%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%@Comment{ The Arithmetic of Dynamical Systems }@Comment{ Joseph H. Silverman }@Comment{ Biblographic Database File - ArithDyn.bib }@COMMENT{Getting these error messages, which are messing up thesetwo items. What is going on?Warning--I'm ignoring arxiv0705.1067's extra "title" field--line 3209 of file ArithDyn.bibWarning--I'm ignoring arxiv0705.1067's extra "note" field--line 3211 of file ArithDyn.bibWarning--I'm ignoring arxiv0705.2182's extra "number" field--line 6144 of file ArithDyn.bibWarning--I'm ignoring arxiv0705.2182's extra "pages" field--line 6145 of file ArithDyn.bib}@comment{SOME PAPERS TO ADD (if not already in database)Minimal subsystems of affine dynamics on local fieldsAi-Hua Fan and Youssef Fares (preprint)R. Oselies and H. Zieschang, Ergodische Eigenschaften der Automorphismen p-adischer Zahlen, Arch. Math., 26 (1975), 144--153.(This is a very early paper on p-adic dynamics)T. Budnytska, Topological conjugacy classes of affine maps, preprintarXiv:0812.4921v1, 2008. (is this p-adic dynamics?)J. L. Chabert, A. H. Fan and Y. Fares, Minimal dynamical systems on a discretevaluation domain, DCDS, Vol. 25, Issue 3 (2009), 777--795.A. H. Fan, M. T. Li, J. Y. Yao and D. Zhou, Strict ergodicity ofaffine p-adic dynamical systems on Zp, Advances in Mathematics 214(2007), 666--700.A. H. Fan, M. T. Li, J. Y. Yao et D. Zhou, p-adic affine dynamical systems andapplications, C. R. Acad. Sci. Paris, Ser. I 342 (2006), 129--134.A. H. Fan and L. M. Liao, On minimal decomposition of p-adicpolynomial dynamical systems, preprint.}@comment{*** Are all of Zieve's dynamics articles joint?Articles preceded bycomment{UPDATE}are preprints or "to appears" that should be updated as appropriate.}@comment{%%% To fix excessive space between bibliography entries, %%% after creating the .bbl file using bibtex, put the following%%% three commands after the \begin{bibliography} command. (May need%%% to adjust the size of \itemsep to look okay.)\setlength{\itemsep}{-4pt}\small\addcontentsline{toc}{chapter}{Bibliography} % not needed for amsbook.sty}@comment{ ------------------------------------------------------------ }@comment{ Material to Add:}@comment{ ----- Blank form for ArXiv papers ---}@comment{UPDATE}@comment{misc {arxiv,    AUTHOR = {},    TITLE = {},     YEAR = {},     NOTE = {\url{}},}}@comment{ ------------------------------------------------------------ }@preamble{   "\def\cprime{$'$} "}@comment{--AAA-- }@book {MR510197,    AUTHOR = {Ahlfors, Lars V.},     TITLE = {Complex {A}nalysis}, PUBLISHER = {McGraw-Hill Book Co.},   ADDRESS = {New York},      YEAR = {1978},     PAGES = {xi+331},      ISBN = {0-07-000657-1},   MRCLASS = {30-01},  MRNUMBER = {MR510197 (80c:30001)},}@comment{UPDATE}@misc {arxiv1010.5583,    AUTHOR = {Ai-Hua, Fan and Liao, Lingmin},    TITLE = {On minimal decomposition of $p$-adic polynomial dynamical systems},     YEAR = {2010},     NOTE = {\url{arXiv:1010.5583}},}@comment{UPDATE}@misc {arxiv0910.4556,    AUTHOR = {Ahmadi, Omran},    TITLE = {A Note on Stable Quadratic Polynomials over Fields           of Characteristic Two},     YEAR = {2009},     NOTE = {\url{arXiv:0910.4556}},}@article {MR0335427,    AUTHOR = {Aho, A. V. and Sloane, N. J. A.},     TITLE = {Some doubly exponential sequences},   JOURNAL = {Fibonacci Quart.},  FJOURNAL = {The Fibonacci Quarterly. Official Organ of the Fibonacci              Association},    VOLUME = {11},      YEAR = {1973},    NUMBER = {4},     PAGES = {429--437},      ISSN = {0015-0517},   MRCLASS = {10A40},  MRNUMBER = {MR0335427 (49 \#209)},MRREVIEWER = {J. B. Roberts},}@comment{http://www.research.att.com/~njas/doc/doubly.html, Publ. List, item 20}@article {arxiv0408170,   AUTHOR = {Aitken, Wayne and Hajir, Farshid and Maire, Christian},    TITLE = {Finitely ramified iterated extensions},   JOURNAL = {IMRN},  FJOURNAL = {International Mathematics Research Notices},     YEAR = {2005},    PAGES = {855--880},   VOLUME = {14},}@comment{\url{ArXiv:math.NT/0408170}}@article {MR2549537,    AUTHOR = {Akbary, Amir and Ghioca, Dragos},     TITLE = {Periods of orbits modulo primes},   JOURNAL = {J. Number Theory},  FJOURNAL = {Journal of Number Theory},    VOLUME = {129},      YEAR = {2009},    NUMBER = {11},     PAGES = {2831--2842},      ISSN = {0022-314X},     CODEN = {JNUTA9},   MRCLASS = {11G35 (11G50 14G25 37Pxx)},  MRNUMBER = {MR2549537},       DOI = {10.1016/j.jnt.2009.03.007},       URL = {http://dx.doi.org/10.1016/j.jnt.2009.03.007},}@article {arxiv0704.3674,    AUTHOR = {Akiyama, Shigeki and Brunotte, Horst and Peth{\H{o}}, Attila              and Steiner, Wolfgang},     TITLE = {Periodicity of certain piecewise affine planar maps},   JOURNAL = {Tsukuba J. Math.},  FJOURNAL = {Tsukuba Journal of Mathematics},    VOLUME = {32},      YEAR = {2008},    NUMBER = {1},     PAGES = {197--251},      ISSN = {0387-4982},   MRCLASS = {37C25 (37E15)},  MRNUMBER = {MR2433023},MRREVIEWER = {Peter Ashwin},}@article {MR1840304,    AUTHOR = {Albeverio, S. and Tirotstsi, B. and Khrennikov, A.              Yu. and de Shmedt, S.},     TITLE = {{$p$}-adic dynamical systems},   JOURNAL = {Teoret. Mat. Fiz.},  FJOURNAL = {Rossi\u\i skaya Akademiya Nauk. Teoreticheskaya i              Matematicheskaya Fizika},    VOLUME = {114},      YEAR = {1998},    NUMBER = {3},     PAGES = {349--365},      ISSN = {0564-6162},   MRCLASS = {37B99 (11S85)},  MRNUMBER = {MR1840304 (2002a:37018)},}@article {MR1928725,    AUTHOR = {Albeverio, S. and Gundlach, M. and Khrennikov, A. and Lindahl,              K.-O.},     TITLE = {On the {M}arkovian behavior of {$p$}-adic random dynamical              systems},   JOURNAL = {Russ. J. Math. Phys.},  FJOURNAL = {Russian Journal of Mathematical Physics},    VOLUME = {8},      YEAR = {2001},    NUMBER = {2},     PAGES = {135--152},      ISSN = {1061-9208},   MRCLASS = {60J10 (37H99 81Q99)},  MRNUMBER = {MR1928725 (2003g:60124)},MRREVIEWER = {Yves Lacroix},}@article {MR2163517,    AUTHOR = {Ali, Nidal},     TITLE = {Stabilit\'e des polyn\^omes},   JOURNAL = {Acta Arith.},  FJOURNAL = {Acta Arithmetica},    VOLUME = {119},      YEAR = {2005},    NUMBER = {1},     PAGES = {53--63},      ISSN = {0065-1036},     CODEN = {AARIA9},   MRCLASS = {11R09 (11T06 12E05)},  MRNUMBER = {MR2163517 (2006h:11125)},MRREVIEWER = {Peter M{\"u}ller},}@comment{The polynomial f is said to be stable if f^m isirreducible for all positive integers m.}@comment{UPDATE}@misc {arxiv0802.2938,    AUTHOR = {Aliev, Iskander and Smyth, Chris },    TITLE = {Power maps and subvarieties of the complex algebraic $n$-torus},     YEAR = {2008},     NOTE = {\url{arXiv:0802.2938}},}@comment{arXiv1007.1635}@article {MR2784670,    AUTHOR = {Amerik, Ekaterina},     TITLE = {Existence of non-preperiodic algebraic points for a rational              self-map of infinite order},   JOURNAL = {Math. Res. Lett.},  FJOURNAL = {Mathematical Research Letters},    VOLUME = {18},      YEAR = {2011},    NUMBER = {2},     PAGES = {251--256},      ISSN = {1073-2780},   MRCLASS = {37Pxx (14Exx)},  MRNUMBER = {2784670},}@article {MR1817715,    AUTHOR = {Amoroso, Francesco and Zannier, Umberto},     TITLE = {A relative {D}obrowolski lower bound over abelian extensions},   JOURNAL = {Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)},  FJOURNAL = {Annali della Scuola Normale Superiore di Pisa. Classe di              Scienze. Serie IV},    VOLUME = {29},      YEAR = {2000},    NUMBER = {3},     PAGES = {711--727},      ISSN = {0391-173X},     CODEN = {PSNAAI},   MRCLASS = {11G50 (11J99 11R47)},  MRNUMBER = {MR1817715 (2003a:11078)},}@article {MR1740514,    AUTHOR = {Amoroso, Francesco and Dvornicich, Roberto},     TITLE = {A lower bound for the height in abelian extensions},   JOURNAL = {J. Number Theory},  FJOURNAL = {Journal of Number Theory},    VOLUME = {80},      YEAR = {2000},    NUMBER = {2},     PAGES = {260--272},      ISSN = {0022-314X},     CODEN = {JNUTA9},   MRCLASS = {11R18 (11G50)},  MRNUMBER = {MR1740514 (2001b:11100)},MRREVIEWER = {Christopher Smyth},}@incollection {arxiv0602083,    AUTHOR = {Anashin, Vladimir},     TITLE = {Ergodic transformations in the space of {$p$}-adic integers}, BOOKTITLE = {$p$-adic mathematical physics},    SERIES = {AIP Conf. Proc.},    VOLUME = {826},     PAGES = {3--24}, PUBLISHER = {Amer. Inst. Phys.},   ADDRESS = {Melville, NY},      YEAR = {2006},   MRCLASS = {37A45 (11S85 37A25)},  MRNUMBER = {MR2258670},}@comment{\url{ArXiv:math.math.DS/0602083}}@book {MR990016,    AUTHOR = {Andr{\'e}, Yves},     TITLE = {{$G$}-functions and geometry},    SERIES = {Aspects of Mathematics, E13}, PUBLISHER = {Friedr. Vieweg \& Sohn},   ADDRESS = {Braunschweig},      YEAR = {1989},     PAGES = {xii+229},      ISBN = {3-528-06317-3},   MRCLASS = {11J82 (11-02 11G10 11J87 12H25)},  MRNUMBER = {990016 (90k:11087)},MRREVIEWER = {C. L. Stewart},}@article {MR2220002,    AUTHOR = {Angl{\`e}s d'Auriac, J.-Ch. and Maillard, J.-M. and Viallet,              C. M.},     TITLE = {On the complexity of some birational transformations},   JOURNAL = {J. Phys. A},  FJOURNAL = {Journal of Physics. A. Mathematical and General},    VOLUME = {39},      YEAR = {2006},    NUMBER = {14},     PAGES = {3641--3654},      ISSN = {0305-4470},     CODEN = {JPHAC5},   MRCLASS = {37F10 (14Q99 20F55 37B99)},  MRNUMBER = {MR2220002 (2006m:37061)},MRREVIEWER = {Eric Bedford},}@book {apostol:utm,     AUTHOR = {Apostol, Tom M.},     TITLE = {Introduction to {A}nalytic {N}umber {T}heory},      NOTE = {Undergraduate Texts in Mathematics}, PUBLISHER = {Springer-Verlag},   ADDRESS = {New York},      YEAR = {1976},     PAGES = {xii+338},   MRCLASS = {10-01 (10AXX 10HXX)},  MRNUMBER = {MR0434929 (55 \#7892)},MRREVIEWER = {E. Grosswald},}@article {MR2163516,    AUTHOR = {Arias de Reyna, J.},     TITLE = {Dynamical zeta functions and {K}ummer congruences},   JOURNAL = {Acta Arith.},  FJOURNAL = {Acta Arithmetica},    VOLUME = {119},      YEAR = {2005},    NUMBER = {1},     PAGES = {39--52},      ISSN = {0065-1036},     CODEN = {AARIA9},   MRCLASS = {11B68 (11A07 11B37 37C30)},  MRNUMBER = {MR2163516 (2006d:11020)},MRREVIEWER = {Thomas Ward},}	@book {MR2777369,    AUTHOR = {Arnold, V. I.},     TITLE = {Dynamics, statistics and projective geometry of {G}alois              fields},      NOTE = {Translated from the Russian,              With words about Arnold by Maxim Kazarian and Ricardo              Uribe-Vargas}, PUBLISHER = {Cambridge University Press},   ADDRESS = {Cambridge},      YEAR = {2011},     PAGES = {x+80},      ISBN = {978-0-521-69290-8},   MRCLASS = {37Pxx (11R32 12E20 37A45)},  MRNUMBER = {2777369},}@article {MR1139553,    AUTHOR = {Arnol{\cprime}d, V. I.},     TITLE = {Dynamics of complexity of intersections},   JOURNAL = {Bol. Soc. Brasil. Mat. (N.S.)},  FJOURNAL = {Boletim da Sociedade Brasileira de Matem\'atica. Nova S\'erie},    VOLUME = {21},      YEAR = {1990},    NUMBER = {1},     PAGES = {1--10},      ISSN = {0100-3569},   MRCLASS = {58C27 (53C99 57R99 58F14)},  MRNUMBER = {1139553 (93c:58031)},MRREVIEWER = {J. S. Joel},       DOI = {10.1007/BF01236277},       URL = {http://dx.doi.org/10.1007/BF01236277},}@comment{includes some motivation for why dynamical degree is interesting}@comment {MR1264116}@article {arrowsmith:padicsiegeldisk,    AUTHOR = {Arrowsmith, David K. and Vivaldi, Franco},     TITLE = {Geometry of {$p$}-adic {S}iegel discs},   JOURNAL = {Phys. D},  FJOURNAL = {Physica D. Nonlinear Phenomena},    VOLUME = {71},      YEAR = {1994},    NUMBER = {1-2},     PAGES = {222--236},      ISSN = {0167-2789},     CODEN = {PDNPDT},   MRCLASS = {11S85 (11S31 58F99)},  MRNUMBER = {MR1264116 (95d:11162)},MRREVIEWER = {Evgenii I. Zelenov},}@comment {MR1218804}@article {arrowsmith:padichorseshoe,    AUTHOR = {Arrowsmith, David K. and Vivaldi, Franco},     TITLE = {Some {$p$}-adic representations of the {S}male horseshoe},   JOURNAL = {Phys. Lett. A},  FJOURNAL = {Physics Letters. A},    VOLUME = {176},      YEAR = {1993},    NUMBER = {5},     PAGES = {292--294},      ISSN = {0375-9601},     CODEN = {PYLAAG},   MRCLASS = {58F03 (11Z50 32P05)},  MRNUMBER = {MR1218804 (94c:58054)},MRREVIEWER = {Evgenii I. Zelenov},}@article {MR2017146,    AUTHOR = {Autissier, Pascal},     TITLE = {Hauteur des correspondances de {H}ecke},   JOURNAL = {Bull. Soc. Math. France},  FJOURNAL = {Bulletin de la Soci\'et\'e Math\'ematique de France},    VOLUME = {131},      YEAR = {2003},    NUMBER = {3},     PAGES = {421--433},      ISSN = {0037-9484},     CODEN = {BSMFAA},   MRCLASS = {11G18 (11F32 11G50 14G40)},  MRNUMBER = {MR2017146 (2004i:11058)},MRREVIEWER = {Robin de Jong},}@comment {\url{ArXiv:math.NT/0404355}}@article {autissierpreprint04,    AUTHOR = {Autissier, Pascal},     TITLE = {Dynamique des correspondances alg\'ebriques et hauteurs},   JOURNAL = {Int. Math. Res. Not.},  FJOURNAL = {International Mathematics Research Notices},      YEAR = {2004},    NUMBER = {69},     PAGES = {3723--3739},      ISSN = {1073-7928},   MRCLASS = {14G40 (11G50)},  MRNUMBER = {MR2099499},MRREVIEWER = {Robin de Jong},}@article {MR1760091,    AUTHOR = {Ayad, Mohamed and McQuillan, Donald L.},     TITLE = {Irreducibility of the iterates of a quadratic polynomial over              a field},   JOURNAL = {Acta Arith.},  FJOURNAL = {Acta Arithmetica},    VOLUME = {93},      YEAR = {2000},    NUMBER = {1},     PAGES = {87--97},      ISSN = {0065-1036},     CODEN = {AARIA9},   MRCLASS = {11C08 (11R09 11T06 12E10)},  MRNUMBER = {MR1760091 (2001c:11031)},MRREVIEWER = {Michel Grandcolas},}@article {MR1845367,    AUTHOR = {Ayad, Mohamed and McQuillan, Donald L.},     TITLE = {Corrections to: ``{I}rreducibility of the iterates of a              quadratic polynomial over a field'' [{A}cta {A}rith.\ \textbf{93}              (2000), no. 1, 87--97]},   JOURNAL = {Acta Arith.},  FJOURNAL = {Acta Arithmetica},    VOLUME = {99},      YEAR = {2001},    NUMBER = {1},     PAGES = {97},      ISSN = {0065-1036},     CODEN = {AARIA9},   MRCLASS = {11R09 (11T06 12E10)},  MRNUMBER = {MR1845367 (2002d:11125)},}@article {MR1242609,    AUTHOR = {Ayad, Mohamed},     TITLE = {P\'eriodicit\'e (mod {$q$}) des suites elliptiques et points              {$S$}-entiers sur les courbes elliptiques},   JOURNAL = {Ann. Inst. Fourier (Grenoble)},  FJOURNAL = {Universit\'e de Grenoble. Annales de l'Institut Fourier},    VOLUME = {43},      YEAR = {1993},    NUMBER = {3},     PAGES = {585--618},      ISSN = {0373-0956},     CODEN = {AIFUA7},   MRCLASS = {11B50 (11B83 11D25 11G05)},  MRNUMBER = {MR1242609 (94f:11009)},MRREVIEWER = {Franck Lepr{\'e}vost},}@comment{--BBB-- }@incollection {MR0277001,    AUTHOR = {Banchoff, Thomas F. and Rosen, Michael I.},     TITLE = {Periodic points of {A}nosov diffeomorphisms}, BOOKTITLE = {Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley,              Calif., 1968)},     PAGES = {17--21}, PUBLISHER = {Amer. Math. Soc.},   ADDRESS = {Providence, R.I.},      YEAR = {1970},   MRCLASS = {57.48 (34.00)},  MRNUMBER = {MR0277001 (43 \#2738)},MRREVIEWER = {J. W. Robbin},}@article {MR2727656,    AUTHOR = {Bandman, Tatiana and Grunewald, Fritz and Kunyavski{\u\i},              Boris},     TITLE = {Geometry and arithmetic of verbal dynamical systems on simple              groups},      NOTE = {With an appendix by Nathan Jones},   JOURNAL = {Groups Geom. Dyn.},  FJOURNAL = {Groups, Geometry, and Dynamics},    VOLUME = {4},      YEAR = {2010},    NUMBER = {4},     PAGES = {607--655},      ISSN = {1661-7207},   MRCLASS = {37Pxx (14Gxx 20Gxx)},  MRNUMBER = {2727656},       DOI = {10.4171/GGD/98},       URL = {http://dx.doi.org/10.4171/GGD/98},}@article {MR0169989,    AUTHOR = {Baker, I. N.},     TITLE = {Fixpoints of polynomials and rational functions},   JOURNAL = {J. London Math. Soc.},    VOLUME = {39},      YEAR = {1964},     PAGES = {615--622},   MRCLASS = {30.10},  MRNUMBER = {MR0169989 (30 \#230)},MRREVIEWER = {W. J. Thron},}@comment{see also Kisaka for representatives of the exceptional maps, andfor characteristic p, see Pezda}@comment{UPDATE}@misc {arxiv0606252,   AUTHOR = {Baker, Matthew},    TITLE = {Uniform structures and {B}erkovich spaces},     YEAR = {2006},     NOTE = {\url{ArXiv:math.NT/0606252}},}@comment{16 pages ? is this ever going to appear?}@article {arxiv0507484,    AUTHOR = {Baker, Matthew},     TITLE = {A lower bound for average values of dynamical {G}reen's              functions},   JOURNAL = {Math. Res. Lett.},  FJOURNAL = {Mathematical Research Letters},    VOLUME = {13},      YEAR = {2006},    NUMBER = {2-3},     PAGES = {245--257},      ISSN = {1073-2780},   MRCLASS = {14Gxx (11Gxx)},  MRNUMBER = {MR2231115},}@comment{\url{ArXiv:math.NT/0507484}}@comment{UPDATE}@article {arxiv0601046,    AUTHOR = {Baker, Matthew},     TITLE = {A finiteness theorem for canonical heights attached to              rational maps over function fields},   JOURNAL = {J. Reine Angew. Math.},  FJOURNAL = {Journal f\"ur die Reine und Angewandte Mathematik. [Crelle's              Journal]},    VOLUME = {626},      YEAR = {2009},     PAGES = {205--233},      ISSN = {0075-4102},     CODEN = {JRMAA8},   MRCLASS = {14Gxx (11G50 11R58)},  MRNUMBER = {MR2492995},}@comment{UPDATE}@misc {arxiv0911.0918,    AUTHOR = {Baker, Matthew and DeMarco, Laura},    TITLE = {Preperiodic points and unlikely intersections},     YEAR = {2009},     NOTE = {\url{arXiv:0911.0918}},}@misc {arxiv0305181,   AUTHOR = {Baker, Matthew and Hsia, Liang-Chung},     TITLE = {Canonical heights, transfinite diameters, and polynomial              dynamics},   JOURNAL = {J. Reine Angew. Math.},  FJOURNAL = {Journal f\"ur die Reine und Angewandte Mathematik},    VOLUME = {585},      YEAR = {2005},     PAGES = {61--92},      ISSN = {0075-4102},     CODEN = {JRMAA8},   MRCLASS = {11G50 (37A45 37F10)},  MRNUMBER = {MR2164622 (2006i:11071)},MRREVIEWER = {Clayton Petsche},}@comment{ArXiv:math.NT/0305181, 28 pages}@article {MR2114791,    AUTHOR = {Baker, Matthew and Ih, Su-ion},     TITLE = {Equidistribution of small subvarieties of an abelian variety},   JOURNAL = {New York J. Math.},  FJOURNAL = {New York Journal of Mathematics},    VOLUME = {10},      YEAR = {2004},     PAGES = {279--285 (electronic)},      ISSN = {1076-9803},   MRCLASS = {11G10 (11G35 14K12 14K15)},  MRNUMBER = {MR2114791 (2005h:11116)},MRREVIEWER = {Robin de Jong},}@article {arxiv0509485,    AUTHOR = {Baker, Matthew and Ih, Su-ion and Rumely, Robert},     TITLE = {A finiteness property of torsion points},   JOURNAL = {Algebra Number Theory},  FJOURNAL = {Algebra \& Number Theory},    VOLUME = {2},      YEAR = {2008},    NUMBER = {2},     PAGES = {217--248},      ISSN = {1937-0652},   MRCLASS = {11G50 (11G05 11J71 11J86)},  MRNUMBER = {MR2377370 (2009g:11078)},MRREVIEWER = {Clayton Petsche},}@comment{UPDATE}@misc {arxiv0407433,   AUTHOR = {Baker, Matthew and Rumely, Robert},    TITLE = {Analysis and dynamics on the {B}erkovich projective line},     YEAR = {2004},     NOTE = {\url{ArXiv:math.NT/0407433}},}@comment{150 pages, these notes have been subsumed by their book}@book {bakerrumelyberkbook,    AUTHOR = {Baker, Matthew and Rumely, Robert},     TITLE = {Potential theory and dynamics on the {B}erkovich projective              line},    SERIES = {Mathematical Surveys and Monographs},    VOLUME = {159}, PUBLISHER = {American Mathematical Society},   ADDRESS = {Providence, RI},      YEAR = {2010},     PAGES = {xxxiv+428},      ISBN = {978-0-8218-4924-8},   MRCLASS = {32H50 (14G20 14G22 20F65 31C15 31C45 37Bxx 37Pxx)},  MRNUMBER = {MR2599526},}@comment{UPDATE}@misc {bakerrumelymontel2005,   AUTHOR = {Baker, Matthew and Rumely, Robert},    TITLE = {Montel's theorem for the {B}erkovich projective line and the                 {B}erkovich {J}ulia set of a rational map},     YEAR = {2005},     NOTE = {Preprint},}@comment{31 pages ? is this ever going to appear?}@article {arxiv0407427,    AUTHOR = {Baker, Matt and Rumely, Robert},     TITLE = {Harmonic analysis on metrized graphs},   JOURNAL = {Canad. J. Math.},  FJOURNAL = {Canadian Journal of Mathematics. Journal Canadien de              Math\'ematiques},    VOLUME = {59},      YEAR = {2007},    NUMBER = {2},     PAGES = {225--275},      ISSN = {0008-414X},     CODEN = {CJMAAB},   MRCLASS = {43A99 (05C12 58C40)},  MRNUMBER = {MR2310616 (2008h:43019)},MRREVIEWER = {Charles Favre},}@article {arxiv0407426,   AUTHOR = {Baker, Matthew and Rumely, Robert},     TITLE = {Equidistribution of small points, rational dynamics, and              potential theory},   JOURNAL = {Ann. Inst. Fourier (Grenoble)},  FJOURNAL = {Universit\'e de Grenoble. Annales de l'Institut Fourier},    VOLUME = {56},      YEAR = {2006},    NUMBER = {3},     PAGES = {625--688},      ISSN = {0373-0956},     CODEN = {AIFUA7},   MRCLASS = {11G50 (37A45 37F10)},  MRNUMBER = {MR2244226}, }@comment{ArXiv:math.NT/0407426, 53 pages}@article {baragarluijk06,    AUTHOR = {Baragar, Arthur and van Luijk, Ronald},     TITLE = {{$K3$} surfaces with {P}icard number three and canonical              vector heights},   JOURNAL = {Math. Comp.},  FJOURNAL = {Mathematics of Computation},    VOLUME = {76},      YEAR = {2007},    NUMBER = {259},     PAGES = {1493--1498 (electronic)},      ISSN = {0025-5718},     CODEN = {MCMPAF},   MRCLASS = {14J28 (11G50 14C22 14G10 14J10)},  MRNUMBER = {MR2299785 (2008e:14055)},MRREVIEWER = {Matthias Sch{\"u}tt},}@article {MR2059748,    AUTHOR = {Baragar, Arthur},     TITLE = {Canonical vector heights on {$K3$} surfaces with {P}icard              number three---an argument for nonexistence},   JOURNAL = {Math. Comp.},  FJOURNAL = {Mathematics of Computation},    VOLUME = {73},      YEAR = {2004},    NUMBER = {248},     PAGES = {2019--2025 (electronic)},      ISSN = {0025-5718},     CODEN = {MCMPAF},   MRCLASS = {14J28 (11G50 14G25)},  MRNUMBER = {MR2059748 (2005e:14058)},MRREVIEWER = {Yuri Tschinkel},}@article {MR2011389,    AUTHOR = {Baragar, Arthur},     TITLE = {Canonical vector heights on algebraic {$K3$} surfaces with              {P}icard number two},   JOURNAL = {Canad. Math. Bull.},  FJOURNAL = {Canadian Mathematical Bulletin. 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Appl.},    VOLUME = {132},     PAGES = {245--251}, PUBLISHER = {Birkh\"auser},   ADDRESS = {Basel},      YEAR = {2002},   MRCLASS = {37E99 (11S85 37A99)},  MRNUMBER = {MR1924983 (2003g:37083)},}@incollection {MR1830516,    AUTHOR = {Khrennikov, Andrei},     TITLE = {Ergodic and non-ergodic behaviour for dynamical systems in              rings of {$p$}-adic integers}, BOOKTITLE = {Fourth Italian-Latin American Conference on Applied and              Industrial Mathematics (Havana, 2001)},     PAGES = {404--409}, PUBLISHER = {Inst. Cybern. Math. Phys., Havana},      YEAR = {2001},   MRCLASS = {37A45 (11S80 37A05)},  MRNUMBER = {MR1830516 (2002b:37008)},MRREVIEWER = {Anatoly N. 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Syst.},  FJOURNAL = {Discrete and Continuous Dynamical Systems. Series A},    VOLUME = {30},      YEAR = {2011},    NUMBER = {1},     PAGES = {209--218},      ISSN = {1078-0947},   MRCLASS = {37Pxx (11S85 37Axx)},  MRNUMBER = {2773138},       DOI = {10.3934/dcds.2011.30.209},       URL = {http://dx.doi.org/10.3934/dcds.2011.30.209},}@article {MR1326795,    AUTHOR = {Kisaka, Masashi},     TITLE = {On some exceptional rational maps},   JOURNAL = {Proc. Japan Acad. Ser. A Math. Sci.},  FJOURNAL = {Japan Academy. Proceedings. Series A. Mathematical Sciences},    VOLUME = {71},      YEAR = {1995},    NUMBER = {2},     PAGES = {35--38},      ISSN = {0386-2194},     CODEN = {PJAADT},   MRCLASS = {30D05 (58F23)},  MRNUMBER = {MR1326795 (96a:30029)},MRREVIEWER = {I. N. Baker},}@comment{gives conjugacy classes of maps for I.N. Baker's descriptionof rational maps lacking a complex periodic point of some order}@article {kiwi2006,    AUTHOR = {Kiwi, Jan},     TITLE = {Puiseux series polynomial dynamics and iteration of complex cubic                     polynomials},   JOURNAL = {Ann. Inst. Fourier (Grenoble)},  FJOURNAL = {Universit\'e de Grenoble. 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Baker},       URL = {http://www.numdam.org/item?id=ASENS_1983_4_16_2_193_0},}@phdthesis{manesthesis,     AUTHOR = {Manes, Michelle},     TITLE = {Arithmetic Dynamics and Moduli Spaces of Rational Maps},   SCHOOL = {Brown University},     YEAR = {2007},}@article {MR2407816,    AUTHOR = {Manes, Michelle},     TITLE = {{$\mathbb{Q}$}-rational cycles for degree-2 rational maps having              an automorphism},   JOURNAL = {Proc. Lond. Math. Soc. (3)},  FJOURNAL = {Proceedings of the London Mathematical Society. Third Series},    VOLUME = {96},      YEAR = {2008},    NUMBER = {3},     PAGES = {669--696},      ISSN = {0024-6115},   MRCLASS = {14G25 (11G99)},  MRNUMBER = {MR2407816 (2009a:14029)},MRREVIEWER = {Dipendra Prasad},}@article {MR2524186,    AUTHOR = {Manes, Michelle},     TITLE = {Moduli spaces for families of rational maps on {$\mathbb{P}^1$}},   JOURNAL = {J. Number Theory},  FJOURNAL = {Journal of Number Theory},    VOLUME = {129},      YEAR = {2009},    NUMBER = {7},     PAGES = {1623--1663},      ISSN = {0022-314X},     CODEN = {JNUTA9},   MRCLASS = {14D20 (11C08 11F03 11G99)},  MRNUMBER = {MR2524186},       DOI = {10.1016/j.jnt.2009.02.010},       URL = {http://dx.doi.org/10.1016/j.jnt.2009.02.010},}@article {manesyasu2011,    AUTHOR = {Manes, Michelle and Yasufuku, Yu},     TITLE = {Explicit descriptions of quadratic maps on $\mathbb{P}^1$},   JOURNAL = {Acta Arith.},  FJOURNAL = {Acta Arithmetica},    VOLUME = {148},      YEAR = {2011},    NUMBER = {3},     PAGES = {257--267},}@phdthesis{marcellothesis,     AUTHOR = {Marcello, Sandra},     TITLE = {Sur la dynamique arithm\'etique des automorphismes affines},   SCHOOL = {Universit\'e Paris 7},     YEAR = {2000},}@article {marcello:CR2000,    AUTHOR = {Marcello, Sandra},     TITLE = {Sur les propri\'et\'es arithm\'etiques des it\'er\'es              d'auto\-mor\-phismes r\'eguliers},   JOURNAL = {C.~R.~Acad. Sci. Paris S\'er. I Math.},  FJOURNAL = {Comptes Rendus de l'Acad\'emie des Sciences. S\'erie I.              Math\'ematique},    VOLUME = {331},      YEAR = {2000},    NUMBER = {1},     PAGES = {11--16},      ISSN = {0764-4442},     CODEN = {CASMEI},   MRCLASS = {11G50},  MRNUMBER = {2001d:11072},}@article {MR1988948,    AUTHOR = {Marcello, Sandra},     TITLE = {Sur la dynamique arithm\'etique des automorphismes de l'espace              affine},   JOURNAL = {Bull. Soc. Math. France},  FJOURNAL = {Bulletin de la Soci\'et\'e Math\'ematique de France},    VOLUME = {131},      YEAR = {2003},    NUMBER = {2},     PAGES = {229--257},      ISSN = {0037-9484},     CODEN = {BSMFAA},   MRCLASS = {11G50 (32M17 37F10)},  MRNUMBER = {MR1988948 (2004d:11053)},MRREVIEWER = {Yuri Tschinkel},}@misc {arxiv0310434,    AUTHOR = {Marcello, Sandra},    TITLE = {G\'eom\'etrie, points rationnels et it\'er\'es                    des automorphismes de l'espace affine},     YEAR = {2003},     NOTE = {\url{ArXiv:math.NT/0310434}},}@misc {arxiv0310454,    AUTHOR = {Marcello, Sandra},    TITLE = {Sur des invariants g\'eom\'etriques associ\'es                   aux automorphismes du plan affine},     YEAR = {2003},     NOTE = {\url{ArXiv:math.AG/0310454}},}@misc {arxiv0310417,    AUTHOR = {Marcello, Sandra},    TITLE = {Sur la dynamique p-adique arithm\'etique des                  automorphismes de l'espace affine},     YEAR = {2003},     NOTE = {\url{ArXiv:math.NT/0310417}},}@comment{?? the preceding three arxiv articles by Marcello are unlikelyto ever be published}@article {masser:lehmer,    AUTHOR = {Masser, D. W.},     TITLE = {Counting points of small height on elliptic curves},   JOURNAL = {Bull. Soc. Math. France},  FJOURNAL = {Bulletin de la Soci\'et\'e Math\'ematique de France},    VOLUME = {117},      YEAR = {1989},    NUMBER = {2},     PAGES = {247--265},      ISSN = {0037-9484},     CODEN = {BSMFAA},   MRCLASS = {11G05 (11D25)},  MRNUMBER = {MR1015810 (90k:11068)},MRREVIEWER = {Michel Laurent},}@article {MR2412783,    AUTHOR = {Masser, David and Zannier, Umberto},     TITLE = {Torsion anomalous points and families of elliptic curves},   JOURNAL = {C. R. Math. Acad. Sci. Paris},  FJOURNAL = {Comptes Rendus Math\'ematique. Acad\'emie des Sciences. 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Publications              Math\'ematiques},    NUMBER = {47},      YEAR = {1977},     PAGES = {33--186 (1978)},      ISSN = {0073-8301},     CODEN = {PMIHA6},   MRCLASS = {14G25 (10D05)},  MRNUMBER = {MR488287 (80c:14015)},MRREVIEWER = {M. Ohta},}@incollection {mcmillan:integmap,    AUTHOR = {McMillan, E. M.},     TITLE = {A problem in the stability of periodic systems}, BOOKTITLE = {Topics in Modern Physics: A Tribute to E. U. Condon},     PAGES = {219--244}, PUBLISHER = {Colorado Assoc. Univ. Press},   ADDRESS = {Boulder, CO},      YEAR = {1971},}@comment{The first integrable map of the plane was introduced byE. M. McMillan \ref[in Topics in modern physics: a tribute toE. U. Condon, 219--244, Colorado Assoc. Univ. Press, Boulder, CO,1971], and contained 6 arbitrary parameters. This map was generalizedto an 18-parameter map, that is now commonly called the QRT map.}@article{mcmullen:rootfinding,     AUTHOR = {McMullen, Curtis T.},     TITLE = {Families of rational maps and iterative root-finding              algorithms},   JOURNAL = {Ann. of Math. (2)},  FJOURNAL = {Annals of Mathematics. Second Series},    VOLUME = {125},      YEAR = {1987},    NUMBER = {3},     PAGES = {467--493},      ISSN = {0003-486X},     CODEN = {ANMAAH},   MRCLASS = {58F08 (58C15 65H05)},  MRNUMBER = {MR890160 (88i:58082)},MRREVIEWER = {Michael Hurley},}@book {MR1312365,    AUTHOR = {McMullen, Curtis T.},     TITLE = {Complex dynamics and renormalization},    SERIES = {Annals of Mathematics Studies},    VOLUME = {135}, PUBLISHER = {Princeton University Press},   ADDRESS = {Princeton, NJ},      YEAR = {1994},     PAGES = {x+214},      ISBN = {0-691-02982-2; 0-691-02981-4},   MRCLASS = {58F23 (30D05)},  MRNUMBER = {1312365 (96b:58097)},MRREVIEWER = {Gregery T. Buzzard},}@article {MR1896103,    AUTHOR = {McMullen, Curtis T.},     TITLE = {Dynamics on {$K3$} surfaces: {S}alem numbers and {S}iegel              disks},   JOURNAL = {J. Reine Angew. Math.},  FJOURNAL = {Journal f\"ur die Reine und Angewandte Mathematik},    VOLUME = {545},      YEAR = {2002},     PAGES = {201--233},      ISSN = {0075-4102},     CODEN = {JRMAA8},   MRCLASS = {37F25 (11F06 32H50 32J15 32Q15 37C40)},  MRNUMBER = {MR1896103 (2003a:37057)},MRREVIEWER = {Serge Cantat},}@article {MR1713286,    AUTHOR = {McMullen, Curtis T.},     TITLE = {From dynamics on surfaces to rational points on curves},   JOURNAL = {Bull. Amer. Math. Soc. (N.S.)},  FJOURNAL = {American Mathematical Society. Bulletin. New Series},    VOLUME = {37},      YEAR = {2000},    NUMBER = {2},     PAGES = {119--140},      ISSN = {0273-0979},     CODEN = {BAMOAD},   MRCLASS = {11G30 (32G15 37E30 57M60)},  MRNUMBER = {MR1713286 (2000j:11093)},MRREVIEWER = {Joseph H. Silverman},}@article {MR1620850,    AUTHOR = {McMullen, Curtis T. and Sullivan, Dennis P.},     TITLE = {Quasiconformal homeomorphisms and dynamics. {III}. {T}he              {T}eichm\"uller space of a holomorphic dynamical system},   JOURNAL = {Adv. Math.},  FJOURNAL = {Advances in Mathematics},    VOLUME = {135},      YEAR = {1998},    NUMBER = {2},     PAGES = {351--395},      ISSN = {0001-8708},     CODEN = {ADMTA4},   MRCLASS = {58F23 (30C62 30D05 30F60)},  MRNUMBER = {MR1620850 (99e:58145)},MRREVIEWER = {Athanase Papadopoulos},}@comment{UPDATE}@misc {arxiv0901.2352,    AUTHOR = {Medvedev, Alice and Scanlon, Thomas},    TITLE = {Polynomial dynamics},     YEAR = {2009},     NOTE = {\url{arXiv:0901.2352}},}@article{merel:uniformbound,     AUTHOR = {Merel, Lo{\"{\i}}c},     TITLE = {Bornes pour la torsion des courbes elliptiques sur les corps              de nombres},   JOURNAL = {Invent. 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Horrocks},}@article{milnor:quadraticmaps,     AUTHOR = {Milnor, John},     TITLE = {Geometry and dynamics of quadratic rational maps},      NOTE = {With an appendix by the author and Lei Tan},   JOURNAL = {Experiment. Math.},  FJOURNAL = {Experimental Mathematics},    VOLUME = {2},      YEAR = {1993},    NUMBER = {1},     PAGES = {37--83},      ISSN = {1058-6458},   MRCLASS = {58F20 (30D05 32S70 58F23)},  MRNUMBER = {MR1246482 (96b:58094)},MRREVIEWER = {J. S. Joel},}@book{milnor:dynamicslectures,     AUTHOR = {Milnor, John},     TITLE = {Dynamics in {O}ne {C}omplex {V}ariable}, PUBLISHER = {Friedr. Vieweg \& Sohn},   ADDRESS = {Braunschweig},      YEAR = {1999},     PAGES = {viii+257},      ISBN = {3-528-03130-1},   MRCLASS = {37Fxx (30D05 37-01)},  MRNUMBER = {MR1721240 (2002i:37057)},MRREVIEWER = {Dierk Schleicher},}@article {MR1806289,    AUTHOR = {Milnor, John},     TITLE = {On rational maps with two critical points},   JOURNAL = {Experiment. Math.},  FJOURNAL = {Experimental Mathematics},    VOLUME = {9},      YEAR = {2000},    NUMBER = {4},     PAGES = {481--522},      ISSN = {1058-6458},   MRCLASS = {37F50 (37F20)},  MRNUMBER = {MR1806289 (2001k:37074)},MRREVIEWER = {Peter Ha{\"{\i}}ssinsky},}@incollection {milnor:lattesmaps,    AUTHOR = {Milnor, John},     TITLE = {On {L}att\`es maps}, BOOKTITLE = {Dynamics on the {R}iemann sphere},     PAGES = {9--43}, PUBLISHER = {Eur. Math. Soc., Z\"urich},      YEAR = {2006},   MRCLASS = {37F10 (30D05 37-02)},  MRNUMBER = {MR2348953},MRREVIEWER = {Feliks Przytycki},     NOTE = {\url{ArXiv:math.DS/0402147},             Stony Brook IMS Preprint \#2004/01},}@incollection {MR2508263,    AUTHOR = {Milnor, John},     TITLE = {Cubic polynomial maps with periodic critical orbit. {I}}, BOOKTITLE = {Complex dynamics},     PAGES = {333--411}, PUBLISHER = {A K Peters},   ADDRESS = {Wellesley, MA},      YEAR = {2009},   MRCLASS = {37Fxx (30D05 32Hxx)},  MRNUMBER = {2508263},}@phdthesis{mimarthesis1997,     AUTHOR = {Mimar, A.},     TITLE = {On the preperiodic points of an endomorphism of $\mathbb{P}^1          \times\mathbb{P}^1$ which lie on a curve},   SCHOOL = {Columbia University},     YEAR = {19997},}@comment{UPDATE}@comment{Open mapping theorem over any valued field}@misc {arxiv1010.0341,    AUTHOR = {Moret-Bailly, Laurent},    TITLE = {Un th\'eor\`eme de l'application ouverte sur les corps                  valu\'es alg\'ebriquement clos},     YEAR = {2010},     NOTE = {\url{arXiv:1010.0341}},}@book {MR2158954,    AUTHOR = {Morris, Dave Witte},     TITLE = {Ratner's {T}heorems on {U}nipotent {F}lows},    SERIES = {Chicago Lectures in Mathematics}, PUBLISHER = {University of Chicago Press},   ADDRESS = {Chicago, IL},      YEAR = {2005},     PAGES = {xii+203},      ISBN = {0-226-53983-0; 0-226-53984-9},   MRCLASS = {37A17 (22E40 22F30 37A25 37A30 37A35)},  MRNUMBER = {MR2158954 (2006h:37006)},MRREVIEWER = {Bachir Bekka},}@article {morton:quadraticmaps,    AUTHOR = {Morton, Patrick},     TITLE = {Arithmetic properties of periodic points of quadratic maps},   JOURNAL = {Acta Arith.},  FJOURNAL = {Acta Arithmetica},    VOLUME = {62},      YEAR = {1992},    NUMBER = {4},     PAGES = {343--372},      ISSN = {0065-1036},     CODEN = {AARIA9},   MRCLASS = {12E10 (12F10)},  MRNUMBER = {MR1199627 (93k:12004)},MRREVIEWER = {R. W. K. Odoni},}@article {morton:quadraticmaps2,    AUTHOR = {Morton, Patrick},     TITLE = {Arithmetic properties of periodic points of quadratic maps.              {II}},   JOURNAL = {Acta Arith.},  FJOURNAL = {Acta Arithmetica},    VOLUME = {87},      YEAR = {1998},    NUMBER = {2},     PAGES = {89--102},      ISSN = {0065-1036},     CODEN = {AARIA9},   MRCLASS = {11G30 (11F03 14G05)},  MRNUMBER = {MR1665198 (2000c:11103)},MRREVIEWER = {Nobuhiko Ishida},}@article {morton:galoisperiodicpoints,    AUTHOR = {Morton, Patrick},     TITLE = {Galois groups of periodic points},   JOURNAL = {J. Algebra},  FJOURNAL = {Journal of Algebra},    VOLUME = {201},      YEAR = {1998},    NUMBER = {2},     PAGES = {401--428},      ISSN = {0021-8693},     CODEN = {JALGA4},   MRCLASS = {12F10},  MRNUMBER = {MR1612390 (99a:12005)},MRREVIEWER = {Tara L. Smith},}@article {morton:finitefielddynamics,    AUTHOR = {Morton, Patrick},     TITLE = {Periods of maps on irreducible polynomials over finite fields},   JOURNAL = {Finite Fields Appl.},  FJOURNAL = {Finite Fields and their Applications},    VOLUME = {3},      YEAR = {1997},    NUMBER = {1},     PAGES = {11--24},      ISSN = {1071-5797},   MRCLASS = {11G99 (11R58)},  MRNUMBER = {MR1429040 (98d:11075)},MRREVIEWER = {Joseph H. Silverman},}@article {morton:dynamicalmoduli,    AUTHOR = {Morton, Patrick},     TITLE = {On certain algebraic curves related to polynomial maps},   JOURNAL = {Compositio Math.},  FJOURNAL = {Compositio Mathematica},    VOLUME = {103},      YEAR = {1996},    NUMBER = {3},     PAGES = {319--350},      ISSN = {0010-437X},     CODEN = {CMPMAF},   MRCLASS = {14H10 (11G99 11R32 58F03)},  MRNUMBER = {MR1414593 (97m:14030)},MRREVIEWER = {Joseph H. Silverman},}@article {MR2771135,    AUTHOR = {Morton, Patrick},     TITLE = {Corrigendum: `{\it {O}n certain algebraic curves related to              polynomial maps}, {C}ompositio {M}ath. 103 (1996), 319--350'},   JOURNAL = {Compos. Math.},  FJOURNAL = {Compositio Mathematica},    VOLUME = {147},      YEAR = {2011},    NUMBER = {1},     PAGES = {332--334},      ISSN = {0010-437X},   MRCLASS = {14H10 (11G99 11R32 37Pxx)},  MRNUMBER = {2771135},       DOI = {10.1112/S0010437X1000480X},       URL = {http://dx.doi.org/10.1112/S0010437X1000480X},}@article {morton:cubicextensions,    AUTHOR = {Morton, Patrick},     TITLE = {Characterizing cyclic cubic extensions by automorphism              polynomials},   JOURNAL = {J. Number Theory},  FJOURNAL = {Journal of Number Theory},    VOLUME = {49},      YEAR = {1994},    NUMBER = {2},     PAGES = {183--208},      ISSN = {0022-314X},     CODEN = {JNUTA9},   MRCLASS = {12F10 (11R16)},  MRNUMBER = {MR1305089 (95j:12006)},MRREVIEWER = {Joseph H. Silverman},}@article {mortonpatel:galoisperiodicpoints,    AUTHOR = {Morton, Patrick and Patel, Pratiksha},     TITLE = {The {G}alois theory of periodic points of polynomial maps},   JOURNAL = {Proc. London Math. Soc. (3)},  FJOURNAL = {Proceedings of the London Mathematical Society. Third Series},    VOLUME = {68},      YEAR = {1994},    NUMBER = {2},     PAGES = {225--263},      ISSN = {0024-6115},     CODEN = {PLMTAL},   MRCLASS = {11R32 (39B12)},  MRNUMBER = {MR1253503 (94i:11090)},MRREVIEWER = {Joseph H. Silverman},}@article {mortonsilverman:rationalperiodicpoints,    AUTHOR = {Morton, Patrick and Silverman, Joseph H.},     TITLE = {Rational periodic points of rational functions},   JOURNAL = {Internat. Math. Res. 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Math.},  FJOURNAL = {Journal f\"ur die Reine und Angewandte Mathematik},    VOLUME = {461},      YEAR = {1995},     PAGES = {81--122},      ISSN = {0075-4102},     CODEN = {JRMAA8},   MRCLASS = {11G99},  MRNUMBER = {MR1324210 (96b:11090)},MRREVIEWER = {Jeffrey Lin Thunder},}@article {mortonvivaldi:bifurcations,    AUTHOR = {Morton, Patrick and Vivaldi, Franco},     TITLE = {Bifurcations and discriminants for polynomial maps},   JOURNAL = {Nonlinearity},  FJOURNAL = {Nonlinearity},    VOLUME = {8},      YEAR = {1995},    NUMBER = {4},     PAGES = {571--584},      ISSN = {0951-7715},     CODEN = {NONLE5},   MRCLASS = {11C08 (11R21 30D05 58F14 58F23)},  MRNUMBER = {MR1342504 (96k:11028)},MRREVIEWER = {Takehiko Morita},}@article {moussa:integersinjuliasets,     AUTHOR = {Moussa, Pierre and Geronimo, Jeffrey S. and Bessis, Daniel},     TITLE = {Ensembles de {J}ulia et propri\'et\'es de localisation des              familles it\'er\'ees d'entiers alg\'ebriques},   JOURNAL = {C.~R.~Acad. Sci. Paris S\'er. I Math.},  FJOURNAL = {Comptes Rendus des S\'eances de l'Acad\'emie des Sciences.              S\'erie I. Math\'ematique},    VOLUME = {299},      YEAR = {1984},    NUMBER = {8},     PAGES = {281--284},      ISSN = {0249-6291},     CODEN = {CASMEI},   MRCLASS = {58F22 (11R06)},  MRNUMBER = {MR761246 (86f:58133)},MRREVIEWER = {M. Rees},}@comment{ The authors prove that algebraic integers in the filledJulia set are preperiodic. They formulate a Lehmer-type conjecture forGreen's functions associated to polynomials.  }@incollection {MR848378,    AUTHOR = {Moussa, Pierre},     TITLE = {Ensembles de {J}ulia et propri\'et\'es de localisation des              entiers alg\'ebriques}, BOOKTITLE = {Seminar on {N}umber {T}heory, 1984--1985 (Talence, 1984/1985)},     PAGES = {Exp.\ No.\ 21, 10}, PUBLISHER = {Univ. Bordeaux I},   ADDRESS = {Talence},      YEAR = {1985},   MRCLASS = {11R04 (11J68 30D05 58F08)},  MRNUMBER = {MR848378 (87i:11141)},MRREVIEWER = {I. N. Baker},}@incollection {moussa:juliasets,     AUTHOR = {Moussa, Pierre},     TITLE = {Diophantine properties of {J}ulia sets}, BOOKTITLE = {Chaotic {D}ynamics and {F}ractals (Atlanta, Ga., 1985)},    SERIES = {Notes Rep. Math. Sci. Engrg.},    VOLUME = {2},     PAGES = {215--227}, PUBLISHER = {Academic Press},   ADDRESS = {Orlando, FL},      YEAR = {1986},   MRCLASS = {58F11 (30D05)},  MRNUMBER = {MR858017},}@article {MR2067949,    AUTHOR = {Mukhamedov, Farruh and Rozikov, Utkir},     TITLE = {On rational {$p$}-adic dynamical systems},   JOURNAL = {Methods Funct. Anal. Topology},  FJOURNAL = {Methods of Functional Analysis and Topology},    VOLUME = {10},      YEAR = {2004},    NUMBER = {2},     PAGES = {21--31},      ISSN = {1029-3531},   MRCLASS = {37F10 (11S85 30D05 30G06 54H20)},  MRNUMBER = {MR2067949 (2005b:37091)},}@comment{Detailed analysis of dynamics of (x+a)/(bx+c)}@article {MR2423943,    AUTHOR = {Mukhamedov, F.},     TITLE = {On the chaotic behavior of a cubic {$p$}-adic dynamical              system},   JOURNAL = {Mat. Zametki},  FJOURNAL = {Rossi\u\i skaya Akademiya Nauk. Matematicheskie Zametki},    VOLUME = {83},      YEAR = {2008},    NUMBER = {3},     PAGES = {468--471},      ISSN = {0025-567X},   MRCLASS = {37E99 (11S80 11S85 37B25 37F10)},  MRNUMBER = {MR2423943 (2009c:37038)},MRREVIEWER = {Anatoly N. Kochubei},}@comment{p-adic dynamics of x^3+ax^3}@comment{UPDATE}@misc {arxiv:0712.4049,    AUTHOR = {Mukhamedov, Farrukh and Rozikov, Utkir},    TITLE = {On one polynomial $p$-adic dynamical system},     YEAR = {2007}, LANGUAGE = {Russian},     NOTE = {\url{ arXiv:0712.4049}},}@incollection {arxiv0608573,    AUTHOR = {Mukhamedov, Farrukh and Mendes, Jos{\'e} F. F.},     TITLE = {On chaos of a cubic {$p$}-adic dynamical system}, BOOKTITLE = {Differential equations, chaos and variational problems},    SERIES = {Progr. Nonlinear Differential Equations Appl.},    VOLUME = {75},     PAGES = {305--315}, PUBLISHER = {Birkh\"auser},   ADDRESS = {Basel},      YEAR = {2008},   MRCLASS = {37Bxx (12Jxx 37Exx 54H20)},  MRNUMBER = {MR2409113},}@article {arxiv0702697,    AUTHOR = {Mukhamedov, Farrukh and Mendes, Jos{\'e} F. F.},     TITLE = {On the chaotic behavior of a generalized logistic {$p$}-adic              dynamical system},   JOURNAL = {J. 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Math.},  FJOURNAL = {Inventiones Mathematicae},    VOLUME = {3},      YEAR = {1967},     PAGES = {215--244},      ISSN = {0020-9910},   MRCLASS = {14.51},  MRNUMBER = {MR0219542 (36 \#2622)},MRREVIEWER = {J. W. S. Cassels},}@article {MR0219541,    AUTHOR = {Mumford, D.},     TITLE = {On the equations defining abelian varieties. {II}},   JOURNAL = {Invent. Math.},  FJOURNAL = {Inventiones Mathematicae},    VOLUME = {3},      YEAR = {1967},     PAGES = {75--135},      ISSN = {0020-9910},   MRCLASS = {14.51},  MRNUMBER = {MR0219541 (36 \#2621)},MRREVIEWER = {J. W. S. Cassels},}@article {MR0204427,    AUTHOR = {Mumford, D.},     TITLE = {On the equations defining abelian varieties. {I}},   JOURNAL = {Invent. Math.},  FJOURNAL = {Inventiones Mathematicae},    VOLUME = {1},      YEAR = {1966},     PAGES = {287--354},      ISSN = {0020-9910},   MRCLASS = {14.51},  MRNUMBER = {MR0204427 (34 \#4269)},MRREVIEWER = {J. W. S. 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Popp},}@book {mumford:geometricinvarianttheory,     AUTHOR = {Mumford, David and Fogarty, John and Kirwan, Frances},     TITLE = {Geometric {I}nvariant {T}heory},    SERIES = {Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results              in Mathematics and Related Areas (2)]},    VOLUME = {34},   EDITION = {Third}, PUBLISHER = {Springer-Verlag},   ADDRESS = {Berlin},      YEAR = {1994},     PAGES = {xiv+292},      ISBN = {3-540-56963-4},   MRCLASS = {14D25 (58E05 58F05)},  MRNUMBER = {MR1304906 (95m:14012)},MRREVIEWER = {Yi Hu},}@book {MR1803093,    AUTHOR = {Murty, M. Ram},     TITLE = {Problems in {A}nalytic {N}umber {T}heory},    SERIES = {Graduate Texts in Mathematics},    VOLUME = {206}, PUBLISHER = {Springer-Verlag},   ADDRESS = {New York},      YEAR = {2001},     PAGES = {xvi+452},      ISBN = {0-387-95143-1},   MRCLASS = {11-01 (11M06 11N05)},  MRNUMBER = {MR1803093 (2001k:11002)},MRREVIEWER = {D. R. Heath-Brown},}@comment{--NNN-- }@article {narkpreprint03,    AUTHOR = {Narkiewicz, W. and Marszalek, Roman},    TITLE = {Finite polynomial orbits in quadratic rings},   JOURNAL = {J. Ramanujan Math. Soc.},  FJOURNAL = {Journal of the Ramanujan Mathematical Society},    VOLUME = {12},      YEAR = {2006},    NUMBER = {1},     PAGES = {91--130},}@article {narkiewiczcubic,    AUTHOR = {Narkiewicz, W.},       TITLE = {Polynomial cycles in cubic fields of negative discriminant},   JOURNAL = {Funct. Approx. Comment. Math.},  FJOURNAL = {Uniwersytet im. Adama Mickiewicza w Poznaniu. Wydzia\l\              Matematyki i Informatyki. Functiones et Approximatio              Commentarii Mathematici},    VOLUME = {35},      YEAR = {2006},     PAGES = {261--270},}@article {MR2040692,    AUTHOR = {Narkiewicz, W.},     TITLE = {Polynomial cycles in certain rings of rationals},   JOURNAL = {J. Th\'eor. Nombres Bordeaux},  FJOURNAL = {Journal de Th\'eorie des Nombres de Bordeaux},    VOLUME = {14},      YEAR = {2002},    NUMBER = {2},     PAGES = {529--552},      ISSN = {1246-7405},   MRCLASS = {11C08 (11R09 13A99)},  MRNUMBER = {MR2040692 (2005a:11029)},MRREVIEWER = {Jean-Luc Chabert},}@incollection {MR2001e:11025,    AUTHOR = {Narkiewicz, W.},     TITLE = {Finite polynomial orbits. {A} survey}, BOOKTITLE = {Algebraic {N}umber {T}heory and Diophantine {A}nalysis (Graz, 1998)},     PAGES = {331--338}, PUBLISHER = {de Gruyter},   ADDRESS = {Berlin},      YEAR = {2000},   MRCLASS = {11C08 (12F05)},  MRNUMBER = {2001e:11025},MRREVIEWER = {Mireille Car},}}@article {narkiewicz:survey,    AUTHOR = {Narkiewicz, W.},     TITLE = {Arithmetics of dynamical systems: a survey},      NOTE = {Number theory (Liptovsk\'y J\'an, 1995)},   JOURNAL = {Tatra Mt. Math. 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Second Series},    VOLUME = {82},      YEAR = {1965},     PAGES = {249--331},      ISSN = {0003-486X},   MRCLASS = {14.49 (14.51)},  MRNUMBER = {MR0179173 (31 \#3424)},MRREVIEWER = {M. Nagata},}@unpublished {nevinsrogers:quadmaps,   AUTHOR = {M. Nevins and T.D. Rogers},    TITLE = {Quadratic maps as dynamical systems on the $p$-adic numbers},     YEAR = {March 2000},     NOTE = {unpublished, \url{www.maths.ex.ac.uk/~mwatkins/zeta/nevins.pdf}},}@comment{Nevins indicated that this project was shelved becauseNilsson published a paper with more general versions of the maintheorems.}@book {MR546290,    AUTHOR = {Newstead, P. E.},     TITLE = {Introduction to {M}oduli {P}roblems and {O}rbit {S}paces},    SERIES = {Tata Institute of Fundamental Research Lectures on Mathematics              and Physics},    VOLUME = {51}, PUBLISHER = {Tata Institute of Fundamental Research},   ADDRESS = {Bombay},      YEAR = {1978},     PAGES = {vi+183},      ISBN = {0-387-08851-2},   MRCLASS = {14-02 (14D20)},  MRNUMBER = {MR546290 (81k:14002)},MRREVIEWER = {G. Horrocks},}@incollection {MR2042407,    AUTHOR = {Niederreiter, Harald and Shparlinski, Igor E.},     TITLE = {Dynamical systems generated by rational functions}, BOOKTITLE = {Applied Algebra, Algebraic Algorithms and Error-Correcting              Codes (Toulouse, 2003)},    SERIES = {Lecture Notes in Comput. Sci.},    VOLUME = {2643},     PAGES = {6--17}, PUBLISHER = {Springer},   ADDRESS = {Berlin},      YEAR = {2003},   MRCLASS = {94A45 (11T71 37B10 94-02)},  MRNUMBER = {MR2042407 (2005a:94047)},}@comment{Summary: "We consider dynamical systems generated byiterations of rational functions over finite fields and residue classrings. We present a survey of recent developments and outline severalopen problems."}@article {MR2026728,    AUTHOR = {Nilsson, Marcus},     TITLE = {Fuzzy cycles of {$p$}-adic monomial dynamical systems},   JOURNAL = {Far East J. Dyn. 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J.},  FJOURNAL = {Michigan Mathematical Journal},    VOLUME = {59},      YEAR = {2010},    NUMBER = {3},     PAGES = {505--515},      ISSN = {0026-2285},   MRCLASS = {37Pxx (11S80 30D05 30G06 37Fxx)},  MRNUMBER = {2745750},       DOI = {10.1307/mmj/1291213954},       URL = {http://dx.doi.org/10.1307/mmj/1291213954},}@comment{UPDATE}@misc {arxiv1106.3367,    AUTHOR = {Okuyama, Y{\^u}suke},    TITLE = {Feketeness, equidistribution and critical orbits in                 non-archimedean dynamics},     YEAR = {2011},     NOTE = {\url{arXiv:1106.3367}},}@comment{UPDATE}@misc {arxiv1106.3363,    AUTHOR = {Okuyama, Y{\^u}suke},    TITLE = {Repelling periodic points and logarithmic              equidistribution in non-archimedean dynamics},     YEAR = {2011},     NOTE = {\url{arXiv:1106.3363}},}@incollection {MR1472499,    AUTHOR = {Oort, Frans},     TITLE = {Canonical liftings and dense sets of {CM}-points}, BOOKTITLE = {Arithmetic geometry ({C}ortona, 1994)},    SERIES = {Sympos. Math., XXXVII},     PAGES = {228--234}, PUBLISHER = {Cambridge Univ. Press},   ADDRESS = {Cambridge},      YEAR = {1997},   MRCLASS = {14K22 (11G15)},  MRNUMBER = {1472499 (98g:14056)},MRREVIEWER = {Rutger Noot},}@article {MR0369301,    AUTHOR = {Oselies, Reinhard and Zieschang, Heiner},     TITLE = {Ergodische {E}igenschaften der {A}utomorphismen {$p$}-adischer              {Z}ahlen},   JOURNAL = {Arch. Math. (Basel)},    VOLUME = {26},      YEAR = {1975},     PAGES = {144--153},   MRCLASS = {10K10 (28A65)},  MRNUMBER = {MR0369301 (51 \#5536)},MRREVIEWER = {Roland Fischer},}@comment{UPDATE}@misc {arxiv0906.3854,    AUTHOR = {Ostafe, Alina},    TITLE = {Multivariate Permutation Polynomial Systems             and Nonlinear Pseudorandom Number Generators},     YEAR = {2009},     NOTE = {\url{arXiv:0906.3854}},}@comment{UPDATE}@misc {arxiv1109.0575,    AUTHOR = {Ostafe, Alina and Shparlinski, Igor},    TITLE = {Degree Growth, Linear Independence and Periods of a Class                  of Rational Dynamical Systems},     YEAR = {2011},     NOTE = {\url{arXiv:1109.0575}},}@comment{UPDATE}@misc {arxiv0902.3884,    AUTHOR = {Ostafe, Alina and Shparlinski, Igor},    TITLE = {On the degree growth in some polynomial dynamical systems and          nonlinear pseudorandom number generators},     YEAR = {2009},     NOTE = {\url{arXiv:0902.3884}},}@comment{UPDATE}@misc {arxiv0909.3972,    AUTHOR = {Ostafe, Alina and Shparlinski, Igor},    TITLE = {On the Length of Critical Orbits of Stable Quadratic Polynomials},     YEAR = {2009},     NOTE = {\url{arXiv:0909.3972}},}@comment{arxiv1001.1504}@article {MR2765701,    AUTHOR = {Ostafe, Alina and Shparlinski, Igor E.},     TITLE = {Pseudorandomness and dynamics of {F}ermat quotients},   JOURNAL = {SIAM J. Discrete Math.},  FJOURNAL = {SIAM Journal on Discrete Mathematics},    VOLUME = {25},      YEAR = {2011},    NUMBER = {1},     PAGES = {50--71},      ISSN = {0895-4801},   MRCLASS = {11A07 (11L40 37A45 37Pxx)},  MRNUMBER = {2765701},       DOI = {10.1137/100798466},       URL = {http://dx.doi.org/10.1137/100798466},}@comment{--PPP-- }@article {arxiv0901.2646,    AUTHOR = {Pakapongpun, Apisit and Ward, Thomas},     TITLE = {Functorial orbit counting},   JOURNAL = {J. Integer Seq.},  FJOURNAL = {Journal of Integer Sequences},    VOLUME = {12},      YEAR = {2009},    NUMBER = {2},     PAGES = {Article 09.2.4, 20},      ISSN = {1530-7638},   MRCLASS = {11B83},  MRNUMBER = {MR2486259},}@comment{UPDATE}@misc {arxiv1001.0314,    AUTHOR = {Pakapongpun, Apisit and Ward, Thomas},    TITLE = {Orbits for products of maps},     YEAR = {2010},     NOTE = {\url{arXiv:1001.0314}},}@comment{UPDATE}@misc {arxiv1108.1900,    AUTHOR = {Pakovich, Fedor},    TITLE = {On semiconjugate rational functions },     YEAR = {2011},     NOTE = {\url{arXiv:1108.1900}},}@article {MR2434164,    AUTHOR = {Pakovich, Fedor},     TITLE = {Conservative polynomials and yet another action of {${\rm              Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$} on plane trees},   JOURNAL = {J. Th\'eor. Nombres Bordeaux},  FJOURNAL = {Journal de Th\'eorie des Nombres de Bordeaux},    VOLUME = {20},      YEAR = {2008},    NUMBER = {1},     PAGES = {205--218},      ISSN = {1246-7405},   MRCLASS = {11C08 (11R32 20E08)},  MRNUMBER = {2434164 (2009e:11048)},MRREVIEWER = {Michael E. Zieve},       URL = {http://jtnb.cedram.org/item?id=JTNB_2008__20_1_205_0},}@article {pal07,    AUTHOR = {P{\'a}l, Ambrus},     TITLE = {On the torsion of the {M}ordell-{W}eil group of the {J}acobian              of {D}rinfeld modular curves},   JOURNAL = {Doc. Math.},  FJOURNAL = {Documenta Mathematica},    VOLUME = {10},      YEAR = {2005},     PAGES = {131--198 (electronic)},      ISSN = {1431-0635},   MRCLASS = {11G18 (11G09)},  MRNUMBER = {MR2148073 (2006c:11070)},MRREVIEWER = {Mihran Papikian},}@comment{UPDATE}@misc {arxiv1103.5473,    AUTHOR = {Panti, Giovanni},    TITLE = {Denominator-preserving maps},     YEAR = {2011},     NOTE = {\url{arXiv:1103.5473}},}@comment{arxiv0911.1913}@article {MR2645156,    AUTHOR = {Pazuki, Fabien},     TITLE = {Zhang's conjecture and squares of abelian surfaces},   JOURNAL = {C. R. Math. Acad. Sci. Paris},  FJOURNAL = {Comptes Rendus Math\'ematique. Acad\'emie des Sciences. Paris},    VOLUME = {348},      YEAR = {2010},    NUMBER = {9-10},     PAGES = {483--486},      ISSN = {1631-073X},   MRCLASS = {37P05 (11G15 14K22 37F10)},  MRNUMBER = {2645156 (2011f:37163)},MRREVIEWER = {Joseph H. Silverman},       DOI = {10.1016/j.crma.2010.03.014},       URL = {http://dx.doi.org/10.1016/j.crma.2010.03.014},}@comment{UPDATE}@misc {arxiv1106.1784,    AUTHOR = {Pazuki, F.},    TITLE = {Polarized morphisms between abelian varieties},     YEAR = {2011},     NOTE = {\url{arXiv:1106.1784}},}@incollection {MR1913468,    AUTHOR = {Peinado, A. and Montoya, F. and Mu{\~n}oz, J. and Yuste, A.              J.},     TITLE = {Maximal periods of {$x\sp 2+c$} in {$\mathbb{F}\sb q$}}, BOOKTITLE = {Applied Algebra, Algebraic Algorithms and Error-Correcting              Codes (Melbourne, 2001)},    SERIES = {Lecture Notes in Comput. Sci.},    VOLUME = {2227},     PAGES = {219--228}, PUBLISHER = {Springer},   ADDRESS = {Berlin},      YEAR = {2001},   MRCLASS = {94A55 (11T71 94A60)},  MRNUMBER = {MR1913468 (2003d:94065)},}@comment{Summary: "The orbits produced by the iterations of the mapping$x\mapsto x^2+c$, defined over $\mathbb{F}_q$, are studied. Several upperbounds for their periods are obtained, depending on the coefficient$c$ and the number of elements $q$."}@article {percival:chaoticmotion,    AUTHOR = {Percival, Ian and Vivaldi, Franco},     TITLE = {Arithmetical properties of strongly chaotic motions},   JOURNAL = {Phys. D},  FJOURNAL = {Physica D. Nonlinear Phenomena},    VOLUME = {25},      YEAR = {1987},    NUMBER = {1-3},     PAGES = {105--130},      ISSN = {0167-2789},     CODEN = {PDNPDT},   MRCLASS = {58F13},  MRNUMBER = {MR887460 (88g:58120)},MRREVIEWER = {Richard Cushman},}@article {MR2779260,    AUTHOR = {Petersen, Karl and Varchenko, Alexander},     TITLE = {The {E}uler adic dynamical system and path counts in the              {E}uler graph},   JOURNAL = {Tokyo J. 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J.},  FJOURNAL = {Indiana University Mathematics Journal},    VOLUME = {46},      YEAR = {1997},    NUMBER = {3},     PAGES = {897--932},      ISSN = {0022-2518},     CODEN = {IUMJAB},   MRCLASS = {32H30 (32H50)},  MRNUMBER = {1488341 (98h:32046)},MRREVIEWER = {Jeffrey Diller},       DOI = {10.1512/iumj.1997.46.1441},       URL = {http://dx.doi.org/10.1512/iumj.1997.46.1441},}@comment{This article defines the various dynamical degrees defined in terms of pull-backs of linear spaces}@article{russowalde:quadraticmaps,    Author = {Russo, P. and Walde, R.},    TITLE = {Rational periodic points of the quadratic                     function ${Q}_c(x)=x^2+c$},  JOURNAL = {Am. Math. 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Soc.},  FJOURNAL = {Bulletin of the London Mathematical Society},    VOLUME = {42},      YEAR = {2010},    NUMBER = {3},     PAGES = {381--387},      ISSN = {0024-6093},   MRCLASS = {37Pxx (11S15)},  MRNUMBER = {2651931},       DOI = {10.1112/blms/bdp130},       URL = {http://dx.doi.org/10.1112/blms/bdp130},}@comment{UPDATE}@misc {arxiv:1102.0568,    AUTHOR = {Sarkis, Ghassan and Specter, Joel},    TITLE = {Galois Extensions of Height-One Commuting Dynamical Systems},     YEAR = {2011},     NOTE = {\url{arXiv:1102.0568}},}@comment{UPDATE}@misc {arxiv:0807.4162,    AUTHOR = {Scanlon, Thomas},    TITLE = {Analytic relations on a dynamical orbit},     YEAR = {2008},     NOTE = {\url{arXiv:0807.4162}},}@article {MR1328699,    AUTHOR = {Schmidt, Wolfgang and Steinmetz, Norbert},     TITLE = {The polynomials associated with a {J}ulia set},   JOURNAL = {Bull. London Math. Soc.},  FJOURNAL = {The Bulletin of the London Mathematical Society},    VOLUME = {27},      YEAR = {1995},    NUMBER = {3},     PAGES = {239--241},      ISSN = {0024-6093},     CODEN = {LMSBBT},   MRCLASS = {30D05},  MRNUMBER = {MR1328699 (96e:30061)},MRREVIEWER = {Lo Yang},}@comment{For any Julia set (of a polynomial) $J$ which is neither acircle nor a straight line segment, there exists a polynomial $P$ suchthat any polynomial with Julia set $J$ can be written in the form$\sigma P^n$, where $\sigma$ is a rotation mapping $J$ onto itself,and $n$ is a positive integer.}@book {MR568710,    AUTHOR = {Schmidt, Wolfgang},     TITLE = {Diophantine {A}pproximation},    SERIES = {Lecture Notes in Mathematics},    VOLUME = {785}, PUBLISHER = {Springer},   ADDRESS = {Berlin},      YEAR = {1980},     PAGES = {x+299},      ISBN = {3-540-09762-7},   MRCLASS = {10Fxx (10-02)},  MRNUMBER = {MR568710 (81j:10038)},MRREVIEWER = {A. J. van der Poorten},}@article{schanuel:heights,    AUTHOR = {S. Schanuel},    TITLE = {Heights in number fields},  JOURNAL = {Bull. Soc. Math. France},   VOLUME = {107},     YEAR = {1979},    PAGES = {443--449},	}@PHDTHESIS{schleicherphd94, AUTHOR  = {D. Schleicher}, TITLE   = {Internal addresses in the {M}andelbrot set and irreducibility of polynomials}, SCHOOL  = {Cornell University}, ADDRESS = {Ithaca NY}, YEAR    = {1994},}@book{schmidt:algebraicdynamics,     AUTHOR = {Schmidt, Klaus},     TITLE = {Dynamical {S}ystems of {A}lgebraic {O}rigin},    SERIES = {Progress in Mathematics},    VOLUME = {128}, PUBLISHER = {Birkh\"auser Verlag},   ADDRESS = {Basel},      YEAR = {1995},     PAGES = {xviii+310},      ISBN = {3-7643-5174-8},   MRCLASS = {28D15 (11K55 22D40 28D20)},  MRNUMBER = {MR1345152 (97c:28041)},MRREVIEWER = {Mahesh Nerurkar},}@article {MR2135559,    AUTHOR = {Schweizer, Andreas},     TITLE = {On periodic points under the iteration of additive              polynomials},   JOURNAL = {Manuscripta Math.},  FJOURNAL = {Manuscripta Mathematica},    VOLUME = {113},      YEAR = {2004},    NUMBER = {1},     PAGES = {25--34},      ISSN = {0025-2611},     CODEN = {MSMHB2},   MRCLASS = {11C08 (11G09 37E15)},  MRNUMBER = {MR2135559 (2005m:11039)},MRREVIEWER = {Julie Tzu-Yueh Wang},}@article {MR0236190,    AUTHOR = {Serre, Jean-Pierre and Tate, John},     TITLE = {Good reduction of abelian varieties},   JOURNAL = {Ann. of Math. (2)},  FJOURNAL = {Annals of Mathematics. Second Series},    VOLUME = {88},      YEAR = {1968},     PAGES = {492--517},      ISSN = {0003-486X},   MRCLASS = {14.51},  MRNUMBER = {MR0236190 (38 \#4488)},MRREVIEWER = {M. J. Greenberg},}@book {MR554237,    AUTHOR = {Serre, Jean-Pierre},     TITLE = {Local {F}ields},    SERIES = {Graduate Texts in Mathematics},    VOLUME = {67},      NOTE = {Translated from the French by Marvin Jay Greenberg}, PUBLISHER = {Springer-Verlag},   ADDRESS = {New York},      YEAR = {1979},     PAGES = {viii+241},      ISBN = {0-387-90424-7},   MRCLASS = {12Bxx},  MRNUMBER = {MR554237 (82e:12016)},}@book {MR1757192,    AUTHOR = {Serre, Jean-Pierre},     TITLE = {Lectures on the {M}ordell-{W}eil {T}heorem},    SERIES = {Aspects of Mathematics},   EDITION = {Third},      NOTE = {Translated from the French and edited by Martin Brown from              notes by Michel Waldschmidt,              with a foreword by Brown and Serre}, PUBLISHER = {Friedr. Vieweg \& Sohn},   ADDRESS = {Braunschweig},      YEAR = {1997},     PAGES = {x+218},      ISBN = {3-528-28968-6},   MRCLASS = {11G10 (11D41 11G30 14G25)},  MRNUMBER = {MR1757192 (2000m:11049)},}@comment{UPDATE}@misc {arxiv1101.3482,    AUTHOR = {Sha, Min},    TITLE = {On the Cycle Structure of Repeated            Exponentiation Modulo a Prime Power},     YEAR = {2011},     NOTE = {\url{arXiv:1101.3482}},}@article{arxiv0910.5550,    AUTHOR = {Sha, Min and Hu, Su},    TITLE = {Monomial Dynamical Systems of Dimension One over Finite Fields},   JOURNAL = {Acta Arith.},  FJOURNAL = {Acta Arithmetica},    VOLUME = {148},      YEAR = {2011},    NUMBER = {4},     PAGES = {309--331},}@article {shimura:fldofdef,   AUTHOR = {Shimura, Goro},    TITLE = {On the field of definition for a field of automorphic             functions. {I}, {I}{I}, {I}{I}{I}},  JOURNAL = {Ann. of Math. (2)},   VOLUME = {80, 81, 83},     YEAR = {1964, 1965, 1966},    PAGES = {160--189, 124--165, 377--385},}@book {MR1291394,    AUTHOR = {Shimura, Goro},     TITLE = {Introduction to the {A}rithmetic {T}heory of {A}utomorphic              {F}unctions},    SERIES = {Publications of the Mathematical Society of Japan},    VOLUME = {11},      NOTE = {Reprint of the 1971 original,              Kan\^o Memorial Lectures, 1}, PUBLISHER = {Princeton University Press},   ADDRESS = {Princeton, NJ},      YEAR = {1994},     PAGES = {xiv+271},      ISBN = {0-691-08092-5},   MRCLASS = {11Fxx (11-02 11G05 11G40)},  MRNUMBER = {MR1291394 (95e:11048)},}@article {MR917803,    AUTHOR = {Shparlinski{\u\i}, I. E.},     TITLE = {The number of different prime divisors of recurrent sequences},   JOURNAL = {Mat. Zametki},  FJOURNAL = {Akademiya Nauk SSSR. Matematicheskie Zametki},    VOLUME = {42},      YEAR = {1987},    NUMBER = {4},     PAGES = {494--507, 622},      ISSN = {0025-567X},   MRCLASS = {11N99 (11B37)},  MRNUMBER = {MR917803 (89k:11092)},MRREVIEWER = {Grigori Kolesnik},}@article {arxiv0607779,    AUTHOR = {Shparlinski, Igor},    TITLE = {On Some Dynamical Systems in Finite Fields and Residue Rings},   JOURNAL = {Discrete Contin. Dyn. Syst.},  FJOURNAL = {Discrete and Continuous Dynamical Systems. Series A},    VOLUME = {17},      YEAR = {2007},     PAGES = {901--917},}@incollection {sibony:panoramas,    AUTHOR = {Sibony, Nessim},     TITLE = {Dynamique des applications rationnelles de {$\mathbb{P}\sp k$}}, BOOKTITLE = {Dynamique et g\'eom\'etrie complexes (Lyon, 1997)},    SERIES = {Panor. Synth\`eses},    VOLUME = {8},     PAGES = {ix--x, xi--xii, 97--185}, PUBLISHER = {Soc. Math. France},   ADDRESS = {Paris},      YEAR = {1999},   MRCLASS = {32H50 (37F10 37F50)},  MRNUMBER = {MR1760844 (2001e:32026)},MRREVIEWER = {Mattias Jonsson},}@incollection {MR2052279,    AUTHOR = {Sidorov, Nikita},     TITLE = {Arithmetic dynamics}, BOOKTITLE = {Topics in Dynamics and Ergodic Theory},    SERIES = {London Math. Soc. Lecture Note Ser.},    VOLUME = {310},     PAGES = {145--189}, PUBLISHER = {Cambridge Univ. Press},   ADDRESS = {Cambridge},      YEAR = {2003},   MRCLASS = {37B10 (11K16 37A45)},  MRNUMBER = {MR2052279},MRREVIEWER = {Manfred Einsiedler},}@comment{From MR: In this survey paper the author describes a collection ofresults dealing with arithmetic expansions of various dynamicalsystems---a (to be developed) branch of symbolic dynamics the authorcalls arithmetic dynamics.}@article{siegel:hyperellipticintegerpoints,    AUTHOR = {Siegel, C.L.},    TITLE = {The integer solutions of the equation $y^2=ax^n+bx^{n-1}+\cdots+k$},  JOURNAL = {J. London Math. Soc.},   VOLUME = {1},     YEAR = {1926},    PAGES = {66--68}}@incollection {siegeldiophappr,   AUTHOR = {Siegel, C.L.},     TITLE = {{\"U}ber einige {A}nwendungen diophantischer {A}pproximationen}, BOOKTITLE = {Collected Works},     PAGES = {209--266}, PUBLISHER = {Springer},   ADDRESS = {Berlin},      YEAR = {1966},}@book{silverman:aec,     AUTHOR = {Silverman, Joseph H.},     TITLE = {The {A}rithmetic of {E}lliptic {C}urves},    SERIES = {Graduate Texts in Mathematics},    VOLUME = {106},      NOTE = {Corrected reprint of the 1986 original}, PUBLISHER = {Springer-Verlag},   ADDRESS = {New York},      YEAR = {1992},     PAGES = {xii+400},      ISBN = {0-387-96203-4},   MRCLASS = {11G05 (14H52)},  MRNUMBER = {MR1329092 (95m:11054)},}@book {MR2514094,    AUTHOR = {Silverman, Joseph H.},     TITLE = {The {A}rithmetic of {E}lliptic {C}urves},    SERIES = {Graduate Texts in Mathematics},    VOLUME = {106},   EDITION = {Second}, PUBLISHER = {Springer},   ADDRESS = {Dordrecht},      YEAR = {2009},     PAGES = {xx+513},      ISBN = {978-0-387-09493-9},   MRCLASS = {11-02 (11G05 14Gxx 14Kxx 14K15)},  MRNUMBER = {MR2514094},       DOI = {10.1007/978-0-387-09494-6},       URL = {http://dx.doi.org/10.1007/978-0-387-09494-6},}@book{silverman:ataec,     AUTHOR = {Silverman, Joseph H.},     TITLE = {Advanced {T}opics in the {A}rithmetic of {E}lliptic {C}urves},    SERIES = {Graduate Texts in Mathematics},    VOLUME = {151}, PUBLISHER = {Springer-Verlag},   ADDRESS = {New York},      YEAR = {1994},     PAGES = {xiv+525},      ISBN = {0-387-94328-5},   MRCLASS = {11G05 (11G07 11G15 11G40 14H52)},  MRNUMBER = {MR1312368 (96b:11074)},MRREVIEWER = {Henri Darmon},}@book {MR2316407,    AUTHOR = {Silverman, Joseph H.},     TITLE = {The {A}rithmetic of {D}ynamical {S}ystems},    SERIES = {Graduate Texts in Mathematics},    VOLUME = {241}, PUBLISHER = {Springer},   ADDRESS = {New York},      YEAR = {2007},     PAGES = {x+511},      ISBN = {978-0-387-69903-5},   MRCLASS = {11-01 (11G99 37Bxx 37Fxx)},  MRNUMBER = {MR2316407},}@article {silverman:K3htseries,    AUTHOR = {Silverman, Joseph H.},     TITLE = {A zeta function over a recurrent sequence},   JOURNAL = {Amer. Math. Monthly},  FJOURNAL = {The American Mathematical Monthly},    VOLUME = {106},      YEAR = {1999},    NUMBER = {7},     PAGES = {686--688},      NOTE = {Problem 10486 with solutions},}@article {silverman:integerpointsongenus1,    AUTHOR = {Silverman, Joseph H.},     TITLE = {Integer points on curves of genus {$1$}},   JOURNAL = {J. London Math. Soc. (2)},  FJOURNAL = {The Journal of the London Mathematical Society. Second Series},    VOLUME = {28},      YEAR = {1983},    NUMBER = {1},     PAGES = {1--7},      ISSN = {0024-6107},     CODEN = {JLMSAK},   MRCLASS = {10B10 (10J06 14G25 14K15)},  MRNUMBER = {MR703458 (84g:10033)},MRREVIEWER = {Michael Laska},}@article {MR703488,    AUTHOR = {Silverman, Joseph H.},     TITLE = {Heights and the specialization map for families of abelian              varieties},   JOURNAL = {J. Reine Angew. Math.},  FJOURNAL = {Journal f\"ur die Reine und Angewandte Mathematik},    VOLUME = {342},      YEAR = {1983},     PAGES = {197--211},      ISSN = {0075-4102},     CODEN = {JRMAA8},   MRCLASS = {14K15 (14D10 14G25)},  MRNUMBER = {MR703488 (84k:14033)},MRREVIEWER = {Gerd Faltings},}@article {MR895285,    AUTHOR = {Silverman, Joseph H.},     TITLE = {A quantitative version of {S}iegel's theorem: integral points              on elliptic curves and {C}atalan curves},   JOURNAL = {J. Reine Angew. Math.},  FJOURNAL = {Journal f\"ur die Reine und Angewandte Mathematik},    VOLUME = {378},      YEAR = {1987},     PAGES = {60--100},      ISSN = {0075-4102},     CODEN = {JRMAA8},   MRCLASS = {11G05 (14G25)},  MRNUMBER = {MR895285 (89g:11047)},MRREVIEWER = {Toshihiro Hadano},}@book {MR1171452,    AUTHOR = {Silverman, Joseph H. and Tate, John},     TITLE = {Rational {P}oints on {E}lliptic {C}urves},    SERIES = {Undergraduate Texts in Mathematics}, PUBLISHER = {Springer-Verlag},   ADDRESS = {New York},      YEAR = {1992},     PAGES = {x+281},      ISBN = {0-387-97825-9},   MRCLASS = {11-01 (11D25 11G05 14H52)},  MRNUMBER = {MR1171452 (93g:11003)},MRREVIEWER = {Andrew Bremner},}@article{silverman:computecanonicalht,    AUTHOR = {Silverman, Joseph H.},     TITLE = {Computing heights on elliptic curves},   JOURNAL = {Math. Comp.},  FJOURNAL = {Mathematics of Computation},    VOLUME = {51},      YEAR = {1988},    NUMBER = {183},     PAGES = {339--358},      ISSN = {0025-5718},     CODEN = {MCMPAF},   MRCLASS = {11G05 (11D25 11Y40 14G25 14K15)},  MRNUMBER = {MR942161 (89d:11049)},MRREVIEWER = {Philippe Satg{\'e}},}@article{silverman:K3heights,     AUTHOR = {Silverman, Joseph H.},     TITLE = {Rational points on {$K3$} surfaces: a new canonical height},   JOURNAL = {Invent. Math.},  FJOURNAL = {Inventiones Mathematicae},    VOLUME = {105},      YEAR = {1991},    NUMBER = {2},     PAGES = {347--373},      ISSN = {0020-9910},     CODEN = {INVMBH},   MRCLASS = {14G05 (11G35 14J20 14J28)},  MRNUMBER = {MR1115546 (92k:14025)},MRREVIEWER = {I. Dolgachev},}@article{silverman:dynamicalintegerpoints,     AUTHOR = {Silverman, Joseph H.},     TITLE = {Integer points, {D}iophantine approximation, and iteration of              rational maps},   JOURNAL = {Duke Math. J.},  FJOURNAL = {Duke Mathematical Journal},    VOLUME = {71},      YEAR = {1993},    NUMBER = {3},     PAGES = {793--829},      ISSN = {0012-7094},     CODEN = {DUMJAO},   MRCLASS = {11G99 (11J99)},  MRNUMBER = {MR1240603 (95e:11070)},MRREVIEWER = {Jeffrey Lin Thunder},}@article{silverman:henonmap,     AUTHOR = {Silverman, Joseph H.},     TITLE = {Geometric and arithmetic properties of the {H}\'enon map},   JOURNAL = {Math. Z.},  FJOURNAL = {Mathematische Zeitschrift},    VOLUME = {215},      YEAR = {1994},    NUMBER = {2},     PAGES = {237--250},      ISSN = {0025-5874},     CODEN = {MAZEAX},   MRCLASS = {14G25 (11G99 14E05 58F20)},  MRNUMBER = {MR1259460 (95f:14040)},MRREVIEWER = {Takeshi Ooe},}@article{silverman:fieldofdef,     AUTHOR = {Silverman, Joseph H.},     TITLE = {The field of definition for dynamical systems on {$\mathbb{P}\sp              1$}},   JOURNAL = {Compositio Math.},  FJOURNAL = {Compositio Mathematica},    VOLUME = {98},      YEAR = {1995},    NUMBER = {3},     PAGES = {269--304},      ISSN = {0010-437X},     CODEN = {CMPMAF},   MRCLASS = {11G99 (11R27 14E22)},  MRNUMBER = {MR1351830 (96j:11090)},MRREVIEWER = {Laurent Denis},}@article{silverman:polynomialiterate,     AUTHOR = {Silverman, Joseph H.},     TITLE = {Rational functions with a polynomial iterate},   JOURNAL = {J. Algebra},  FJOURNAL = {Journal of Algebra},    VOLUME = {180},      YEAR = {1996},    NUMBER = {1},     PAGES = {102--110},      ISSN = {0021-8693},     CODEN = {JALGA4},   MRCLASS = {11G99},  MRNUMBER = {MR1375568 (96j:11091)},MRREVIEWER = {Jeffrey Lin Thunder},}@article {MR1388892,    AUTHOR = {Silverman, Joseph H.},     TITLE = {Computing canonical heights with little (or no) factorization},   JOURNAL = {Math. Comp.},  FJOURNAL = {Mathematics of Computation},    VOLUME = {66},      YEAR = {1997},    NUMBER = {218},     PAGES = {787--805},      ISSN = {0025-5718},     CODEN = {MCMPAF},   MRCLASS = {11G05 (11Y16)},  MRNUMBER = {MR1388892 (97f:11040)},MRREVIEWER = {Alain Kraus},}@article{silverman:modulirationalmaps,     AUTHOR = {Silverman, Joseph H.},     TITLE = {The space of rational maps on {$\mathbb{P}\sp 1$}},   JOURNAL = {Duke Math. J.},  FJOURNAL = {Duke Mathematical Journal},    VOLUME = {94},      YEAR = {1998},    NUMBER = {1},     PAGES = {41--77},      ISSN = {0012-7094},     CODEN = {DUMJAO},   MRCLASS = {14D10 (14L30 26A18 37E99 37F10)},  MRNUMBER = {MR1635900 (2000m:14010)},}@article{silverman:arithdistfunctions,     AUTHOR = {Silverman, Joseph H.},     TITLE = {Arithmetic distance functions and height functions in              {D}iophantine geometry},   JOURNAL = {Math. Ann.},  FJOURNAL = {Mathematische Annalen},    VOLUME = {279},      YEAR = {1987},    NUMBER = {2},     PAGES = {193--216},      ISSN = {0025-5831},     CODEN = {MAANA},   MRCLASS = {11G99 (11D99 14G99)},  MRNUMBER = {MR919501 (89a:11066)},MRREVIEWER = {Andrew Bremner},}@article {silverman:htregauts,    AUTHOR = {Silverman, Joseph H.},    TITLE = {Height Bounds and Preperiodic Points  for          Families of Jointly Regular Affine Maps},   JOURNAL = {Quart. J. Pure Appl. Math.},    VOLUME = {2},      YEAR = {2006},     PAGES = {135--145},}@article {arXiv:0707.1505,    AUTHOR = {Silverman, Joseph H.},     TITLE = {Variation of periods modulo {$p$} in arithmetic dynamics},   JOURNAL = {New York J. Math.},  FJOURNAL = {New York Journal of Mathematics},    VOLUME = {14},      YEAR = {2008},     PAGES = {601--616},      ISSN = {1076-9803},   MRCLASS = {11G35 (11G50 11R45 14G40 37F10)},  MRNUMBER = {MR2448661},MRREVIEWER = {Thomas Ward},}@comment{UPDATE}@article {arxiv0908.3835,    AUTHOR = {Silverman, Joseph H.},    TITLE = {Height Estimates for Equidimensional Dominant Rational Maps},     YEAR = {2011},   JOURNAL = {J. Ramanujan Math. 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P.},     TITLE = {Integrable mappings},   JOURNAL = {Uspekhi Mat. Nauk},  FJOURNAL = {Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo.              Uspekhi Matematicheskikh Nauk},    VOLUME = {46},      YEAR = {1991},    NUMBER = {5(281)},     PAGES = {3--45, 190},      ISSN = {0042-1316},   MRCLASS = {58F07 (14K99 39A12 70H05)},  MRNUMBER = {MR1160332 (93e:58096)},MRREVIEWER = {J. S. Joel},}@article {MR2273670,    AUTHOR = {Vi{\^e}t-Anh Nguy{\^e}n},     TITLE = {Algebraic degrees for iterates of meromorphic self-maps of              {${\mathbb P}^k$}},   JOURNAL = {Publ. 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Phys.},    VOLUME = {47},     PAGES = {294--301}, PUBLISHER = {Springer},   ADDRESS = {Berlin},      YEAR = {1990},   MRCLASS = {58F13 (11R11 58F11)},  MRNUMBER = {MR1058474 (91i:58093)},MRREVIEWER = {F. Schweiger},}@incollection {vivaldi:arithmeticchaos,    AUTHOR = {Vivaldi, Franco},     TITLE = {The arithmetic of chaos}, BOOKTITLE = {Chaos, Noise and Fractals (Como, 1986)},    SERIES = {Malvern Phys. Ser.},    VOLUME = {3},     PAGES = {187--199}, PUBLISHER = {Hilger},   ADDRESS = {Bristol},      YEAR = {1987},   MRCLASS = {58F11 (11R11 11R29)},  MRNUMBER = {MR918575 (89b:58123)},MRREVIEWER = {F. Schweiger},}@article {vivaldi:arithmeticanosov,    AUTHOR = {Vivaldi, Franco},     TITLE = {Arithmetical theory of {A}nosov diffeomorphisms},   JOURNAL = {Proc. Roy. Soc. London Ser. A},  FJOURNAL = {Proceedings of the Royal Society. London. Series A.              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Dolgachev},}@article {MR1367580,    AUTHOR = {Voutier, Paul},     TITLE = {An effective lower bound for the height of algebraic numbers},   JOURNAL = {Acta Arith.},  FJOURNAL = {Acta Arithmetica},    VOLUME = {74},      YEAR = {1996},    NUMBER = {1},     PAGES = {81--95},      ISSN = {0065-1036},     CODEN = {AARIA9},   MRCLASS = {11J25 (11R06)},  MRNUMBER = {MR1367580 (96j:11098)},MRREVIEWER = {D. Boyd},}@comment{--WWW-- }@incollection {MR1861987,    AUTHOR = {Waldschmidt, Michel},     TITLE = {Algebraic dynamics and transcendental numbers}, BOOKTITLE = {Noise, oscillators and algebraic randomness (Chapelle des              Bois, 1999)},    SERIES = {Lecture Notes in Phys.},    VOLUME = {550},     PAGES = {372--378}, PUBLISHER = {Springer},   ADDRESS = {Berlin},      YEAR = {2000},   MRCLASS = {37F10 (11J81 37F99)},  MRNUMBER = {MR1861987 (2002h:37074)},}@incollection {MR1352278,    AUTHOR = {Wang, Lan},     TITLE = {Rational points and canonical heights on {$K3$}-surfaces in              {$\mathbb{P}\sp 1\times\mathbb{P}\sp 1\times\mathbb{P}\sp 1$}}, BOOKTITLE = {Recent Developments in the Inverse Galois Problem (Seattle,              WA, 1993)},    SERIES = {Contemp. Math.},    VOLUME = {186},     PAGES = {273--289}, PUBLISHER = {Amer. Math. Soc.},   ADDRESS = {Providence, RI},      YEAR = {1995},   MRCLASS = {14G05 (11G35 14J28)},  MRNUMBER = {MR1352278 (97a:14023)},MRREVIEWER = {Gregory S. Call},}@article {ward:sintegerdynamics,   AUTHOR = {Ward, T. B.},    TITLE = {Almost all ${S}$-integer dynamical systems have many periodic             points},  JOURNAL = {Ergodic Theory Dynam. Systems}, FJOURNAL = {Ergodic Theory and Dynamical Systems},   VOLUME = {18},     YEAR = {1998},   NUMBER = {2},}@book {MR1421575,    AUTHOR = {Washington, Lawrence C.},     TITLE = {Introduction to cyclotomic fields},    SERIES = {Graduate Texts in Mathematics},    VOLUME = {83}, PUBLISHER = {Springer-Verlag},   ADDRESS = {New York},      YEAR = {1997},     PAGES = {xiv+487},      ISBN = {0-387-94762-0},   MRCLASS = {11R18 (11-01 11-02 11R23)},  MRNUMBER = {MR1421575 (97h:11130)},MRREVIEWER = {T. Mets{\"a}nkyl{\"a}},}@article {MR925498,    AUTHOR = {Wehler, Joachim},     TITLE = {{$K3$}-surfaces with {P}icard number {$2$}},   JOURNAL = {Arch. 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Math.}, FJOURNAL = {Experimental Mathematics},   VOLUME = {7},     YEAR = {1998},   NUMBER = {4},    PAGES = {333--342},}@comment{--XXX-- }@comment{--YYY-- }@article {arxiv0612424,    AUTHOR = {Yuan, Xinyi},     TITLE = {Big line bundles over arithmetic varieties},   JOURNAL = {Invent. Math.},  FJOURNAL = {Inventiones Mathematicae},    VOLUME = {173},      YEAR = {2008},    NUMBER = {3},     PAGES = {603--649},      ISSN = {0020-9910},     CODEN = {INVMBH},   MRCLASS = {14G40 (11G50 11H06 14C20)},  MRNUMBER = {MR2425137},}@comment{Abstract: We prove a Hilbert-Samuel type result of arithmeticbig line bundles in Arakelov geometry, which is an analogue of aclassical theorem of Siu. An application of this result givesequidistribution of small points over algebraic dynamical systems,following the work of Szpiro-Ullmo-Zhang. We also generalizeChambert-Loir's non-archimedean equidistribution.}@comment{UPDATE}@misc {yuanzhang09,    AUTHOR = {Yuan, Xinyi and  Zhang, Shou-Wu},    TITLE = {Calabi theorem and algebraic dynamics},     YEAR = {November 14, 2009},     NOTE = {\url{www.math.harvard.edu/~yxy/preprints/calabisemigroup.pdf}},}@comment{--ZZZ-- }@article {MR669663,    AUTHOR = {Zagier, Don},     TITLE = {On the number of {M}arkoff numbers below a given bound},   JOURNAL = {Math. Comp.},  FJOURNAL = {Mathematics of Computation},    VOLUME = {39},      YEAR = {1982},    NUMBER = {160},     PAGES = {709--723},      ISSN = {0025-5718},     CODEN = {MCMPAF},   MRCLASS = {10F20 (10A20 10B10)},  MRNUMBER = {MR669663 (83k:10062)},MRREVIEWER = {Richard T. Bumby},}@incollection {zhang:distalgdyn,    AUTHOR = {Zhang, Shou-Wu},     TITLE = {Distributions in algebraic dynamics}, BOOKTITLE = {Differential Geometry: A Tribute to Professor {S}.-{S}. {C}hern,                   Surv. Differ. Geom., Vol. X},     PAGES = {381--430}, PUBLISHER = {Int. Press},   ADDRESS = {Boston, MA},      YEAR = {2006},}@comment{This is an overview of a lot of different topics inarithmetic dynamics, including canonical metrics, equidistribution ofperiodic and small points, dynamical Lehmer and Manin-Mumfordconjectures, etc. There is also a video of a talk by Zhang at MSRIwith this title available at www.msri.org/communications/vmath/VMathVideos/VideoInfo/2488/show_video}@article {MR1609518,    AUTHOR = {Zhang, Shou-Wu},     TITLE = {Equidistribution of small points on abelian varieties},   JOURNAL = {Ann. of Math. (2)},  FJOURNAL = {Annals of Mathematics. 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