Abstract
We find good dynamical compactifications for arbitrary polynomial mappings of $\mathbf{C}^2$ and use them to show that the degree growth sequence satisfies a linear integral recursion formula. For maps of low topological degree we prove that the Green function is well behaved. For maps of maximum topological degree, we give normal forms.
-
[BD]
E. Bedford and J. Diller, "Energy and invariant measures for birational surface maps," Duke Math. J., vol. 128, iss. 2, pp. 331-368, 2005.
@article {BD, MRKEY = {2140266},
AUTHOR = {Bedford, Eric and Diller, Jeffrey},
TITLE = {Energy and invariant measures for birational surface maps},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {128},
YEAR = {2005},
NUMBER = {2},
PAGES = {331--368},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {32H50 (37F10)},
MRNUMBER = {2140266},
MRREVIEWER = {Vincent Guedj},
DOI = {10.1215/S0012-7094-04-12824-6},
ZBLNUMBER = {1076.37031},
} -
[BK1] E. Bedford and K. Kim, "On the degree growth of birational mappings in higher dimension," J. Geom. Anal., vol. 14, iss. 4, pp. 567-596, 2004.
@article {BK1, MRKEY = {2111418},
AUTHOR = {Bedford, Eric and Kim, Kyounghee},
TITLE = {On the degree growth of birational mappings in higher dimension},
JOURNAL = {J. Geom. Anal.},
FJOURNAL = {The Journal of Geometric Analysis},
VOLUME = {14},
YEAR = {2004},
NUMBER = {4},
PAGES = {567--596},
ISSN = {1050-6926},
MRCLASS = {14E05 (32H50 37F10)},
MRNUMBER = {2111418},
MRREVIEWER = {Serge Cantat},
ZBLNUMBER = {1067.37054},
} -
[BK2]
E. Bedford and K. Kim, "Periodicities in linear fractional recurrences: degree growth of birational surface maps," Michigan Math. J., vol. 54, iss. 3, pp. 647-670, 2006.
@article {BK2, MRKEY = {2280499},
AUTHOR = {Bedford, Eric and Kim, Kyounghee},
TITLE = {Periodicities in linear fractional recurrences: degree growth of birational surface maps},
JOURNAL = {Michigan Math. J.},
FJOURNAL = {Michigan Mathematical Journal},
VOLUME = {54},
YEAR = {2006},
NUMBER = {3},
PAGES = {647--670},
ISSN = {0026-2285},
MRCLASS = {32H50 (37B40 37C25 37F10)},
MRNUMBER = {2280499},
MRREVIEWER = {Serge Cantat},
DOI = {10.1307/mmj/1163789919},
ZBLNUMBER = {1171.37023},
} -
[BK3]
E. Bedford and K. Kim, "Linear recurrences in the degree sequences of monomial mappings," Ergodic Theory Dynam. Systems, vol. 28, iss. 5, pp. 1369-1375, 2008.
@article {BK3, MRKEY = {2449533},
AUTHOR = {Bedford, Eric and Kim, Kyounghee},
TITLE = {Linear recurrences in the degree sequences of monomial mappings},
JOURNAL = {Ergodic Theory Dynam. Systems},
FJOURNAL = {Ergodic Theory and Dynamical Systems},
VOLUME = {28},
YEAR = {2008},
NUMBER = {5},
PAGES = {1369--1375},
ISSN = {0143-3857},
MRCLASS = {32H50 (37F10)},
MRNUMBER = {2449533},
MRREVIEWER = {Charles Favre},
DOI = {10.1017/S0143385708000242},
ZBLNUMBER = {1161.37032},
} -
[BS1]
E. Bedford and J. Smillie, "Polynomial diffeomorphisms of ${\bf C}^2$: currents, equilibrium measure and hyperbolicity," Invent. Math., vol. 103, iss. 1, pp. 69-99, 1991.
@article {BS1, MRKEY = {1079840},
AUTHOR = {Bedford, Eric and Smillie, John},
TITLE = {Polynomial diffeomorphisms of {${\bf C}\sp 2$}: currents, equilibrium measure and hyperbolicity},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {103},
YEAR = {1991},
NUMBER = {1},
PAGES = {69--99},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {32H50 (32C30 32L30 58F11 58F15 58F23)},
MRNUMBER = {1079840},
MRREVIEWER = {Feliks Przytycki},
DOI = {10.1007/BF01239509},
ZBLNUMBER = {0721.58037},
} -
[bellon-viallet]
M. P. Bellon and C. -M. Viallet, "Algebraic entropy," Comm. Math. Phys., vol. 204, iss. 2, pp. 425-437, 1999.
@article {bellon-viallet, MRKEY = {1704282},
AUTHOR = {Bellon, M. P. and Viallet, C.-M.},
TITLE = {Algebraic entropy},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {204},
YEAR = {1999},
NUMBER = {2},
PAGES = {425--437},
ISSN = {0010-3616},
CODEN = {CMPHAY},
MRCLASS = {37D45 (37A35 37C99 39A12)},
MRNUMBER = {1704282},
MRREVIEWER = {Remo Badii},
DOI = {10.1007/s002200050652},
ZBLNUMBER = {0987.37007},
} -
[BF]
A. M. Bonifant and J. E. Fornaess, "Growth of degree for iterates of rational maps in several variables," Indiana Univ. Math. J., vol. 49, iss. 2, pp. 751-778, 2000.
@article {BF, MRKEY = {1793690},
AUTHOR = {Bonifant, Araceli M. and Forn{æ}ss, John Erik},
TITLE = {Growth of degree for iterates of rational maps in several variables},
JOURNAL = {Indiana Univ. Math. J.},
FJOURNAL = {Indiana University Mathematics Journal},
VOLUME = {49},
YEAR = {2000},
NUMBER = {2},
PAGES = {751--778},
ISSN = {0022-2518},
CODEN = {IUMJAB},
MRCLASS = {32H50 (37F10)},
MRNUMBER = {1793690},
MRREVIEWER = {Vincent Guedj},
DOI = {10.1512/iumj.2000.49.1813},
ZBLNUMBER = {0987.37033},
} -
[deggrowth]
S. Boucksom, C. Favre, and M. Jonsson, "Degree growth of meromorphic surface maps," Duke Math. J., vol. 141, iss. 3, pp. 519-538, 2008.
@article {deggrowth, MRKEY = {2387430},
AUTHOR = {Boucksom, S{é}bastien and Favre, Charles and Jonsson, Mattias},
TITLE = {Degree growth of meromorphic surface maps},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {141},
YEAR = {2008},
NUMBER = {3},
PAGES = {519--538},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {32H04 (14E05 32J27)},
MRNUMBER = {2387430},
MRREVIEWER = {Ma{\l}gorzata Stawiska},
DOI = {10.1215/00127094-2007-004},
ZBLNUMBER = {1185.32009},
} -
[briend-duval]
J. Briend and J. Duval, "Deux caractérisations de la mesure d’équilibre d’un endomorphisme de ${ P}^k(\Bbb C)$," Publ. Math. Inst. Hautes Études Sci., vol. 93, pp. 145-159, 2001.
@article {briend-duval, MRKEY = {1863737},
AUTHOR = {Briend, Jean-Yves and Duval, Julien},
TITLE = {Deux caractérisations de la mesure d'équilibre d'un endomorphisme de {${\rm P}\sp k(\bold C)$}},
JOURNAL = {Publ. Math. Inst. Hautes Études Sci.},
FJOURNAL = {Publications Mathématiques. Institut de Hautes Études Scientifiques},
VOLUME = {93},
YEAR = {2001},
PAGES = {145--159},
ISSN = {0073-8301},
MRCLASS = {32H50 (37A35 37F10)},
MRNUMBER = {186373},
MRREVIEWER = {Mattias Jonsson},
DOI = {10.1007/s10240-001-8190-4},
ZBLNUMBER = {1010.37004},
} -
[CPR]
A. Campillo, O. Piltant, and A. J. Reguera-López, "Cones of curves and of line bundles on surfaces associated with curves having one place at infinity," Proc. London Math. Soc., vol. 84, iss. 3, pp. 559-580, 2002.
@article {CPR, MRKEY = {1888423},
AUTHOR = {Campillo, Antonio and Piltant, Olivier and Reguera-L{ó}pez, Ana J.},
TITLE = {Cones of curves and of line bundles on surfaces associated with curves having one place at infinity},
JOURNAL = {Proc. London Math. Soc.},
FJOURNAL = {Proceedings of the London Mathematical Society. Third Series},
VOLUME = {84},
YEAR = {2002},
NUMBER = {3},
PAGES = {559--580},
ISSN = {0024-6115},
CODEN = {PLMTAL},
MRCLASS = {14C20 (14J26)},
MRNUMBER = {1888423},
MRREVIEWER = {Cindy De Volder},
DOI = {10.1112/S0024611502013394},
ZBLNUMBER = {1074.14505},
} -
[cantat] S. Cantat, Sur les groupes de transformations birationnelles des surfaces.
@misc{cantat,
author={Cantat, S.},
TITLE={Sur les groupes de transformations birationnelles des surfaces},
NOTE={{\it Ann. of Math.} {\bf 173} (2011), to appear; available at \href{http://perso.univ-rennes1.fr/serge.cantat/publications.html}{http://perso.univ-rennes1.fr/serge.cantat/publications.html}},
} -
[cox] D. A. Cox, "The homogeneous coordinate ring of a toric variety," J. Algebraic Geom., vol. 4, iss. 1, pp. 17-50, 1995.
@article {cox, MRKEY = {1299003},
AUTHOR = {Cox, David A.},
TITLE = {The homogeneous coordinate ring of a toric variety},
JOURNAL = {J. Algebraic Geom.},
FJOURNAL = {Journal of Algebraic Geometry},
VOLUME = {4},
YEAR = {1995},
NUMBER = {1},
PAGES = {17--50},
ISSN = {1056-3911},
MRCLASS = {14M25},
MRNUMBER = {1299003},
MRREVIEWER = {Mina Teicher},
ZBLNUMBER = {0846.14032},
} -
[DTV] H. de Thélin and G. Vigny, Entropy of meromorphic maps and dynamics of birational maps.
@misc{DTV,
author={{de} Th{é}lin, H. and Vigny, G.},
TITLE={Entropy of meromorphic maps and dynamics of birational maps},
NOTE={{\it Mem. de la SMF} {\bf 122} (2010), to appear},
ARXIV={0806.4874},
} -
[DF]
J. Diller and C. Favre, "Dynamics of bimeromorphic maps of surfaces," Amer. J. Math., vol. 123, iss. 6, pp. 1135-1169, 2001.
@article {DF, MRKEY = {1867314},
AUTHOR = {Diller, Jeffrey and Favre, C.},
TITLE = {Dynamics of bimeromorphic maps of surfaces},
JOURNAL = {Amer. J. Math.},
FJOURNAL = {American Journal of Mathematics},
VOLUME = {123},
YEAR = {2001},
NUMBER = {6},
PAGES = {1135--1169},
ISSN = {0002-9327},
CODEN = {AJMAAN},
MRCLASS = {32H50 (37F10)},
MRNUMBER = {1867314},
MRREVIEWER = {Serge Cantat},
URL = {http://muse.jhu.edu/journals/american_journal_of_mathematics/v123/123.6diller.pdf},
ZBLNUMBER = {1112.37308},
} -
[DDG1] J. Diller, R. Dujardin, and V. Guedj, " Dynamics of meromorphic maps with small topological degree I: From cohomology to currents," Indiana Univ. Math. J., vol. 59, pp. 521-562, 2010.
@article{DDG1,
author = {Diller, Jeffrey and Dujardin, Romain and Guedj, Vincent},
TITLE = {{\rm Dynamics of meromorphic maps with small topological degree I: From cohomology to currents}},
JOURNAL={Indiana Univ. Math. J.},
VOLUME={59},
YEAR={2010},
PAGES={521--562},
} -
[DDG2] J. Diller, R. Dujardin, and V. Guedj, Dynamics of meromorphic maps with small topological degree II: Energy and invariant measure.
@misc{DDG2,
author = {Diller, Jeffrey and Dujardin, Romain and Guedj, Vincent},
TITLE = {Dynamics of meromorphic maps with small topological degree {II}: Energy and invariant measure},
NOTE={{\it Comment. Math. Helv.},
to appear},
} -
[DDG3] J. Diller, R. Dujardin, and V. Guedj, "Dynamics of meromorphic maps with small topological degree III: Geometric currents and ergodic theory," Ann. Sci. Éc. Norm. Supér., vol. 43, iss. 2, pp. 235-278, 2010.
@article {DDG3, MRKEY = {2662665},
AUTHOR = {Diller, Jeffrey and Dujardin, Romain and Guedj, Vincent},
TITLE = {Dynamics of meromorphic maps with small topological degree {III}: Geometric currents and ergodic theory},
JOURNAL = {Ann. Sci. Éc. Norm. Supér.},
FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
VOLUME = {43},
YEAR = {2010},
NUMBER = {2},
PAGES = {235--278},
ISSN = {0012-9593},
MRCLASS = {37Fxx (32Hxx 32Uxx 37Axx 37Cxx)},
MRNUMBER = {2662665},
} -
[DDS]
T. C. Dinh, R. Dujardin, and N. Sibony, "On the dynamics near infinity of some polynomial mappings in ${\Bbb C}^2$," Math. Ann., vol. 333, iss. 4, pp. 703-739, 2005.
@article {DDS, MRKEY = {2195140},
AUTHOR = {Dinh, Tien Cuong and Dujardin, Romain and Sibony, Nessim},
TITLE = {On the dynamics near infinity of some polynomial mappings in {${\Bbb C}\sp 2$}},
JOURNAL = {Math. Ann.},
FJOURNAL = {Mathematische Annalen},
VOLUME = {333},
YEAR = {2005},
NUMBER = {4},
PAGES = {703--739},
ISSN = {0025-5831},
CODEN = {MAANA},
MRCLASS = {37F10 (32H50 32U40)},
MRNUMBER = {2195140},
MRREVIEWER = {Eric Bedford},
DOI = {10.1007/s00208-005-0661-3},
ZBLNUMBER = {1079.37040},
} -
[DSallure]
T. Dinh and N. Sibony, "Dynamique des applications d’allure polynomiale," J. Math. Pures Appl., vol. 82, iss. 4, pp. 367-423, 2003.
@article {DSallure, MRKEY = {1992375},
AUTHOR = {Dinh, Tien-Cuong and Sibony, Nessim},
TITLE = {Dynamique des applications d'allure polynomiale},
JOURNAL = {J. Math. Pures Appl.},
FJOURNAL = {Journal de Mathématiques Pures et Appliquées. Neuvième Série},
VOLUME = {82},
YEAR = {2003},
NUMBER = {4},
PAGES = {367--423},
ISSN = {0021-7824},
CODEN = {JMPAAM},
MRCLASS = {37F10 (32H50 32U05 37A25 37C40)},
MRNUMBER = {1992375},
MRREVIEWER = {Jean-Yves Briend},
DOI = {10.1016/S0021-7824(03)00026-6},
ZBLNUMBER = {1033.37023},
} -
[DSarkiv] T. Dinh and N. Sibony, "Dynamique des applications polynomiales semi-régulières," Ark. Mat., vol. 42, iss. 1, pp. 61-85, 2004.
@article {DSarkiv, MRKEY = {2056545},
AUTHOR = {Dinh, Tien-Cuong and Sibony, Nessim},
TITLE = {Dynamique des applications polynomiales semi-régulières},
JOURNAL = {Ark. Mat.},
FJOURNAL = {Arkiv för Matematik},
VOLUME = {42},
YEAR = {2004},
NUMBER = {1},
PAGES = {61--85},
ISSN = {0004-2080},
CODEN = {AKMTAJ},
MRCLASS = {37F10 (32H50)},
MRNUMBER = {2172806},
MRREVIEWER = {Ma{\l}gorzata Stawiska},
ZBLNUMBER = {1059.37033},
} -
[DSgeom]
T. Dinh and N. Sibony, "Geometry of currents, intersection theory and dynamics of horizontal-like maps," Ann. Inst. Fourier $($Grenoble$)$, vol. 56, iss. 2, pp. 423-457, 2006.
@article {DSgeom, MRKEY = {2226022},
AUTHOR = {Dinh, Tien-Cuong and Sibony, Nessim},
TITLE = {Geometry of currents, intersection theory and dynamics of horizontal-like maps},
JOURNAL = {Ann. Inst. Fourier $($Grenoble$)$},
FJOURNAL = {Université de Grenoble. Annales de l'Institut Fourier},
VOLUME = {56},
YEAR = {2006},
NUMBER = {2},
PAGES = {423--457},
ISSN = {0373-0956},
CODEN = {AIFUA7},
MRCLASS = {32U40 (32H50 37F10)},
MRNUMBER = {2226022},
MRREVIEWER = {Ma{\l}gorzata Stawiska},
URL = {http://aif.cedram.org/item?id=AIF_2006__56_2_423_0},
ZBLNUMBER = {1089.37036},
} -
[DSdecay]
T. Dinh and N. Sibony, "Decay of correlations and the central limit theorem for meromorphic maps," Comm. Pure Appl. Math., vol. 59, iss. 5, pp. 754-768, 2006.
@article {DSdecay, MRKEY = {2172806},
AUTHOR = {Dinh, Tien-Cuong and Sibony, Nessim},
TITLE = {Decay of correlations and the central limit theorem for meromorphic maps},
JOURNAL = {Comm. Pure Appl. Math.},
FJOURNAL = {Communications on Pure and Applied Mathematics},
VOLUME = {59},
YEAR = {2006},
NUMBER = {5},
PAGES = {754--768},
ISSN = {0010-3640},
CODEN = {CPAMA},
MRCLASS = {37F10 (32H50 37A05 37A50 37C40)},
MRNUMBER = {2172806},
MRREVIEWER = {Laura G. DeMarco},
DOI = {10.1002/cpa.20119},
ZBLNUMBER = {1137.37023},
} -
[DSsuper]
T. Dinh and N. Sibony, "Super-potentials of positive closed currents, intersection theory and dynamics," Acta Math., vol. 203, iss. 1, pp. 1-82, 2009.
@article {DSsuper, MRKEY = {2545825},
AUTHOR = {Dinh, Tien-Cuong and Sibony, Nessim},
TITLE = {Super-potentials of positive closed currents, intersection theory and dynamics},
JOURNAL = {Acta Math.},
FJOURNAL = {Acta Mathematica},
VOLUME = {203},
YEAR = {2009},
NUMBER = {1},
PAGES = {1--82},
ISSN = {0001-5962},
CODEN = {ACMAA8},
MRCLASS = {32Uxx (32Cxx)},
MRNUMBER = {2545825},
DOI = {10.1007/s11511-009-0038-7},
} -
[dolgachev]
I. Dolgachev, "Weighted projective varieties," in Group Actions and Vector Fields (Vancouver, B.C., 1981), New York: Springer-Verlag, 1982, vol. 956, pp. 34-71.
@incollection {dolgachev, MRKEY = {704986},
AUTHOR = {Dolgachev, Igor},
TITLE = {Weighted projective varieties},
BOOKTITLE = {Group Actions and Vector Fields {\rm ({V}ancouver, {B}.{C}., 1981)}},
SERIES = {Lecture Notes in Math.},
VOLUME = {956},
PAGES = {34--71},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1982},
MRCLASS = {14L32 (14A05 14B05)},
MRNUMBER = {704986},
ZBLNUMBER = {0516.14014},
DOI={10.1007/BFb0101508},
} -
[henon-like]
R. Dujardin, "Hénon-like mappings in $\Bbb C^2$," Amer. J. Math., vol. 126, iss. 2, pp. 439-472, 2004.
@article {henon-like, MRKEY = {2045508},
AUTHOR = {Dujardin, Romain},
TITLE = {Hénon-like mappings in {$\Bbb C\sp 2$}},
JOURNAL = {Amer. J. Math.},
FJOURNAL = {American Journal of Mathematics},
VOLUME = {126},
YEAR = {2004},
NUMBER = {2},
PAGES = {439--472},
ISSN = {0002-9327},
CODEN = {AJMAAN},
MRCLASS = {32H50 (37A35 37F10)},
MRNUMBER = {204550},
MRREVIEWER = {Charles Favre},
URL = {http://muse.jhu.edu/journals/american_journal_of_mathematics/v126/126.2dujardin.pdf},
ZBLNUMBER = {1064.37035},
DOI={10.1353/ajm.2004.0010},
} -
[laminar]
R. Dujardin, "Laminar currents and birational dynamics," Duke Math. J., vol. 131, iss. 2, pp. 219-247, 2006.
@article {laminar, MRKEY = {2219241},
AUTHOR = {Dujardin, Romain},
TITLE = {Laminar currents and birational dynamics},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {131},
YEAR = {2006},
NUMBER = {2},
PAGES = {219--247},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {37F10 (32H50 32U40 37A25 37B40 37D99)},
MRNUMBER = {2219241},
MRREVIEWER = {Eric Bedford},
DOI = {10.1215/S0012-7094-06-13122-8},
ZBLNUMBER = {1099.37037},
} -
[F-rigid]
C. Favre, "Classification of 2-dimensional contracting rigid germs and Kato surfaces. I," J. Math. Pures Appl., vol. 79, iss. 5, pp. 475-514, 2000.
@article {F-rigid, MRKEY = {1759437},
AUTHOR = {Favre, Charles},
TITLE = {Classification of 2-dimensional contracting rigid germs and {K}ato surfaces. {I}},
JOURNAL = {J. Math. Pures Appl.},
FJOURNAL = {Journal de Mathématiques Pures et Appliquées. Neuvième Série},
VOLUME = {79},
YEAR = {2000},
NUMBER = {5},
PAGES = {475--514},
ISSN = {0021-7824},
CODEN = {JMPAAM},
MRCLASS = {32H50 (32B10 32J05 37F99)},
MRNUMBER = {1759437},
MRREVIEWER = {Eric Bedford},
DOI = {10.1016/S0021-7824(00)00162-8},
ZBLNUMBER = {0983.32023},
} -
[valtree] C. Favre and M. Jonsson, The Valuative Tree, New York: Springer-Verlag, 2004, vol. 1853.
@book {valtree, MRKEY = {2097722},
AUTHOR = {Favre, Charles and Jonsson, Mattias},
TITLE = {The Valuative Tree},
SERIES = {Lecture Notes in Math.},
VOLUME = {1853},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {2004},
PAGES = {xiv+234},
ISBN = {3-540-22984-1},
MRCLASS = {13A18 (12J20 13F30 14H20 37F99)},
MRNUMBER = {2097722},
MRREVIEWER = {Karen E. Smith},
ZBLNUMBER = {1064.14024},
} -
[pshsing]
C. Favre and M. Jonsson, "Valuative analysis of planar plurisubharmonic functions," Invent. Math., vol. 162, iss. 2, pp. 271-311, 2005.
@article {pshsing, MRKEY = {2199007},
AUTHOR = {Favre, Charles and Jonsson, Mattias},
TITLE = {Valuative analysis of planar plurisubharmonic functions},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {162},
YEAR = {2005},
NUMBER = {2},
PAGES = {271--311},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {32U05 (13A18 32U25)},
MRNUMBER = {2199007},
MRREVIEWER = {Romain Dujardin},
DOI = {10.1007/s00222-005-0443-2},
ZBLNUMBER = {1089.32032},
} -
[eigenval]
C. Favre and M. Jonsson, "Eigenvaluations," Ann. Sci. École Norm. Sup., vol. 40, iss. 2, pp. 309-349, 2007.
@article {eigenval, MRKEY = {2339287},
AUTHOR = {Favre, Charles and Jonsson, Mattias},
TITLE = {Eigenvaluations},
JOURNAL = {Ann. Sci. École Norm. Sup.},
FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
VOLUME = {40},
YEAR = {2007},
NUMBER = {2},
PAGES = {309--349},
ISSN = {0012-9593},
CODEN = {ASENAH},
MRCLASS = {37F10 (32H50)},
MRNUMBER = {2339287},
MRREVIEWER = {Romain Dujardin},
DOI = {10.1016/j.ansens.2007.01.002},
ZBLNUMBER = {1135.37018},
} -
[FS1]
J. E. Fornaess and N. Sibony, "Complex Hénon mappings in ${\bf C}^2$ and Fatou-Bieberbach domains," Duke Math. J., vol. 65, iss. 2, pp. 345-380, 1992.
@article {FS1, MRKEY = {1150591},
AUTHOR = {Forn{æ}ss, John Erik and Sibony, Nessim},
TITLE = {Complex {H}énon mappings in {${\bf C}\sp 2$} and {F}atou-{B}ieberbach domains},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {65},
YEAR = {1992},
NUMBER = {2},
PAGES = {345--380},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {32H50},
MRNUMBER = {1150591},
MRREVIEWER = {Marco Abate},
DOI = {10.1215/S0012-7094-92-06515-X},
ZBLNUMBER = {0761.32015},
} -
[FS2] J. E. Fornaess and N. Sibony, "Complex dynamics in higher dimension. II," in Modern Methods in Complex Analysis, Princeton, NJ: Princeton Univ. Press, 1995, vol. 137, pp. 135-182.
@incollection {FS2, MRKEY = {1369137},
AUTHOR = {Fornaess, John Erik and Sibony, Nessim},
TITLE = {Complex dynamics in higher dimension. {II}},
BOOKTITLE = {Modern Methods in Complex Analysis},
SERIES = {Ann. of Math. Stud.},
VOLUME = {137},
PAGES = {135--182},
PUBLISHER = {Princeton Univ. Press},
ADDRESS = {Princeton, NJ},
YEAR = {1995},
MRCLASS = {32H50 (32C30 58F23)},
MRNUMBER = {1369137},
MRREVIEWER = {Eric Bedford},
ZBLNUMBER = {0847.58059},
} -
[FriedlandMilnor]
S. Friedland and J. Milnor, "Dynamical properties of plane polynomial automorphisms," Ergodic Theory Dynam. Systems, vol. 9, iss. 1, pp. 67-99, 1989.
@article {FriedlandMilnor, MRKEY = {991490},
AUTHOR = {Friedland, Shmuel and Milnor, John},
TITLE = {Dynamical properties of plane polynomial automorphisms},
JOURNAL = {Ergodic Theory Dynam. Systems},
FJOURNAL = {Ergodic Theory and Dynamical Systems},
VOLUME = {9},
YEAR = {1989},
NUMBER = {1},
PAGES = {67--99},
ISSN = {0143-3857},
MRCLASS = {58F99},
MRNUMBER = {991490},
MRREVIEWER = {M. Lyubich},
DOI = {10.1017/S014338570000482X},
ZBLNUMBER = {0651.58027},
} -
[fulton] W. Fulton, Introduction to Toric Varieties, Princeton, NJ: Princeton Univ. Press, 1993, vol. 131.
@book {fulton, MRKEY = {1234037},
AUTHOR = {Fulton, William},
TITLE = {Introduction to Toric Varieties},
SERIES = {Annals of Math. Studies},
VOLUME = {131},
PUBLISHER = {Princeton Univ. Press},
ADDRESS = {Princeton, NJ},
YEAR = {1993},
PAGES = {xii+157},
ISBN = {0-691-00049-2},
MRCLASS = {14M25 (14-02 14J30)},
MRNUMBER = {1234037},
MRREVIEWER = {T. Oda},
ZBLNUMBER = {0813.14039},
} -
[giz] M. H. Gizatullin, "Rational $G$-surfaces," Izv. Akad. Nauk SSSR Ser. Mat., vol. 44, iss. 1, pp. 110-144, 239, 1980.
@article {giz, MRKEY = {563788},
AUTHOR = {Gizatullin, M. H.},
TITLE = {Rational {$G$}-surfaces},
JOURNAL = {Izv. Akad. Nauk SSSR Ser. Mat.},
FJOURNAL = {Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya},
VOLUME = {44},
YEAR = {1980},
NUMBER = {1},
PAGES = {110--144, 239},
ISSN = {0373-2436},
MRCLASS = {14J10 (14L30 14M20)},
MRNUMBER = {563788},
MRREVIEWER = {Gerhard Pfister},
ZBLNUMBER = {0428.14022},
} -
[guedjpoly]
V. Guedj, "Dynamics of polynomial mappings of $\Bbb C^2$," Amer. J. Math., vol. 124, iss. 1, pp. 75-106, 2002.
@article {guedjpoly, MRKEY = {1879000},
AUTHOR = {Guedj, Vincent},
TITLE = {Dynamics of polynomial mappings of {$\Bbb C\sp 2$}},
JOURNAL = {Amer. J. Math.},
FJOURNAL = {American Journal of Mathematics},
VOLUME = {124},
YEAR = {2002},
NUMBER = {1},
PAGES = {75--106},
ISSN = {0002-9327},
CODEN = {AJMAAN},
MRCLASS = {32H50 (32C30 37A25 37F10)},
MRNUMBER = {1879000},
MRREVIEWER = {John E. Forn{æ}ss},
URL = {http://muse.jhu.edu/journals/american_journal_of_mathematics/v124/124.1guedj.pdf},
ZBLNUMBER = {01731124},
} -
[guedjquad]
V. Guedj, "Dynamics of quadratic polynomial mappings of $\Bbb C^2$," Michigan Math. J., vol. 52, iss. 3, pp. 627-648, 2004.
@article {guedjquad, MRKEY = {2097402},
AUTHOR = {Guedj, Vincent},
TITLE = {Dynamics of quadratic polynomial mappings of {$\Bbb C\sp 2$}},
JOURNAL = {Michigan Math. J.},
FJOURNAL = {The Michigan Mathematical Journal},
VOLUME = {52},
YEAR = {2004},
NUMBER = {3},
PAGES = {627--648},
ISSN = {0026-2285},
MRCLASS = {37F10 (32H50 37A35)},
MRNUMBER = {2097402},
MRREVIEWER = {Jeffrey Diller},
DOI = {10.1307/mmj/1100623417},
ZBLNUMBER = {1073.37053},
} -
[guedjannals]
V. Guedj, "Ergodic properties of rational mappings with large topological degree," Ann. of Math., vol. 161, iss. 3, pp. 1589-1607, 2005.
@article {guedjannals, MRKEY = {2179389},
AUTHOR = {Guedj, Vincent},
TITLE = {Ergodic properties of rational mappings with large topological degree},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {161},
YEAR = {2005},
NUMBER = {3},
PAGES = {1589--1607},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {32H50 (32H30 37A20 37F10)},
MRNUMBER = {2179389},
MRREVIEWER = {Jean-Yves Briend},
DOI = {10.4007/annals.2005.161.1589},
ZBLNUMBER = {1088.37020},
} -
[propp]
B. Hasselblatt and J. Propp, "Degree-growth of monomial maps," Ergodic Theory Dynam. Systems, vol. 27, iss. 5, pp. 1375-1397, 2007.
@article {propp, MRKEY = {2358970},
AUTHOR = {Hasselblatt, Boris and Propp, James},
TITLE = {Degree-growth of monomial maps},
JOURNAL = {Ergodic Theory Dynam. Systems},
FJOURNAL = {Ergodic Theory and Dynamical Systems},
VOLUME = {27},
YEAR = {2007},
NUMBER = {5},
PAGES = {1375--1397},
ISSN = {0143-3857},
MRCLASS = {37F45 (32H50 37B40)},
MRNUMBER = {2358970},
MRREVIEWER = {Mattias Jonsson},
DOI = {10.1017/S0143385707000168},
ZBLNUMBER = {2358970},
} -
[HO]
J. H. Hubbard and R. W. Oberste-Vorth, "Hénon mappings in the complex domain. I. The global topology of dynamical space," Inst. Hautes Études Sci. Publ. Math., iss. 79, pp. 5-46, 1994.
@article {HO, MRKEY = {1307296},
AUTHOR = {Hubbard, John H. and Oberste-Vorth, Ralph W.},
TITLE = {Hénon mappings in the complex domain. {I}. {T}he global topology of dynamical space},
JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
NUMBER = {79},
YEAR = {1994},
PAGES = {5--46},
ISSN = {0073-8301},
CODEN = {PMIHA6},
MRCLASS = {58F23 (32H50)},
MRNUMBER = {1307296},
MRREVIEWER = {Eric Bedford},
ZBLNUMBER={0839.54029},
DOI={10.1007/BF02698886},
URL = {http://www.numdam.org/item?id=PMIHES_1994__79__5_0},
} -
[HP1] J. H. Hubbard and P. Papadopol, "Newton’s method applied to two quadratic equations in $\Bbb C^2$ viewed as a global dynamical system," Mem. Amer. Math. Soc., vol. 891, p. vi, 2008.
@article {HP1, MRKEY = {2376103},
AUTHOR = {Hubbard, John H. and Papadopol, Peter},
TITLE = {Newton's method applied to two quadratic equations in {$\Bbb C\sp 2$} viewed as a global dynamical system},
JOURNAL = {Mem. Amer. Math. Soc.},
FJOURNAL = {Memoirs of the American Mathematical Society},
VOLUME = {891},
YEAR = {2008},
PAGES = {vi+146},
ISSN = {0065-9266},
CODEN = {MAMCAU},
MRCLASS = {37F10 (32H50 37N40 65P99)},
MRNUMBER = {2376103},
MRREVIEWER = {Eric Bedford},
ZBLNUMBER={1133.37018},
} -
[HPV]
J. Hubbard, P. Papadopol, and V. Veselov, "A compactification of Hénon mappings in ${\bf C}^2$ as dynamical systems," Acta Math., vol. 184, iss. 2, pp. 203-270, 2000.
@article {HPV, MRKEY = {1768111},
AUTHOR = {Hubbard, John and Papadopol, Peter and Veselov, Vladimir},
TITLE = {A compactification of {H}énon mappings in {${\bf C}\sp 2$} as dynamical systems},
JOURNAL = {Acta Math.},
FJOURNAL = {Acta Mathematica},
VOLUME = {184},
YEAR = {2000},
NUMBER = {2},
PAGES = {203--270},
ISSN = {0001-5962},
CODEN = {ACMAA8},
MRCLASS = {37F10 (32H50)},
MRNUMBER = {1768111},
MRREVIEWER = {Adrien Douady},
DOI = {10.1007/BF02392629},
ZBLNUMBER = {0987.37035},
} -
[manin] Y. I. Manin, Cubic Forms. Algebra, Geometry, Arithmetic, Second ed., Amsterdam: North-Holland Publishing Co., 1986, vol. 4.
@book {manin, MRKEY = {833513},
AUTHOR = {Manin, Yu. I.},
TITLE = {Cubic Forms. {A}lgebra, Geometry, Arithmetic},
SERIES = {North-Holland Math. Library},
VOLUME = {4},
EDITION = {Second},
PUBLISHER = {North-Holland Publishing Co.},
ADDRESS = {Amsterdam},
YEAR = {1986},
PAGES = {x+326},
ISBN = {0-444-87823-8},
MRCLASS = {11Gxx (14Gxx 14J20)},
MRNUMBER = {833513},
ZBLNUMBER = {0582.14010},
} -
[Moh]
T. T. Moh, "On analytic irreducibility at $\infty $ of a pencil of curves," Proc. Amer. Math. Soc., vol. 44, pp. 22-24, 1974.
@article {Moh, MRKEY = {0357409},
AUTHOR = {Moh, T. T.},
TITLE = {On analytic irreducibility at {$\infty $} of a pencil of curves},
JOURNAL = {Proc. Amer. Math. Soc.},
FJOURNAL = {Proceedings of the American Mathematical Society},
VOLUME = {44},
YEAR = {1974},
PAGES = {22--24},
ISSN = {0002-9939},
MRCLASS = {14H45 (14C20)},
MRNUMBER = {0357409},
MRREVIEWER = {D Gallarati},
DOI = {10.2307/2039219},
ZBLNUMBER = {0309.14011},
} -
[Nguyen] V. Nguyên, "Algebraic degrees for iterates of meromorphic self-maps of ${\Bbb P}^k$," Publ. Mat., vol. 50, iss. 2, pp. 457-473, 2006.
@article {Nguyen, MRKEY = {2273670},
AUTHOR = {Vi{ê}t-Anh Nguy{ê}n},
TITLE = {Algebraic degrees for iterates of meromorphic self-maps of {${\Bbb P}\sp k$}},
JOURNAL = {Publ. Mat.},
FJOURNAL = {Publicacions Matemàtiques},
VOLUME = {50},
YEAR = {2006},
NUMBER = {2},
PAGES = {457--473},
ISSN = {0214-1493},
MRCLASS = {37F10 (32H50)},
MRNUMBER = {2273670},
ZBLNUMBER = {1112.37035},
} -
[RS]
A. Russakovskii and B. Shiffman, "Value distribution for sequences of rational mappings and complex dynamics," Indiana Univ. Math. J., vol. 46, iss. 3, pp. 897-932, 1997.
@article {RS, MRKEY = {1488341},
AUTHOR = {Russakovskii, Alexander and Shiffman, Bernard},
TITLE = {Value distribution for sequences of rational mappings and complex dynamics},
JOURNAL = {Indiana Univ. Math. 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},
MRREVIEWER = {Jeffrey Diller},
DOI = {10.1512/iumj.1997.46.1441},
ZBLNUMBER = {0901.58023},
} -
[Sib] N. Sibony, "Dynamique des applications rationnelles de $\Bbb P^k$," in Dynamique et Géométrie Complexes (Lyon, 1997), Paris: Soc. Math. France, 1999, vol. 8, p. ix-x, xi.
@incollection {Sib, MRKEY = {1760844},
AUTHOR = {Sibony, Nessim},
TITLE = {Dynamique des applications rationnelles de {$\bold P\sp k$}},
BOOKTITLE = {Dynamique et Géométrie Complexes {\rm ({L}yon, 1997)}},
SERIES = {Panor. Synthèses},
VOLUME = {8},
PAGES = {ix--x, xi--xii, 97--185},
PUBLISHER = {Soc. Math. France},
ADDRESS = {Paris},
YEAR = {1999},
MRCLASS = {32H50 (37F10 37F50)},
MRNUMBER = {1760844},
MRREVIEWER = {Mattias Jonsson},
ZBLNUMBER={1020.37026},
} -
[Vigny] G. Vigny, "Dynamics semi-conjugated to a subshift for some polynomial mappings in $\Bbb C^2$," Publ. Mat., vol. 51, iss. 1, pp. 201-222, 2007.
@article {Vigny, MRKEY = {2307152},
AUTHOR = {Vigny, Gabriel},
TITLE = {Dynamics semi-conjugated to a subshift for some polynomial mappings in {$\Bbb C\sp 2$}},
JOURNAL = {Publ. Mat.},
FJOURNAL = {Publicacions Matemàtiques},
VOLUME = {51},
YEAR = {2007},
NUMBER = {1},
PAGES = {201--222},
ISSN = {0214-1493},
MRCLASS = {37F10 (32H50 37B10)},
MRNUMBER = {2307152},
MRREVIEWER = {Eric Bedford},
ZBLNUMBER = {1125.37035},
} -
[ZS] O. Zariski and P. Samuel, Commutative Algebra, Vol. 2, New York: Springer-Verlag, 1975, vol. 29.
@book{ZS,
author={Zariski, O. and Samuel, P.},
TITLE={Commutative Algebra, Vol. 2},
SERIES={Grad. Texts Math.},
VOLUME={29},
PUBLISHER={Springer-Verlag},
ADDRESS={New York},
YEAR={1975},
MRNUMBER={0389876},
ZBLNUMBER={0322.13001},
}