.1515/crll.1995.461.81
Yûsuke Okuyama, Repelling periodic points and logarithmic equidistribution in non-archimedean dynamics, Acta Arith. 152 (2012), no. 3, 267–277. MR 2885787, DOI 10.4064/aa152-3-3
Yûsuke Okuyama, Quantitative approximations of the Lyapunov exponent of a rational function over valued fields, Math. Z. 280 (2015), no. 3-4, 691–706. MR 3369346, DOI 10.1007/s00209-015-1443-6
Reinhard Oselies and Heiner Zieschang, Ergodische Eigenschaften der Automorphismen $p$-adischer Zahlen, Arch. Math. (Basel) 26 (1975), 144–153 (German). MR 369301, DOI 10.1007/BF01229718
T. Pezda, Polynomial cycles in certain local domains, Acta Arith. 66 (1994), no. 1, 11–22. MR 1262650, DOI 10.4064/aa-66-1-11-22
Mary Rees, Ergodic rational maps with dense critical point forward orbit, Ergodic Theory Dynam. Systems 4 (1984), no. 2, 311–322. MR 766108, DOI 10.1017/S0143385700002467
Mary Rees, Positive measure sets of ergodic rational maps, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 3, 383–407. MR 870689
Juan Rivera-Letelier, Dynamique des fonctions rationnelles sur des corps locaux, Astérisque 287 (2003), xv, 147–230 (French, with English and French summaries). Geometric methods in dynamics. II. MR 2040006
Juan Rivera-Letelier, Espace hyperbolique $p$-adique et dynamique des fonctions rationnelles, Compositio Math. 138 (2003), no. 2, 199–231 (French, with English summary). MR 2018827, DOI 10.1023/A:1026136530383
J. Rivera-Letelier, Sur la structure des ensembles de Fatou $p$-adiques, arxiv:math.DS/0412180 (2004).
Juan Rivera-Letelier, Points périodiques des fonctions rationnelles dans l’espace hyperbolique $p$-adique, Comment. Math. Helv. 80 (2005), no. 3, 593–629 (French, with English summary). MR 2165204, DOI 10.4171/CMH/27
Juan Rivera-Letelier, Wild recurrent critical points, J. London Math. Soc. (2) 72 (2005), no. 2, 305–326. MR 2156656, DOI 10.1112/S0024610705006885
J. E. Rivera-Letelier, There are rational and irrational ultrametric wandering domains, preprint (2011).
J. E. Rivera-Letelier, personal communication, 2012.
Philippe Robba, Fonctions analytiques sur les corps valués ultramétriques complets, Prolongement analytique et algèbres de Banach ultramétriques, Astérisque, No. 10, Société Mathématique de France, Paris, 1973, pp. 109–218, 219–220 (French, with English summary). MR 0357841
Alain M. Robert, A course in $p$-adic analysis, Graduate Texts in Mathematics, vol. 198, Springer-Verlag, New York, 2000. MR 1760253, DOI 10.1007/978-1-4757-3254-2
H. L. Royden and P. Fitzpatrick, Real Analysis, 4th ed., Prentice Hall, New York, 2010.
Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR 924157
Robert Rumely, The minimal resultant locus, Acta Arith. 169 (2015), no. 3, 251–290. MR 3361223, DOI 10.4064/aa169-3-3
W. H. Schikhof, Ultrametric calculus, Cambridge Studies in Advanced Mathematics, vol. 4, Cambridge University Press, Cambridge, 1984. An introduction to $p$-adic analysis. MR 791759
Jean-Pierre Serre, Algebraic groups and class fields, Graduate Texts in Mathematics, vol. 117, Springer-Verlag, New York, 1988. Translated from the French. MR 918564, DOI 10.1007/978-1-4612-1035-1
A. N. Shiryayev, Probability, Graduate Texts in Mathematics, vol. 95, Springer-Verlag, New York, 1984. Translated from the Russian by R. P. Boas. MR 737192, DOI 10.1007/978-1-4899-0018-0
Mitsuhiro Shishikura, On the quasiconformal surgery of rational functions, Ann. Sci. École Norm. Sup. (4) 20 (1987), no. 1, 1–29. MR 892140
Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR 817210, DOI 10.1007/978-1-4757-1920-8
Joseph H. Silverman, The arithmetic of dynamical systems, Graduate Texts in Mathematics, vol. 241, Springer, New York, 2007. MR 2316407, DOI 10.1007/978-0-387-69904-2
T. Silverman, A non-archimedean $\lambda$-lemma, arXiv:1712.01372 (2017).
Christopher F. Woodcock and Nigel P. Smart, $p$-adic chaos and random number generation, Experiment. Math. 7 (1998), no. 4, 333–342. MR 1678087
Dennis Sullivan, Quasiconformal homeomorphisms and dynamics. I. Solution of the Fatou-Julia problem on wandering domains, Ann. of Math. (2) 122 (1985), no. 3, 401–418. MR 819553, DOI 10.2307/1971308
John Tate, Rigid analytic spaces, Invent. Math. 12 (1971), 257–289. MR 306196, DOI 10.1007/BF01403307
E. Thiran, D. Verstegen, and J. Weyers, $p$-adic dynamics, J. Statist. Phys. 54 (1989), no. 3-4, 893–913. MR 988564, DOI 10.1007/BF01019780
Eugenio Trucco, Wandering Fatou components and algebraic Julia sets, Bull. Soc. Math. France 142 (2014), no. 3, 411–464 (English, with English and French summaries). MR 3295719, DOI 10.24033/bsmf.2670
A. Thuillier, Théorie du potential sur les courbes en géométrie analytique non archimédienne. Applications à la théorie d’Arakelov, University of Rennes Ph.D. thesis, 2005.
D. Verstegen, $p$-adic dynamical systems, Number theory and physics (Les Houches, 1989) Springer Proc. Phys., vol. 47, Springer, Berlin, 1990, pp. 235–242. MR 1058468, DOI 10.1007/978-3-642-75405-0_{2}5
Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR 648108
Jean-Christophe Yoccoz, Linéarisation des germes de difféomorphismes holomorphes de $(\textbf {C}, 0)$, C. R. Acad. Sci. Paris Sér. I Math. 306 (1988), no. 1, 55–58 (French, with English summary). MR 929279

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Dynamics in One Non-Archimedean Variable

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