Spectral interpretations of dynamical degrees and applications

Abstract

We prove that dynamical degrees of rational self-maps on projective varieties can be interpreted as spectral radii of naturally defined operators on suitable Banach spaces. Generalizing Shokurov’s notion of $b$-divisors, we consider the space of $b$-classes of higher codimension cycles, and endow this space with various Banach norms. Building on these constructions, we design a natural extension to higher dimensions of the Picard-Manin space introduced by Cantat and Boucksom-Favre-Jonsson in the case of surfaces. We prove a version of the Hodge index theorem, and a surprising compactness result in this Banach space. We use these two theorems to infer a precise control of the sequence of degrees of iterates of a map under the assumption $\lambda _1^2>\lambda _2$ on the dynamical degrees. As a consequence, we obtain that the dynamical degrees of an automorphism of the affine $3$-space are all algebraic numbers.

  • [MR1704016] N. Abarenkova, J. Anglès d’Auriac, S. Boukraa, S. Hassani, and J. -M. Maillard, "From Yang-Baxter equations to dynamical zeta functions for birational transformations," in Statistical Physics on the Eve of the 21st Century, World Sci. Publ., River Edge, NJ, 1999, vol. 14, pp. 436-490.
    @INCOLLECTION{MR1704016,
      author = {Abarenkova, N. and Anglès d'Auriac, J.-Ch. and Boukraa, S. and Hassani, S. and Maillard, J.-M.},
      title = {From {Y}ang-{B}axter equations to dynamical zeta functions for birational transformations},
      booktitle = {Statistical Physics on the Eve of the 21st Century},
      series = {Ser. Adv. Statist. Mech.},
      volume = {14},
      pages = {436--490},
      publisher = {World Sci. Publ., River Edge, NJ},
      year = {1999},
      mrclass = {37C30 (37D45 37E99 37N20 82B20 82B23)},
      mrnumber = {1704016},
      zblnumber = {},
      }
  • [MR1732080] Go to document N. Abarenkova, J. Anglès d’Auriac, S. Boukraa, S. Hassani, and J. -M. Maillard, "Topological entropy and Arnold complexity for two-dimensional mappings," Phys. Lett. A, vol. 262, iss. 1, pp. 44-49, 1999.
    @ARTICLE{MR1732080,
      author = {Abarenkova, N. and Anglès d'Auriac, J.-Ch. and Boukraa, S. and Hassani, S. and Maillard, J.-M.},
      title = {Topological entropy and {A}rnold complexity for two-dimensional mappings},
      journal = {Phys. Lett. A},
      fjournal = {Physics Letters. A},
      volume = {262},
      year = {1999},
      number = {1},
      pages = {44--49},
      issn = {0375-9601},
      mrclass = {37E30 (37B40 37C30 37C35)},
      mrnumber = {1732080},
      mrreviewer = {Joan Carles Tatjer},
      doi = {10.1016/S0375-9601(99)00662-3},
      url = {https://doi.org/10.1016/S0375-9601(99)00662-3},
      zblnumber = {0936.37005},
      }
  • [MR1694727] Go to document N. Abarenkova, J. Anglès d’Auriac, S. Boukraa, and J. -M. Maillard, "Growth-complexity spectrum of some discrete dynamical systems," Phys. D, vol. 130, iss. 1-2, pp. 27-42, 1999.
    @ARTICLE{MR1694727,
      author = {Abarenkova, N. and Anglès d'Auriac, J.-Ch. and Boukraa, S. and Maillard, J.-M.},
      title = {Growth-complexity spectrum of some discrete dynamical systems},
      journal = {Phys. D},
      fjournal = {Physica D. Nonlinear Phenomena},
      volume = {130},
      year = {1999},
      number = {1-2},
      pages = {27--42},
      issn = {0167-2789},
      mrclass = {37N20 (37D45 37M99)},
      mrnumber = {1694727},
      doi = {10.1016/S0167-2789(99)00014-7},
      url = {https://doi.org/10.1016/S0167-2789(99)00014-7},
      zblnumber = {0940.37017},
      }
  • [MR2582437] Go to document E. Amerik, "A computation of invariants of a rational self-map," Ann. Fac. Sci. Toulouse Math. (6), vol. 18, iss. 3, pp. 445-457, 2009.
    @ARTICLE{MR2582437,
      author = {Amerik, Ekaterina},
      title = {A computation of invariants of a rational self-map},
      journal = {Ann. Fac. Sci. Toulouse Math. (6)},
      fjournal = {Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série 6},
      volume = {18},
      year = {2009},
      number = {3},
      pages = {445--457},
      issn = {0240-2963},
      mrclass = {14E05},
      mrnumber = {2582437},
      mrreviewer = {Marko Roczen},
      doi = {10.5802/afst.1211},
      url = {https://doi.org/10.5802/afst.1211},
      zblnumber = {1204.14005},
      }
  • [MR2220002] Go to document J. Anglès d’Auriac, J. -M. Maillard, and C. M. Viallet, "On the complexity of some birational transformations," J. Phys. A, vol. 39, iss. 14, pp. 3641-3654, 2006.
    @ARTICLE{MR2220002,
      author = {Anglès 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},
      mrclass = {37F10 (14Q99 20F55 37B99)},
      mrnumber = {2220002},
      mrreviewer = {Eric Bedford},
      doi = {10.1088/0305-4470/39/14/009},
      url = {https://doi.org/10.1088/0305-4470/39/14/009},
      zblnumber = {1086.14504},
      }
  • [MR3277202] Go to document E. Bedford, S. Cantat, and K. Kim, "Pseudo-automorphisms with no invariant foliation," J. Mod. Dyn., vol. 8, iss. 2, pp. 221-250, 2014.
    @ARTICLE{MR3277202,
      author = {Bedford, Eric and Cantat, Serge and Kim, Kyounghee},
      title = {Pseudo-automorphisms with no invariant foliation},
      journal = {J. Mod. Dyn.},
      fjournal = {Journal of Modern Dynamics},
      volume = {8},
      year = {2014},
      number = {2},
      pages = {221--250},
      issn = {1930-5311},
      mrclass = {32H50 (14E05 37F10)},
      mrnumber = {3277202},
      mrreviewer = {Mattias Jonsson},
      doi = {10.3934/jmd.2014.8.221},
      url = {https://doi.org/10.3934/jmd.2014.8.221},
      zblnumber = {1333.37059},
      }
  • [MR2111418] Go to document 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{MR2111418,
      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},
      doi = {10.1007/BF02922170},
      url = {https://doi.org/10.1007/BF02922170},
      zblnumber = {1067.37054},
      }
  • [bedford_kim_degree_matrix] Go to document E. Bedford and K. Kim, "Degree growth of matrix inversion: birational maps of symmetric, cyclic matrices," Discrete Contin. Dyn. Syst., vol. 21, iss. 4, pp. 977-1013, 2008.
    @ARTICLE{bedford_kim_degree_matrix,
      author = {Bedford, Eric and Kim, Kyounghee},
      title = {Degree growth of matrix inversion: birational maps of symmetric, cyclic matrices},
      journal = {Discrete Contin. Dyn. Syst.},
      fjournal = {Discrete and Continuous Dynamical Systems. Series A},
      volume = {21},
      year = {2008},
      number = {4},
      pages = {977--1013},
      issn = {1078-0947},
      mrclass = {14E07 (37F10)},
      mrnumber = {2399446},
      mrreviewer = {Jeffrey Diller},
      doi = {10.3934/dcds.2008.21.977},
      url = {https://doi.org/10.3934/dcds.2008.21.977},
      zblnumber = {1215.37031},
      }
  • [MR2428100] Go to document E. Bedford, K. Kim, T. T. Truong, N. Abarenkova, and J. Maillard, "Degree complexity of a family of birational maps," Math. Phys. Anal. Geom., vol. 11, iss. 1, pp. 53-71, 2008.
    @ARTICLE{MR2428100,
      author = {Bedford, Eric and Kim, Kyounghee and Truong, Tuyen Trung and Abarenkova, Nina and Maillard, Jean-Marie},
      title = {Degree complexity of a family of birational maps},
      journal = {Math. Phys. Anal. Geom.},
      fjournal = {Mathematical Physics, Analysis and Geometry. An International Journal Devoted to the Theory and Applications of Analysis and Geometry to Physics},
      volume = {11},
      year = {2008},
      number = {1},
      pages = {53--71},
      issn = {1385-0172},
      mrclass = {32H50 (14E05 37F10)},
      mrnumber = {2428100},
      mrreviewer = {Julie Déserti},
      doi = {10.1007/s11040-008-9039-6},
      url = {https://doi.org/10.1007/s11040-008-9039-6},
      zblnumber = {1153.37387},
      }
  • [bedford_truong] Go to document E. Bedford and T. T. Truong, "Degree complexity of birational maps related to matrix inversion," Comm. Math. Phys., vol. 298, iss. 2, pp. 357-368, 2010.
    @ARTICLE{bedford_truong,
      author = {Bedford, Eric and Truong, Tuyen Trung},
      title = {Degree complexity of birational maps related to matrix inversion},
      journal = {Comm. Math. Phys.},
      fjournal = {Communications in Mathematical Physics},
      volume = {298},
      year = {2010},
      number = {2},
      pages = {357--368},
      issn = {0010-3616},
      mrclass = {37F10 (32H50)},
      mrnumber = {2669440},
      mrreviewer = {Julie Déserti},
      doi = {10.1007/s00220-010-1063-2},
      url = {https://doi.org/10.1007/s00220-010-1063-2},
      zblnumber = {1205.15009},
      }
  • [bell_diller_jonsson] Go to document J. P. Bell, J. Diller, and M. Jonsson, "A transcendental dynamical degree," Acta Math., vol. 225, iss. 2, pp. 193-225, 2020.
    @ARTICLE{bell_diller_jonsson,
      author = {Bell, Jason P. and Diller, Jeffrey and Jonsson, Mattias},
      title = {A transcendental dynamical degree},
      journal = {Acta Math.},
      fjournal = {Acta Mathematica},
      volume = {225},
      year = {2020},
      number = {2},
      pages = {193--225},
      issn = {0001-5962},
      mrclass = {37P05 (11J81 14E05 32H50 37C35 37F80)},
      mrnumber = {4205407},
      doi = {10.4310/ACTA.2020.v225.n2.a1},
      url = {https://doi.org/10.4310/ACTA.2020.v225.n2.a1},
      zblnumber = {07335070},
      }
  • [MR1751620] Go to document M. P. Bellon, "Algebraic entropy of birational maps with invariant curves," Lett. Math. Phys., vol. 50, iss. 1, pp. 79-90, 1999.
    @ARTICLE{MR1751620,
      author = {Bellon, M. P.},
      title = {Algebraic entropy of birational maps with invariant curves},
      journal = {Lett. Math. Phys.},
      fjournal = {Letters in Mathematical Physics},
      volume = {50},
      year = {1999},
      number = {1},
      pages = {79--90},
      issn = {0377-9017},
      mrclass = {37E99 (32H50 37B40 37F10)},
      mrnumber = {1751620},
      mrreviewer = {Serge Cantat},
      doi = {10.1023/A:1007634406786},
      url = {https://doi.org/10.1023/A:1007634406786},
      zblnumber = {1052.37508},
      }
  • [MR1704282] Go to document M. P. Bellon and C. -M. Viallet, "Algebraic entropy," Comm. Math. Phys., vol. 204, iss. 2, pp. 425-437, 1999.
    @ARTICLE{MR1704282,
      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},
      mrclass = {37D45 (37A35 37C99 39A12)},
      mrnumber = {1704282},
      mrreviewer = {Remo Badii},
      doi = {10.1007/s002200050652},
      url = {https://doi.org/10.1007/s002200050652},
      zblnumber = {0987.37007},
      }
  • [blanc_cantat] Go to document J. Blanc and S. Cantat, "Dynamical degrees of birational transformations of projective surfaces," J. Amer. Math. Soc., vol. 29, iss. 2, pp. 415-471, 2016.
    @ARTICLE{blanc_cantat,
      author = {Blanc, J. and Cantat, Serge},
      title = {Dynamical degrees of birational transformations of projective surfaces},
      journal = {J. Amer. Math. Soc.},
      fjournal = {Journal of the Amer. Math. Soc.},
      volume = {29},
      year = {2016},
      number = {2},
      pages = {415--471},
      issn = {0894-0347},
      mrclass = {37F10 (14E05 32H50)},
      mrnumber = {3454379},
      mrreviewer = {Fei Hu},
      doi = {10.1090/jams831},
      url = {https://doi.org/10.1090/jams831},
      zblnumber = {1394.14011},
      }
  • [blanc_santen_automorphism] J. Blanc and I. van Santen, Automorphisms of the affine 3-space of degree 3, 2019.
    @MISC{blanc_santen_automorphism,
      author = {Blanc, J. and van Santen, I.},
      title = {Automorphisms of the affine 3-space of degree 3},
      arxiv = {1912.02144},
      year = {2019},
      zblnumber = {},
      }
  • [blanc_santen_dynamical] J. Blanc and I. van Santen, Dynamical degrees of affine-triangular automorphisms of affine spaces, 2019.
    @MISC{blanc_santen_dynamical,
      author = {Blanc, J. and van Santen, I.},
      title = {Dynamical degrees of affine-triangular automorphisms of affine spaces},
      arxiv = {1912.01324},
      year = {2019},
      zblnumber = {},
      }
  • [boucksom_favre_jonsson_deggrowth] Go to document 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{boucksom_favre_jonsson_deggrowth,
      author = {Boucksom, Sébastien 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},
      mrclass = {32H04 (14E05 32J27)},
      mrnumber = {2387430},
      mrreviewer = {Ma\l gorzata Stawiska},
      doi = {10.1215/00127094-2007-004},
      url = {https://doi.org/10.1215/00127094-2007-004},
      zblnumber = {1185.32009},
      }
  • [MR2426355] Go to document S. Boucksom, C. Favre, and M. Jonsson, "Valuations and plurisubharmonic singularities," Publ. Res. Inst. Math. Sci., vol. 44, iss. 2, pp. 449-494, 2008.
    @ARTICLE{MR2426355,
      author = {Boucksom, Sébastien and Favre, Charles and Jonsson, Mattias},
      title = {Valuations and plurisubharmonic singularities},
      journal = {Publ. Res. Inst. Math. Sci.},
      fjournal = {Kyoto Univ.. Research Institute for Mathematical Sciences. Publications},
      volume = {44},
      year = {2008},
      number = {2},
      pages = {449--494},
      issn = {0034-5318},
      mrclass = {32U25 (13A18 14B05 32U05)},
      mrnumber = {2426355},
      mrreviewer = {Romain Dujardin},
      doi = {10.2977/prims/1210167334},
      url = {https://doi.org/10.2977/prims/1210167334},
      zblnumber = {1146.32017},
      }
  • [zbMATH05541351] Go to document S. Boucksom, C. Favre, and M. Jonsson, "Differentiability of volumes of divisors and a problem of Teissier," J. Algebraic Geom., vol. 18, iss. 2, pp. 279-308, 2009.
    @ARTICLE{zbMATH05541351,
      author = {Boucksom, Sébastien and Favre, Charles and Jonsson, Mattias},
      title = {Differentiability of volumes of divisors and a problem of {T}eissier},
      journal = {J. Algebraic Geom.},
      fjournal = {Journal of Algebraic Geometry},
      volume = {18},
      year = {2009},
      number = {2},
      pages = {279--308},
      issn = {1056-3911},
      mrclass = {14C20 (14C17)},
      mrnumber = {2475816},
      mrreviewer = {James McKernan},
      doi = {10.1090/S1056-3911-08-00490-6},
      url = {https://doi.org/10.1090/S1056-3911-08-00490-6},
      zblnumber = {1162.14003},
      }
  • [boucksom_fernex_favre_urbinati] Go to document S. Boucksom, T. de Fernex, C. Favre, and S. Urbinati, "Valuation spaces and multiplier ideals on singular varieties," in Recent Advances in Algebraic Geometry, Cambridge Univ. Press, Cambridge, 2015, vol. 417, pp. 29-51.
    @INCOLLECTION{boucksom_fernex_favre_urbinati,
      author = {Boucksom, S. and de Fernex, T. and Favre, C. and Urbinati, S.},
      title = {Valuation spaces and multiplier ideals on singular varieties},
      booktitle = {Recent Advances in Algebraic Geometry},
      series = {London Math. Soc. Lecture Note Ser.},
      volume = {417},
      pages = {29--51},
      publisher = {Cambridge Univ. Press, Cambridge},
      year = {2015},
      mrclass = {14F18 (13A18 14B05 14C20 14J17)},
      mrnumber = {3380442},
      mrreviewer = {Patrick Graf},
      url = {https://doi.org/10.1017/CBO9781107416000.004},
      doi = {10.1017/CBO9781107416000.004},
      zblnumber = {1330.14025},
      }
  • [boucksom_fernex_favre] Go to document S. Boucksom, T. de Fernex, and C. Favre, "The volume of an isolated singularity," Duke Math. J., vol. 161, iss. 8, pp. 1455-1520, 2012.
    @ARTICLE{boucksom_fernex_favre,
      author = {Boucksom, Sebastien and de Fernex, Tommaso and Favre, Charles},
      title = {The volume of an isolated singularity},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {161},
      year = {2012},
      number = {8},
      pages = {1455--1520},
      issn = {0012-7094},
      mrclass = {14J17 (14B05 14C20 14F18)},
      mrnumber = {2931273},
      mrreviewer = {Cihan Özgür},
      doi = {10.1215/00127094-1593317},
      url = {https://doi.org/10.1215/00127094-1593317},
      zblnumber = {1251.14026},
      }
  • [MR2019672] Go to document S. Boukraa, S. Hassani, and J. -M. Maillard, "Noetherian mappings," Phys. D, vol. 185, iss. 1, pp. 3-44, 2003.
    @ARTICLE{MR2019672,
      author = {Boukraa, S. and Hassani, S. and Maillard, J.-M.},
      title = {Noetherian mappings},
      journal = {Phys. D},
      fjournal = {Physica D. Nonlinear Phenomena},
      volume = {185},
      year = {2003},
      number = {1},
      pages = {3--44},
      issn = {0167-2789},
      mrclass = {37F10 (14E05 37C30 39B12 82C20)},
      mrnumber = {2019672},
      mrreviewer = {Charles Favre},
      doi = {10.1016/S0167-2789(03)00177-5},
      url = {https://doi.org/10.1016/S0167-2789(03)00177-5},
      zblnumber = {1078.37035},
      }
  • [MR1254598] Go to document S. Boukraa, J. -M. Maillard, and G. Rollet, "Almost integrable mappings," Internat. J. Modern Phys. B, vol. 8, iss. 1-2, pp. 137-174, 1994.
    @ARTICLE{MR1254598,
      author = {Boukraa, S. and Maillard, J.-M. and Rollet, G.},
      title = {Almost integrable mappings},
      journal = {Internat. J. Modern Phys. B},
      fjournal = {International Journal of Modern Physics B},
      volume = {8},
      year = {1994},
      number = {1-2},
      pages = {137--174},
      issn = {0217-9792},
      mrclass = {82B23 (14H52 58F07 58F13)},
      mrnumber = {1254598},
      doi = {10.1142/S0217979294000063},
      url = {https://doi.org/10.1142/S0217979294000063},
      zblnumber = {0804.58045},
      }
  • [brezis] Go to document H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.
    @BOOK{brezis,
      author = {Brezis, Haim},
      title = {Functional Analysis, {S}obolev Spaces and Partial Differential Equations},
      series = {Universitext},
      publisher = {Springer, New York},
      year = {2011},
      pages = {xiv+599},
      isbn = {978-0-387-70913-0},
      mrclass = {35-01 (46-01 46E35 46N20 47F05)},
      mrnumber = {2759829},
      mrreviewer = {Vicen\c{t}iu D. R\u{a}dulescu},
      doi = {10.1007/978-0-387-70914-7},
      url = {https://doi.org/10.1007/978-0-387-70914-7},
      zblnumber = {1220.46002},
      }
  • [cantat_bir_surfaces] Go to document S. Cantat, "Sur les groupes de transformations birationnelles des surfaces," Ann. of Math. (2), vol. 174, iss. 1, pp. 299-340, 2011.
    @ARTICLE{cantat_bir_surfaces,
      author = {Cantat, Serge},
      title = {Sur les groupes de transformations birationnelles des surfaces},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {174},
      year = {2011},
      number = {1},
      pages = {299--340},
      issn = {0003-486X},
      mrclass = {14E07 (14E05 32J15 37F10)},
      mrnumber = {2811600},
      mrreviewer = {Andreas Höring},
      doi = {10.4007/annals.2011.174.1.8},
      url = {https://doi.org/10.4007/annals.2011.174.1.8},
      zblnumber = {1233.14011},
      }
  • [cantat_xie_degrees] Go to document S. Cantat and J. Xie, "On degrees of birational mappings," Math. Res. Lett., vol. 27, iss. 2, pp. 319-337, 2020.
    @ARTICLE{cantat_xie_degrees,
      author = {Cantat, Serge and Xie, Junyi},
      title = {On degrees of birational mappings},
      journal = {Math. Res. Lett.},
      fjournal = {Mathematical Research Letters},
      volume = {27},
      year = {2020},
      number = {2},
      pages = {319--337},
      issn = {1073-2780},
      mrclass = {37F80 (14E05 37C35 37P99)},
      mrnumber = {4117078},
      mrreviewer = {Jeffrey Diller},
      doi = {10.4310/MRL.2020.v27.n2.a1},
      url = {https://doi.org/10.4310/MRL.2020.v27.n2.a1},
      zblnumber = {1441.14044},
      }
  • [corti_3_fold] Go to document A. Corti, "3-fold flips after Shokurov," in Flips for 3-Folds and 4-Folds, Oxford Univ. Press, Oxford, 2007, vol. 35, pp. 18-48.
    @INCOLLECTION{corti_3_fold,
      author = {Corti, Alessio},
      title = {3-fold flips after {S}hokurov},
      booktitle = {Flips for 3-Folds and 4-Folds},
      series = {Oxford Lecture Ser. Math. Appl.},
      volume = {35},
      pages = {18--48},
      publisher = {Oxford Univ. Press, Oxford},
      year = {2007},
      mrclass = {14E30},
      mrnumber = {2359340},
      doi = {10.1093/acprof:oso/9780198570615.003.0002},
      url = {https://doi.org/10.1093/acprof:oso/9780198570615.003.0002},
      zblnumber = {1286.14022},
      }
  • [dang_tame] N. Dang, Degree growth for tame automorphisms of an affine quadric threefold, 2018.
    @MISC{dang_tame,
      author = {Dang, Nguyen-Bac},
      title = {Degree growth for tame automorphisms of an affine quadric threefold},
      arxiv = {1810.09094},
      year = {2018},
      zblnumber = {},
      }
  • [dang_degree_normal] Go to document N. Dang, "Degrees of iterates of rational maps on normal projective varieties," Proc. Lond. Math. Soc. (3), vol. 121, iss. 5, pp. 1268-1310, 2020.
    @ARTICLE{dang_degree_normal,
      author = {Dang, Nguyen-Bac},
      title = {Degrees of iterates of rational maps on normal projective varieties},
      journal = {Proc. Lond. Math. Soc. (3)},
      fjournal = {Proceedings of the London Mathematical Society. Third Series},
      volume = {121},
      year = {2020},
      number = {5},
      pages = {1268--1310},
      issn = {0024-6115},
      mrclass = {14E05 (37C35 37P55)},
      mrnumber = {4133708},
      doi = {10.1112/plms.12366},
      url = {https://doi.org/10.1112/plms.12366},
      zblnumber = {1455.14028},
      }
  • [dang_favre] N. Dang and C. Favre, Intersection theory of nef $b$-divisor classes, 2020.
    @MISC{dang_favre,
      author = {Dang, Nguyen-Bac and Favre, C.},
      title = {Intersection theory of nef $b$-divisor classes},
      arxiv = {2007.04549},
      year = {2020},
      zblnumber = {},
      }
  • [de_thelin_vigny_2010] Go to document H. De Thélin and G. Vigny, "Entropy of meromorphic maps and dynamics of birational maps," Mém. Soc. Math. Fr. (N.S.), iss. 122, p. vi, 2010.
    @ARTICLE{de_thelin_vigny_2010,
      author = {De Thélin, Henry and Vigny, Gabriel},
      title = {Entropy of meromorphic maps and dynamics of birational maps},
      journal = {Mém. Soc. Math. Fr. (N.S.)},
      fjournal = {Mémoires de la Société Mathématique de France. Nouvelle Série},
      number = {122},
      year = {2010},
      pages = {vi+98},
      issn = {0249-633X},
      isbn = {978-2-85629-302-7},
      mrclass = {37F10 (32H50 37A25 37A35)},
      mrnumber = {2752759},
      mrreviewer = {Ma\l gorzata Stawiska},
      doi = {10.24033/msmf.434},
      url = {https://doi.org/10.24033/msmf.434},
      zblnumber = {1214.37004},
      }
  • [debarre_higher] Go to document O. Debarre, Higher-Dimensional Algebraic Geometry, Springer-Verlag, New York, 2001.
    @BOOK{debarre_higher,
      author = {Debarre, Olivier},
      title = {Higher-Dimensional Algebraic Geometry},
      series = {Universitext},
      publisher = {Springer-Verlag, New York},
      year = {2001},
      pages = {xiv+233},
      isbn = {0-387-95227-6},
      mrclass = {14-02 (14E30 14Jxx)},
      mrnumber = {1841091},
      mrreviewer = {Mark Gross},
      doi = {10.1007/978-1-4757-5406-3},
      url = {https://doi.org/10.1007/978-1-4757-5406-3},
      zblnumber = {0978.14001},
      }
  • [debarre_ein_lazarsfeld_voisin] Go to document O. Debarre, L. Ein, R. Lazarsfeld, and C. Voisin, "Pseudoeffective and nef classes on abelian varieties," Compos. Math., vol. 147, iss. 6, pp. 1793-1818, 2011.
    @ARTICLE{debarre_ein_lazarsfeld_voisin,
      author = {Debarre, Olivier and Ein, Lawrence and Lazarsfeld, Robert and Voisin, Claire},
      title = {Pseudoeffective and nef classes on abelian varieties},
      journal = {Compos. Math.},
      fjournal = {Compositio Mathematica},
      volume = {147},
      year = {2011},
      number = {6},
      pages = {1793--1818},
      issn = {0010-437X},
      mrclass = {14C25 (14K05)},
      mrnumber = {2862063},
      mrreviewer = {Fumio Hazama},
      doi = {10.1112/S0010437X11005227},
      url = {https://doi.org/10.1112/S0010437X11005227},
      zblnumber = {1234.14008},
      }
  • [deserti_degree_examples] Go to document J. Déserti, "Degree growth of polynomial automorphisms and birational maps: some examples," Eur. J. Math., vol. 4, iss. 1, pp. 200-211, 2018.
    @ARTICLE{deserti_degree_examples,
      author = {D{é}serti, Julie},
      title = {Degree growth of polynomial automorphisms and birational maps: some examples},
      journal = {Eur. J. Math.},
      fjournal = {European Journal of Mathematics},
      volume = {4},
      year = {2018},
      number = {1},
      pages = {200--211},
      issn = {2199-675X},
      mrclass = {32H50 (14E07 37F10)},
      mrnumber = {3782220},
      mrreviewer = {Fei Hu},
      doi = {10.1007/s40879-017-0175-z},
      url = {https://doi.org/10.1007/s40879-017-0175-z},
      zblnumber = {1397.32002},
      }
  • [zbMATH05723568] Go to document 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. (4), vol. 43, iss. 2, pp. 235-278, 2010.
    @ARTICLE{zbMATH05723568,
      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. \'{E}c. Norm. Supér. (4)},
      fjournal = {Annales Scientifiques de l'\'{E}cole Normale Supérieure. Quatrième Série},
      volume = {43},
      year = {2010},
      number = {2},
      pages = {235--278},
      issn = {0012-9593},
      mrclass = {32H50 (32U40 37F10 37F15)},
      mrnumber = {2662665},
      mrreviewer = {Tien Cuong Dinh},
      doi = {10.24033/asens.2120},
      url = {https://doi.org/10.24033/asens.2120},
      zblnumber = {1197.37059},
      }
  • [diller_favre] Go to document J. Diller and C. Favre, "Dynamics of bimeromorphic maps of surfaces," Amer. J. Math., vol. 123, iss. 6, pp. 1135-1169, 2001.
    @ARTICLE{diller_favre,
      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},
      mrclass = {32H50 (37F10)},
      mrnumber = {1867314},
      mrreviewer = {Serge Cantat},
      doi = {10.1353/ajm.2001.0038},
      url = {https://doi.org/10.1353/ajm.2001.0038},
      zblnumber = {1112.37308},
      }
  • [dinh_nguyen_2011] Go to document T. Dinh and V. Nguyên, "Comparison of dynamical degrees for semi-conjugate meromorphic maps," Comment. Math. Helv., vol. 86, iss. 4, pp. 817-840, 2011.
    @ARTICLE{dinh_nguyen_2011,
      author = {Dinh, Tien-Cuong and Nguy\^{e}n, Vi\^{e}t-Anh},
      title = {Comparison of dynamical degrees for semi-conjugate meromorphic maps},
      journal = {Comment. Math. Helv.},
      fjournal = {Commentarii Mathematici Helvetici. A Journal of the Swiss Mathematical Society},
      volume = {86},
      year = {2011},
      number = {4},
      pages = {817--840},
      issn = {0010-2571},
      mrclass = {32H50 (32U40 37F10)},
      mrnumber = {2851870},
      mrreviewer = {Andreas Höring},
      doi = {10.4171/CMH/241},
      url = {https://doi.org/10.4171/CMH/241},
      zblnumber = {1279.32018},
      }
  • [dinh_nguyen_truong] Go to document T. Dinh, V. Nguyên, and T. T. Truong, "On the dynamical degrees of meromorphic maps preserving a fibration," Commun. Contemp. Math., vol. 14, iss. 6, p. 1250042, 2012.
    @ARTICLE{dinh_nguyen_truong,
      author = {Dinh, Tien-Cuong and Nguy\^{e}n, Vi\^{e}t-Anh and Truong, Tuyen Trung},
      title = {On the dynamical degrees of meromorphic maps preserving a fibration},
      journal = {Commun. Contemp. Math.},
      fjournal = {Communications in Contemporary Mathematics},
      volume = {14},
      year = {2012},
      number = {6},
      pages = {1250042, 18},
      issn = {0219-1997},
      mrclass = {37F10 (32H50 32U40)},
      mrnumber = {2989646},
      mrreviewer = {Roland K. W. Roeder},
      doi = {10.1142/S0219199712500423},
      url = {https://doi.org/10.1142/S0219199712500423},
      zblnumber = {1311.37034},
      }
  • [blms.12082] Go to document T. Dinh, V. Nguyên, and T. T. Truong, "Growth of the number of periodic points for meromorphic maps," Bull. Lond. Math. Soc., vol. 49, iss. 6, pp. 947-964, 2017.
    @ARTICLE{blms.12082,
      author = {Dinh, Tien-Cuong and Nguy\^{e}n, Vi\^{e}t-Anh and Truong, Tuyen Trung},
      title = {Growth of the number of periodic points for meromorphic maps},
      journal = {Bull. Lond. Math. Soc.},
      fjournal = {Bulletin of the London Mathematical Society},
      volume = {49},
      year = {2017},
      number = {6},
      pages = {947--964},
      issn = {0024-6093},
      mrclass = {32H50 (32U40 37F10)},
      mrnumber = {3743479},
      mrreviewer = {Matteo Ruggiero},
      doi = {10.1112/blms.12082},
      url = {https://doi.org/10.1112/blms.12082},
      zblnumber = {1401.37054},
      }
  • [dinh_sibony_2004_groupes] Go to document T. Dinh and N. Sibony, "Groupes commutatifs d’automorphismes d’une variété kählérienne compacte," Duke Math. J., vol. 123, iss. 2, pp. 311-328, 2004.
    @ARTICLE{dinh_sibony_2004_groupes,
      author = {Dinh, Tien-Cuong and Sibony, Nessim},
      title = {Groupes commutatifs d'automorphismes d'une variété kählérienne compacte},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {123},
      year = {2004},
      number = {2},
      pages = {311--328},
      issn = {0012-7094},
      mrclass = {32H50 (32J27 32Q15 37B40 37C85 37F10)},
      mrnumber = {2066940},
      mrreviewer = {Mattias Jonsson},
      doi = {10.1215/S0012-7094-04-12323-1},
      url = {https://doi.org/10.1215/S0012-7094-04-12323-1},
      zblnumber = {1065.32012},
      }
  • [dinh_sibony_reg_currents_entropy] Go to document T. Dinh and N. Sibony, "Regularization of currents and entropy," Ann. Sci. École Norm. Sup. (4), vol. 37, iss. 6, pp. 959-971, 2004.
    @ARTICLE{dinh_sibony_reg_currents_entropy,
      author = {Dinh, Tien-Cuong and Sibony, Nessim},
      title = {Regularization of currents and entropy},
      journal = {Ann. Sci. \'{E}cole Norm. Sup. (4)},
      fjournal = {Annales Scientifiques de l'\'{E}cole Normale Supérieure. Quatrième Série},
      volume = {37},
      year = {2004},
      number = {6},
      pages = {959--971},
      issn = {0012-9593},
      mrclass = {32U40 (32C30 32H04 32Q15 37B40)},
      mrnumber = {2119243},
      mrreviewer = {Ma\l gorzata Stawiska},
      doi = {10.1016/j.ansens.2004.09.002},
      url = {https://doi.org/10.1016/j.ansens.2004.09.002},
      zblnumber = {1074.53058},
      }
  • [dinh_sibony_une_borne_sup] Go to document T. Dinh and N. Sibony, "Une borne supérieure pour l’entropie topologique d’une application rationnelle," Ann. of Math. (2), vol. 161, iss. 3, pp. 1637-1644, 2005.
    @ARTICLE{dinh_sibony_une_borne_sup,
      author = {Dinh, Tien-Cuong and Sibony, Nessim},
      title = {Une borne supérieure pour l'entropie topologique d'une application rationnelle},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {161},
      year = {2005},
      number = {3},
      pages = {1637--1644},
      issn = {0003-486X},
      mrclass = {32H50 (37B40 37F05)},
      mrnumber = {2180409},
      mrreviewer = {Jean-Yves Briend},
      doi = {10.4007/annals.2005.161.1637},
      url = {https://doi.org/10.4007/annals.2005.161.1637},
      zblnumber = {1084.54013},
      }
  • [favre_jonsson_eigenvaluations] Go to document C. Favre and M. Jonsson, "Eigenvaluations," Ann. Sci. École Norm. Sup. (4), vol. 40, iss. 2, pp. 309-349, 2007.
    @ARTICLE{favre_jonsson_eigenvaluations,
      author = {Favre, Charles and Jonsson, Mattias},
      title = {Eigenvaluations},
      journal = {Ann. Sci. \'{E}cole Norm. Sup. (4)},
      fjournal = {Annales Scientifiques de l'\'{E}cole Normale Supérieure. Quatrième Série},
      volume = {40},
      year = {2007},
      number = {2},
      pages = {309--349},
      issn = {0012-9593},
      mrclass = {37F10 (32H50)},
      mrnumber = {2339287},
      mrreviewer = {Romain Dujardin},
      doi = {10.1016/j.ansens.2007.01.002},
      url = {https://doi.org/10.1016/j.ansens.2007.01.002},
      zblnumber = {1135.37018},
      }
  • [favre_jonsson_dynamical_compactification] Go to document C. Favre and M. Jonsson, "Dynamical compactifications of ${\bf C}^2$," Ann. of Math. (2), vol. 173, iss. 1, pp. 211-248, 2011.
    @ARTICLE{favre_jonsson_dynamical_compactification,
      author = {Favre, Charles and Jonsson, Mattias},
      title = {Dynamical compactifications of {${\bf C}^2$}},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {173},
      year = {2011},
      number = {1},
      pages = {211--248},
      issn = {0003-486X},
      mrclass = {32H50 (32J05 32U35 37F10)},
      mrnumber = {2753603},
      mrreviewer = {Araceli Bonifant},
      doi = {10.4007/annals.2011.173.1.6},
      url = {https://doi.org/10.4007/annals.2011.173.1.6},
      zblnumber = {1244.32012},
      }
  • [favre_wulcan] Go to document C. Favre and E. Wulcan, "Degree growth of monomial maps and McMullen’s polytope algebra," Indiana Univ. Math. J., vol. 61, iss. 2, pp. 493-524, 2012.
    @ARTICLE{favre_wulcan,
      author = {Favre, Charles and Wulcan, Elizabeth},
      title = {Degree growth of monomial maps and {M}c{M}ullen's polytope algebra},
      journal = {Indiana Univ. Math. J.},
      fjournal = {Indiana Univ. Mathematics Journal},
      volume = {61},
      year = {2012},
      number = {2},
      pages = {493--524},
      issn = {0022-2518},
      mrclass = {32H50 (14E05 52A39)},
      mrnumber = {3043585},
      mrreviewer = {Jan-Li Lin},
      doi = {10.1512/iumj.2012.61.4555},
      url = {https://doi.org/10.1512/iumj.2012.61.4555},
      zblnumber = {1291.37058},
      }
  • [friedland_milnor] Go to document S. Friedland and J. Milnor, "Dynamical properties of plane polynomial automorphisms," Ergodic Theory Dynam. Systems, vol. 9, iss. 1, pp. 67-99, 1989.
    @ARTICLE{friedland_milnor,
      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 = {0991490},
      mrreviewer = {M. Lyubich},
      doi = {10.1017/S014338570000482X},
      url = {https://doi.org/10.1017/S014338570000482X},
      zblnumber = {0651.58027},
      }
  • [MATH06741284] Go to document M. Fulger and B. Lehmann, "Positive cones of dual cycle classes," Algebr. Geom., vol. 4, iss. 1, pp. 1-28, 2017.
    @ARTICLE{MATH06741284,
      author = {Fulger, Mihai and Lehmann, Brian},
      title = {Positive cones of dual cycle classes},
      journal = {Algebr. Geom.},
      fjournal = {Algebraic Geometry},
      volume = {4},
      year = {2017},
      number = {1},
      pages = {1--28},
      issn = {2313-1691},
      mrclass = {14C25 (14C17 14C20 14C40)},
      mrnumber = {3592463},
      mrreviewer = {Alberto Collino},
      doi = {10.14231/AG-2017-001},
      url = {https://doi.org/10.14231/AG-2017-001},
      zblnumber = {1376.14009},
      }
  • [fulton] Go to document W. Fulton, Intersection Theory, Second ed., Springer-Verlag, Berlin, 1998, vol. 2.
    @BOOK{fulton,
      author = {Fulton, William},
      title = {Intersection Theory},
      series = {Ergeb. Math. Grenzgeb.},
      volume = {2},
      edition = {Second},
      publisher = {Springer-Verlag, Berlin},
      year = {1998},
      pages = {xiv+470},
      isbn = {3-540-62046-X; 0-387-98549-2},
      mrclass = {14C17 (14-02)},
      mrnumber = {1644323},
      doi = {10.1007/978-1-4612-1700-8},
      url = {https://doi.org/10.1007/978-1-4612-1700-8},
      zblnumber = {0885.14002},
      }
  • [MR2087743] Go to document B. Grammaticos and A. Ramani, "Discrete Painlevé equations: a review," in Discrete Integrable Systems, Springer, Berlin, 2004, vol. 644, pp. 245-321.
    @INCOLLECTION{MR2087743,
      author = {Grammaticos, B. and Ramani, A.},
      title = {Discrete {P}ainlevé equations: a review},
      booktitle = {Discrete Integrable Systems},
      series = {Lecture Notes in Phys.},
      volume = {644},
      pages = {245--321},
      publisher = {Springer, Berlin},
      year = {2004},
      mrclass = {39A12 (33E17 34M55 37J35 37K20)},
      mrnumber = {2087743},
      mrreviewer = {Andrew Pickering},
      doi = {10.1007/978-3-540-40357-9\_7},
      url = {https://doi.org/10.1007/978-3-540-40357-9_7},
      zblnumber = {1064.39019},
      }
  • [MR2026895] Go to document M. Gromov, "On the entropy of holomorphic maps," Enseign. Math. (2), vol. 49, iss. 3-4, pp. 217-235, 2003.
    @ARTICLE{MR2026895,
      author = {Gromov, Mikhaïl},
      title = {On the entropy of holomorphic maps},
      journal = {Enseign. Math. (2)},
      fjournal = {L'Enseignement Mathématique. Revue Internationale. 2e Série},
      volume = {49},
      year = {2003},
      number = {3-4},
      pages = {217--235},
      issn = {0013-8584},
      mrclass = {37F10 (32H50 37B40)},
      mrnumber = {2026895},
      mrreviewer = {Serge Cantat},
      url = {https://www.e-periodica.ch/digbib/view?pid=ens-001%3A2003%3A49%3A%3A562#566},
      zblnumber = {1080.37051},
      }
  • [MR2139697] Go to document V. Guedj, "Decay of volumes under iteration of meromorphic mappings," Ann. Inst. Fourier (Grenoble), vol. 54, iss. 7, pp. 2369-2386, 2004.
    @ARTICLE{MR2139697,
      author = {Guedj, Vincent},
      title = {Decay of volumes under iteration of meromorphic mappings},
      journal = {Ann. Inst. Fourier (Grenoble)},
      fjournal = {Université de Grenoble. Annales de l'Institut Fourier},
      volume = {54},
      year = {2004},
      number = {7},
      pages = {2369--2386},
      issn = {0373-0956},
      mrclass = {32H50 (37F10)},
      mrnumber = {2139697},
      mrreviewer = {Jeffrey Diller},
      doi = {10.5802/aif.2083},
      url = {https://doi.org/10.5802/aif.2083},
      zblnumber = {1069.32007},
      }
  • [zbMATH05004659] Go to document V. Guedj, "Ergodic properties of rational mappings with large topological degree," Ann. of Math. (2), vol. 161, iss. 3, pp. 1589-1607, 2005.
    @ARTICLE{zbMATH05004659,
      author = {Guedj, Vincent},
      title = {Ergodic properties of rational mappings with large topological degree},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {161},
      year = {2005},
      number = {3},
      pages = {1589--1607},
      issn = {0003-486X},
      mrclass = {32H50 (32H30 37A20 37F10)},
      mrnumber = {2179389},
      mrreviewer = {Jean-Yves Briend},
      doi = {10.4007/annals.2005.161.1589},
      url = {https://doi.org/10.4007/annals.2005.161.1589},
      zblnumber = {1088.37020},
      }
  • [guignard] Go to document Q. Guignard, "A New Proof of Raynaud–Gruson’s Flattening Theorem," Int. Math. Res. Not. IMRN, iss. 9, pp. 6932-6966, 2021.
    @ARTICLE{guignard,
      author = {Guignard, Quentin},
      title = {A {N}ew {P}roof of {R}aynaud--{G}ruson's {F}lattening {T}heorem},
      journal = {Int. Math. Res. Not. IMRN},
      fjournal = {International Mathematics Research Notices. IMRN},
      year = {2021},
      number = {9},
      pages = {6932--6966},
      issn = {1073-7928},
      mrclass = {14G22 (14E15)},
      mrnumber = {4251293},
      doi = {10.1093/imrn/rnz042},
      url = {https://doi.org/10.1093/imrn/rnz042},
      zblnumber = {},
      }
  • [hartshorne] Go to document R. Hartshorne, Algebraic Geometry, Springer-Verlag, New York, 1977, vol. 52.
    @BOOK{hartshorne,
      author = {Hartshorne, Robin},
      title = {Algebraic Geometry},
      note = {Grad. Texts in Math.},
      volume = {52},
      publisher = {Springer-Verlag, New York},
      year = {1977},
      pages = {xvi+496},
      isbn = {0-387-90244-9},
      mrclass = {14-01},
      mrnumber = {0463157},
      mrreviewer = {Robert Speiser},
      doi = {10.1007/978-1-4757-3849-0},
      url = {https://doi.org/10.1007/978-1-4757-3849-0},
      zblnumber = {0531.14001},
      }
  • [hille_phillips] Go to document E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, Amer. Math. Soc., Providence, R. I., 1957, vol. 31.
    @BOOK{hille_phillips,
      author = {Hille, Einar and Phillips, Ralph S.},
      title = {Functional Analysis and Semi-Groups},
      series = {Amer. Math. Soc. Colloq. Publ.},
      volume = {31},
      note = {rev. ed},
      publisher = {Amer. Math. Soc., Providence, R. I.},
      year = {1957},
      pages = {xii+808},
      mrclass = {46.2X},
      mrnumber = {0089373},
      mrreviewer = {M. H. Stone},
      doi = {10.1090/coll/031},
      url = {https://doi.org/10.1090/coll/031},
      zblnumber = {0078.10004},
      }
  • [MR393556] Go to document H. Hironaka, "Flattening theorem in complex-analytic geometry," Amer. J. Math., vol. 97, pp. 503-547, 1975.
    @ARTICLE{MR393556,
      author = {Hironaka, Heisuke},
      title = {Flattening theorem in complex-analytic geometry},
      journal = {Amer. J. Math.},
      fjournal = {American Journal of Mathematics},
      volume = {97},
      year = {1975},
      pages = {503--547},
      issn = {0002-9327},
      mrclass = {32C45},
      mrnumber = {0393556},
      mrreviewer = {Margherita Galbiati},
      doi = {10.2307/2373721},
      url = {https://doi.org/10.2307/2373721},
      zblnumber = {0307.32011},
      }
  • [zbMATH06522286] Go to document F. Hu, J. Keum, and D. Zhang, "Criteria for the existence of equivariant fibrations on algebraic surfaces and hyperkähler manifolds and equality of automorphisms up to powers: a dynamical viewpoint," J. Lond. Math. Soc. (2), vol. 92, iss. 3, pp. 724-735, 2015.
    @ARTICLE{zbMATH06522286,
      author = {Hu, Fei and Keum, JongHae and Zhang, De-Qi},
      title = {Criteria for the existence of equivariant fibrations on algebraic surfaces and hyperkähler manifolds and equality of automorphisms up to powers: a dynamical viewpoint},
      journal = {J. Lond. Math. Soc. (2)},
      fjournal = {Journal of the London Mathematical Society. Second Series},
      volume = {92},
      year = {2015},
      number = {3},
      pages = {724--735},
      issn = {0024-6107},
      mrclass = {32H50 (14J50 32J25 37B40 37C85 53C26)},
      mrnumber = {3431659},
      mrreviewer = {Mattias Jonsson},
      doi = {10.1112/jlms/jdv045},
      url = {https://doi.org/10.1112/jlms/jdv045},
      zblnumber = {1332.14050},
      }
  • [MR2376103] Go to document 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. 191, iss. 891, p. vi, 2008.
    @ARTICLE{MR2376103,
      author = {Hubbard, John H. and Papadopol, Peter},
      title = {Newton's method applied to two quadratic equations in {$\Bbb C^2$} viewed as a global dynamical system},
      journal = {Mem. Amer. Math. Soc.},
      fjournal = {Memoirs of the Amer. Math. Soc.},
      volume = {191},
      year = {2008},
      number = {891},
      pages = {vi+146},
      issn = {0065-9266},
      mrclass = {37F10 (32H50 37N40 65P99)},
      mrnumber = {2376103},
      mrreviewer = {Eric Bedford},
      doi = {10.1090/memo/0891},
      url = {https://doi.org/10.1090/memo/0891},
      zblnumber = {1133.37018},
      }
  • [MR3330767] Go to document M. Jonsson, "Dynamics of Berkovich spaces in low dimensions," in Berkovich Spaces and Applications, Springer, Cham, 2015, vol. 2119, pp. 205-366.
    @INCOLLECTION{MR3330767,
      author = {Jonsson, Mattias},
      title = {Dynamics of {B}erkovich spaces in low dimensions},
      booktitle = {Berkovich Spaces and Applications},
      series = {Lecture Notes in Math.},
      volume = {2119},
      pages = {205--366},
      publisher = {Springer, Cham},
      year = {2015},
      mrclass = {37P50 (14G22 26E30)},
      mrnumber = {3330767},
      mrreviewer = {Benjamin A. Hutz},
      doi = {10.1007/978-3-319-11029-5\_6},
      url = {https://doi.org/10.1007/978-3-319-11029-5_6},
      zblnumber = {1401.37103},
      }
  • [MR3060755] Go to document M. Jonsson and M. Mustactua, "Valuations and asymptotic invariants for sequences of ideals," Ann. Inst. Fourier (Grenoble), vol. 62, iss. 6, pp. 2145-2209, 2013.
    @ARTICLE{MR3060755,
      author = {Jonsson, Mattias and Musta\c{t}\u{a},
      Mircea},
      title = {Valuations and asymptotic invariants for sequences of ideals},
      journal = {Ann. Inst. Fourier (Grenoble)},
      fjournal = {Université de Grenoble. Annales de l'Institut Fourier},
      volume = {62},
      year = {2013},
      number = {6},
      pages = {2145--2209},
      issn = {0373-0956},
      mrclass = {14F18 (12J20 14B05)},
      mrnumber = {3060755},
      mrreviewer = {Carlos Galindo},
      doi = {10.5802/aif.2746},
      url = {https://doi.org/10.5802/aif.2746},
      zblnumber = {1272.14016},
      }
  • [zbMATH03142405] Go to document S. Karlin, "Positive operators," J. Math. Mech., vol. 8, pp. 907-937, 1959.
    @ARTICLE{zbMATH03142405,
      author = {Karlin, Samuel},
      title = {Positive operators},
      journal = {J. Math. Mech.},
      volume = {8},
      year = {1959},
      pages = {907--937},
      mrclass = {46.00},
      mrnumber = {0114138},
      mrreviewer = {F. F. Bonsall},
      doi = {10.1512/iumj.1959.8.58058},
      url = {https://doi.org/10.1512/iumj.1959.8.58058},
      zblnumber = {0087.11002},
      }
  • [kawaguchi_silverman] Go to document S. Kawaguchi and J. H. Silverman, "On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties," J. Reine Angew. Math., vol. 713, pp. 21-48, 2016.
    @ARTICLE{kawaguchi_silverman,
      author = {Kawaguchi, Shu and Silverman, Joseph H.},
      title = {On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties},
      journal = {J. Reine Angew. Math.},
      fjournal = {Journal für die Reine und Angewandte Mathematik. [Crelle's Journal]},
      volume = {713},
      year = {2016},
      pages = {21--48},
      issn = {0075-4102},
      mrclass = {37P55 (37P30)},
      mrnumber = {3483624},
      mrreviewer = {Yu Yasufuku},
      doi = {10.1515/crelle-2014-0020},
      url = {https://doi.org/10.1515/crelle-2014-0020},
      zblnumber = {1393.37115},
      }
  • [MR3570024] Go to document S. Koch and R. K. W. Roeder, "Computing dynamical degrees of rational maps on moduli space," Ergodic Theory Dynam. Systems, vol. 36, iss. 8, pp. 2538-2579, 2016.
    @ARTICLE{MR3570024,
      author = {Koch, Sarah and Roeder, Roland K. W.},
      title = {Computing dynamical degrees of rational maps on moduli space},
      journal = {Ergodic Theory Dynam. Systems},
      fjournal = {Ergodic Theory and Dynamical Systems},
      volume = {36},
      year = {2016},
      number = {8},
      pages = {2538--2579},
      issn = {0143-3857},
      mrclass = {32H50 (14E05 14H10)},
      mrnumber = {3570024},
      mrreviewer = {Romain Dujardin},
      doi = {10.1017/etds.2015.29},
      url = {https://doi.org/10.1017/etds.2015.29},
      zblnumber = {1378.37082},
      }
  • [kollar] Go to document J. Kollár, Lectures on Resolution of Singularities (AM-166), Princeton Univ. Press, 2007, vol. 166.
    @BOOK{kollar,
      author = {Koll{á}r, J.},
      title = {Lectures on Resolution of Singularities (AM-166)},
      series = {Ann. of Math. Stud.},
      publisher = {Princeton Univ. Press},
      year = {2007},
      volume = {166},
      isbn = {978-0-691-12923-5; 0-691-12923-1},
      mrclass = {14E15 (32S45)},
      mrnumber = {2289519},
      mrreviewer = {Dan Abramovich},
      doi = {10.1515/9781400827800},
      url = {https://doi.org/10.1515/9781400827800},
      zblnumber = {1113.14013},
      }
  • [MR3010168] Go to document A. Küronya and C. Maclean, "Zariski decomposition of b-divisors," Math. Z., vol. 273, iss. 1-2, pp. 427-436, 2013.
    @ARTICLE{MR3010168,
      author = {Küronya, Alex and Maclean, Catriona},
      title = {Zariski decomposition of b-divisors},
      journal = {Math. Z.},
      fjournal = {Mathematische Zeitschrift},
      volume = {273},
      year = {2013},
      number = {1-2},
      pages = {427--436},
      issn = {0025-5874},
      mrclass = {14C20},
      mrnumber = {3010168},
      mrreviewer = {Kelly Jabbusch},
      doi = {10.1007/s00209-012-1012-1},
      url = {https://doi.org/10.1007/s00209-012-1012-1},
      zblnumber = {1271.14019},
      }
  • [lamy] S. Lamy and P. Przytycki, Presqu’un immeuble pour le groupe des automorphismes modérés, 2018.
    @MISC{lamy,
      author = {Lamy, S. and Przytycki, P.},
      title = {Presqu'un immeuble pour le groupe des automorphismes modérés},
      arxiv = {1802.00481},
      year = {2018},
      zblnumber = {},
      }
  • [lazarsfeld_positivity_1] Go to document R. Lazarsfeld, Positivity in Algebraic Geometry. I. Classical Setting: Line Bundles and Linear Series, Springer-Verlag, Berlin, 2004, vol. 48.
    @BOOK{lazarsfeld_positivity_1,
      author = {Lazarsfeld, Robert},
      title = {Positivity in Algebraic Geometry. {I.} Classical Setting: Line Bundles and Linear Series},
      series = {Ergeb. Math. Grenzgeb.},
      volume = {48},
      publisher = {Springer-Verlag, Berlin},
      year = {2004},
      pages = {xviii+387},
      isbn = {3-540-22533-1},
      mrclass = {14-02 (14C20)},
      mrnumber = {2095471},
      mrreviewer = {Mihnea Popa},
      doi = {10.1007/978-3-642-18808-4},
      url = {https://doi.org/10.1007/978-3-642-18808-4},
      zblnumber = {1093.14501},
      }
  • [lin] Go to document J. Lin, "Algebraic stability and degree growth of monomial maps," Math. Z., vol. 271, iss. 1-2, pp. 293-311, 2012.
    @ARTICLE{lin,
      author = {Lin, Jan-Li},
      title = {Algebraic stability and degree growth of monomial maps},
      journal = {Math. Z.},
      fjournal = {Mathematische Zeitschrift},
      volume = {271},
      year = {2012},
      number = {1-2},
      pages = {293--311},
      issn = {0025-5874},
      mrclass = {37P05},
      mrnumber = {2917145},
      mrreviewer = {Peter Haïssinsky},
      doi = {10.1007/s00209-011-0864-0},
      url = {https://doi.org/10.1007/s00209-011-0864-0},
      zblnumber = {1247.32018},
      }
  • [MR4030548] Go to document F. Lo Bianco, "On the cohomological action of automorphisms of compact Kähler threefolds," Bull. Soc. Math. France, vol. 147, iss. 3, pp. 469-514, 2019.
    @ARTICLE{MR4030548,
      author = {Lo Bianco, F.},
      title = {On the cohomological action of automorphisms of compact {K}ähler threefolds},
      journal = {Bull. Soc. Math. France},
      fjournal = {Bulletin de la Société Mathématique de France},
      volume = {147},
      year = {2019},
      number = {3},
      pages = {469--514},
      issn = {0037-9484},
      mrclass = {14J50 (14J30 32Q15)},
      mrnumber = {4030548},
      mrreviewer = {G. K. Sankaran},
      doi = {10.24033/bsmf.279},
      url = {https://doi.org/10.24033/bsmf.279},
      zblnumber = {1444.14073},
      }
  • [bianco] Go to document F. Lo Bianco, "On the primitivity of birational transformations of irreducible holomorphic symplectic manifolds," Int. Math. Res. Not. IMRN, iss. 1, pp. 1-32, 2019.
    @ARTICLE{bianco,
      author = {Lo Bianco, Federico},
      title = {On the primitivity of birational transformations of irreducible holomorphic symplectic manifolds},
      journal = {Int. Math. Res. Not. IMRN},
      fjournal = {International Mathematics Research Notices. IMRN},
      year = {2019},
      number = {1},
      pages = {1--32},
      issn = {1073-7928},
      mrclass = {32H50 (14E07 32J27 37F10)},
      mrnumber = {3897424},
      mrreviewer = {Mattias Jonsson},
      doi = {10.1093/imrn/rnx109},
      url = {https://doi.org/10.1093/imrn/rnx109},
      zblnumber = {1425.32017},
      }
  • [urech-lonjou] A. Lonjou and C. Urech, Actions of Cremona groups on $\mathrm{CAT}(0)$ cube complexes, 2020.
    @MISC{urech-lonjou,
      author = {Lonjou, A. and Urech, C.},
      title = {Actions of {C}remona groups on {$\mathrm{CAT}(0)$} cube complexes},
      arxiv = {2001.00783},
      year = {2020},
      note = {{\em Duke Math. J.},
      to appear},
      }
  • [maegawa] Go to document K. Maegawa, "Classification of quadratic polynomial automorphisms of ${\Bbb C}^3$ from a dynamical point of view," Indiana Univ. Math. J., vol. 50, iss. 2, pp. 935-951, 2001.
    @ARTICLE{maegawa,
      author = {Maegawa, Kazutoshi},
      title = {Classification of quadratic polynomial automorphisms of {${\Bbb C}^3$} from a dynamical point of view},
      journal = {Indiana Univ. Math. J.},
      fjournal = {Indiana Univ. Mathematics Journal},
      volume = {50},
      year = {2001},
      number = {2},
      pages = {935--951},
      issn = {0022-2518},
      mrclass = {37F10 (32H50)},
      mrnumber = {1864065},
      mrreviewer = {John E. Fornæ ss},
      doi = {10.1512/iumj.2001.50.1833},
      url = {https://doi.org/10.1512/iumj.2001.50.1833},
      zblnumber = {1046.37029},
      }
  • [manin_cubic] Y. I. Manin, Cubic Forms. Algebra, Geometry, Arithmetic, Second ed., North-Holland Publishing Co., Amsterdam, 1986, vol. 4.
    @BOOK{manin_cubic,
      author = {Manin, Yu. I.},
      title = {Cubic Forms. {A}lgebra, {G}eometry, {A}rithmetic},
      series = {North-Holland Math. Library},
      volume = {4},
      edition = {Second},
      titlenote = {translated from the Russian by M. Hazewinkel},
      publisher = {North-Holland Publishing Co., Amsterdam},
      year = {1986},
      pages = {x+326},
      isbn = {0-444-87823-8},
      mrclass = {11Gxx (14Gxx 14J20)},
      mrnumber = {0833513},
      zblnumber = {0582.14010},
      }
  • [matsuzawa_sano_shibata_surfaces] Go to document Y. Matsuzawa, K. Sano, and T. Shibata, "Arithmetic degrees and dynamical degrees of endomorphisms on surfaces," Algebra Number Theory, vol. 12, iss. 7, pp. 1635-1657, 2018.
    @ARTICLE{matsuzawa_sano_shibata_surfaces,
      author = {Matsuzawa, Yohsuke and Sano, Kaoru and Shibata, Takahiro},
      title = {Arithmetic degrees and dynamical degrees of endomorphisms on surfaces},
      journal = {Algebra Number Theory},
      fjournal = {Algebra \& Number Theory},
      volume = {12},
      year = {2018},
      number = {7},
      pages = {1635--1657},
      issn = {1937-0652},
      mrclass = {14G05 (11G35 11G50 14E05 37P05 37P15 37P30)},
      mrnumber = {3871505},
      mrreviewer = {Benjamin A. Hutz},
      doi = {10.2140/ant.2018.12.1635},
      url = {https://doi.org/10.2140/ant.2018.12.1635},
      zblnumber = {1423.14163},
      }
  • [matsuzawa_sano_shibata] Go to document Y. Matsuzawa, K. Sano, and T. Shibata, "Arithmetic degrees for dynamical systems over function fields of characteristic zero," Math. Z., vol. 290, iss. 3-4, pp. 1063-1083, 2018.
    @ARTICLE{matsuzawa_sano_shibata,
      author = {Matsuzawa, Yohsuke and Sano, Kaoru and Shibata, Takahiro},
      title = {Arithmetic degrees for dynamical systems over function fields of characteristic zero},
      journal = {Math. Z.},
      fjournal = {Mathematische Zeitschrift},
      volume = {290},
      year = {2018},
      number = {3-4},
      pages = {1063--1083},
      issn = {0025-5874},
      mrclass = {37P15 (14E05)},
      mrnumber = {3856845},
      mrreviewer = {Dragos Ghioca},
      doi = {10.1007/s00209-018-2053-x},
      url = {https://doi.org/10.1007/s00209-018-2053-x},
      zblnumber = {1401.37102},
      }
  • [maulik_poonen] Go to document D. Maulik and B. Poonen, "Néron-Severi groups under specialization," Duke Math. J., vol. 161, iss. 11, pp. 2167-2206, 2012.
    @ARTICLE{maulik_poonen,
      author = {Maulik, Davesh and Poonen, Bjorn},
      title = {Néron-{S}everi groups under specialization},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {161},
      year = {2012},
      number = {11},
      pages = {2167--2206},
      issn = {0012-7094},
      mrclass = {14J10 (14C25 14D07)},
      mrnumber = {2957700},
      mrreviewer = {Ekaterina Amerik},
      doi = {10.1215/00127094-1699490},
      url = {https://doi.org/10.1215/00127094-1699490},
      zblnumber = {1248.14011},
      }
  • [MR153682] Go to document D. Mumford, "The topology of normal singularities of an algebraic surface and a criterion for simplicity," Inst. Hautes Études Sci. Publ. Math., iss. 9, pp. 5-22, 1961.
    @ARTICLE{MR153682,
      author = {Mumford, David},
      title = {The topology of normal singularities of an algebraic surface and a criterion for simplicity},
      journal = {Inst. Hautes \'{E}tudes Sci. Publ. Math.},
      fjournal = {Institut des Hautes \'{E}tudes Scientifiques. Publications Mathématiques},
      number = {9},
      year = {1961},
      pages = {5--22},
      issn = {0073-8301},
      mrclass = {14.18 (14.55)},
      mrnumber = {0153682},
      mrreviewer = {T. Matsusaka},
      url = {http://www.numdam.org/item?id=PMIHES_1961__9__5_0},
      zblnumber = {0108.16801},
      }
  • [zbMATH06911551] Go to document K. Oguiso, "A criterion for the primitivity of a birational automorphism of a Calabi-Yau manifold and an application," Math. Res. Lett., vol. 25, iss. 1, pp. 181-198, 2018.
    @ARTICLE{zbMATH06911551,
      author = {Oguiso, Keiji},
      title = {A criterion for the primitivity of a birational automorphism of a {C}alabi-{Y}au manifold and an application},
      journal = {Math. Res. Lett.},
      fjournal = {Mathematical Research Letters},
      volume = {25},
      year = {2018},
      number = {1},
      pages = {181--198},
      issn = {1073-2780},
      mrclass = {14E07 (14J32)},
      mrnumber = {3818619},
      mrreviewer = {Alan Matthew Thompson},
      doi = {10.4310/mrl.2018.v25.n1.a8},
      url = {https://doi.org/10.4310/mrl.2018.v25.n1.a8},
      zblnumber = {1403.14075},
      }
  • [popovici] Go to document D. Popovici, "Sufficient bigness criterion for differences of two nef classes," Math. Ann., vol. 364, iss. 1-2, pp. 649-655, 2016.
    @ARTICLE{popovici,
      author = {Popovici, Dan},
      title = {Sufficient bigness criterion for differences of two nef classes},
      journal = {Math. Ann.},
      fjournal = {Mathematische Annalen},
      volume = {364},
      year = {2016},
      number = {1-2},
      pages = {649--655},
      issn = {0025-5831},
      mrclass = {32J25 (32W20)},
      mrnumber = {3451400},
      mrreviewer = {Jian Xiao},
      doi = {10.1007/s00208-015-1230-z},
      url = {https://doi.org/10.1007/s00208-015-1230-z},
      zblnumber = {1341.32015},
      }
  • [MR3725886] Go to document R. Ramadas, "Hurwitz correspondences on compactifications of $\mathcal M_{0,N}$," Adv. Math., vol. 323, pp. 622-667, 2018.
    @ARTICLE{MR3725886,
      author = {Ramadas, Rohini},
      title = {Hurwitz correspondences on compactifications of {$\mathcal M_{0,N}$}},
      journal = {Adv. Math.},
      fjournal = {Advances in Mathematics},
      volume = {323},
      year = {2018},
      pages = {622--667},
      issn = {0001-8708},
      mrclass = {14H10 (37F20)},
      mrnumber = {3725886},
      mrreviewer = {Dawei Chen},
      doi = {10.1016/j.aim.2017.10.044},
      url = {https://doi.org/10.1016/j.aim.2017.10.044},
      zblnumber = {1387.14068},
      }
  • [MR302645] Go to document M. Raynaud, "Flat modules in algebraic geometry," Compositio Math., vol. 24, iss. 1, pp. 11-31, 1972.
    @ARTICLE{MR302645,
      author = {Raynaud, M.},
      title = {Flat modules in algebraic geometry},
      journal = {Compositio Math.},
      fjournal = {Compositio Mathematica},
      volume = {24},
      number = {1},
      year = {1972},
      pages = {11--31},
      issn = {0010-437X},
      mrclass = {14A05 (13C99)},
      mrnumber = {0302645},
      mrreviewer = {T. Oda},
      url = {http://www.numdam.org/item?id=CM_1972__24_1_11_0},
      zblnumber = {0244.14001},
      }
  • [MR1488341] Go to document 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{MR1488341,
      author = {Russakovskii, Alexander and Shiffman, Bernard},
      title = {Value distribution for sequences of rational mappings and complex dynamics},
      journal = {Indiana Univ. Math. J.},
      fjournal = {Indiana Univ. Mathematics Journal},
      volume = {46},
      year = {1997},
      number = {3},
      pages = {897--932},
      issn = {0022-2518},
      mrclass = {32H30 (32H50)},
      mrnumber = {1488341},
      mrreviewer = {Jeffrey Diller},
      doi = {10.1512/iumj.1997.46.1441},
      url = {https://doi.org/10.1512/iumj.1997.46.1441},
      zblnumber = {0901.58023},
      }
  • [shokurov_3_fold] Go to document V. V. Shokurov, "$3$-fold log models," J. Math. Sci., vol. 81, iss. 3, pp. 2667-2699, 1996.
    @article{shokurov_3_fold,
      author = {Shokurov, V. V.},
      title = {{$3$}-fold log models},
      titlenote = {Algebraic Geometry, 4},
      journal = {J. Math. Sci.},
      fjournal = {Journal of Mathematical Sciences},
      volume = {81},
      year = {1996},
      number = {3},
      pages = {2667--2699},
      issn = {1072-3374},
      mrclass = {14E30 (14E15 14E35)},
      mrnumber = {1420223},
      mrreviewer = {Alessio Corti},
      doi = {10.1007/BF02362335},
      url = {https://doi.org/10.1007/BF02362335},
      zblnumber = {0873.14014},
      }
  • [shokurov_prelimiting] V. V. Shokurov, "Prelimiting flips," Tr. Mat. Inst. Steklova, vol. 240, iss. Biratsion. Geom. Lineĭn. Sist. Konechno Porozhdennye Algebry, pp. 82-219, 2003.
    @ARTICLE{shokurov_prelimiting,
      author = {Shokurov, V. V.},
      title = {Prelimiting flips},
      journal = {Tr. Mat. Inst. Steklova},
      fjournal = {Trudy Matematicheskogo Instituta Imeni V. A. Steklova},
      volume = {240},
      year = {2003},
      number = {Biratsion. Geom. Lineĭn. Sist. Konechno Porozhdennye Algebry},
      pages = {82--219},
      issn = {0371-9685},
      mrclass = {14E30},
      mrnumber = {1993750},
      mrreviewer = {S\'{a}ndor J. Kov\'{a}cs},
      zblnumber = {1082.14019},
      }
  • [trapani] Go to document S. Trapani, "Numerical criteria for the positivity of the difference of ample divisors," Math. Z., vol. 219, iss. 3, pp. 387-401, 1995.
    @ARTICLE{trapani,
      author = {Trapani, Stefano},
      title = {Numerical criteria for the positivity of the difference of ample divisors},
      journal = {Math. Z.},
      fjournal = {Mathematische Zeitschrift},
      volume = {219},
      year = {1995},
      number = {3},
      pages = {387--401},
      issn = {0025-5874},
      mrclass = {14C20 (14F17 32C30 32L20)},
      mrnumber = {1339712},
      mrreviewer = {Claire Voisin},
      doi = {10.1007/BF02572372},
      url = {https://doi.org/10.1007/BF02572372},
      zblnumber = {0828.14002},
      }
  • [MR3255693] Go to document T. T. Truong, "The simplicity of the first spectral radius of a meromorphic map," Michigan Math. J., vol. 63, iss. 3, pp. 623-633, 2014.
    @ARTICLE{MR3255693,
      author = {Truong, Tuyen Trung},
      title = {The simplicity of the first spectral radius of a meromorphic map},
      journal = {Michigan Math. J.},
      fjournal = {Michigan Mathematical Journal},
      volume = {63},
      year = {2014},
      number = {3},
      pages = {623--633},
      issn = {0026-2285},
      mrclass = {32H50 (32H04 53C55)},
      mrnumber = {3255693},
      mrreviewer = {Vincent Guedj},
      doi = {10.1307/mmj/1409932635},
      url = {https://doi.org/10.1307/mmj/1409932635},
      zblnumber = {1308.32020},
      }
  • [tuyen2] Go to document T. T. Truong, "Relative dynamical degrees of correspondences over a field of arbitrary characteristic," J. Reine Angew. Math., vol. 758, pp. 139-182, 2020.
    @ARTICLE{tuyen2,
      author = {Truong, Tuyen Trung},
      title = {Relative dynamical degrees of correspondences over a field of arbitrary characteristic},
      journal = {J. Reine Angew. Math.},
      fjournal = {Journal für die Reine und Angewandte Mathematik. [Crelle's Journal]},
      volume = {758},
      year = {2020},
      pages = {139--182},
      issn = {0075-4102},
      mrclass = {37P05 (14E05 37F05 37F80)},
      mrnumber = {4048444},
      doi = {10.1515/crelle-2017-0052},
      url = {https://doi.org/10.1515/crelle-2017-0052},
      zblnumber = {07148358},
      }
  • [zbMATH06911556] Go to document C. Urech, "Remarks on the degree growth of birational transformations," Math. Res. Lett., vol. 25, iss. 1, pp. 291-308, 2018.
    @ARTICLE{zbMATH06911556,
      author = {Urech, Christian},
      title = {Remarks on the degree growth of birational transformations},
      journal = {Math. Res. Lett.},
      fjournal = {Mathematical Research Letters},
      volume = {25},
      year = {2018},
      number = {1},
      pages = {291--308},
      issn = {1073-2780},
      mrclass = {37P05 (14E07 32H50 37C35)},
      mrnumber = {3818624},
      mrreviewer = {Mattias Jonsson},
      doi = {10.4310/mrl.2018.v25.n1.a13},
      url = {https://doi.org/10.4310/mrl.2018.v25.n1.a13},
      zblnumber = {1393.37111},
      }
  • [vaquie] Go to document M. Vaquié, "Valuations and local uniformization," in Singularity Theory and its Applications, Math. Soc. Japan, Tokyo, 2006, vol. 43, pp. 477-527.
    @INCOLLECTION{vaquie,
      author = {Vaquié,
      Michel},
      title = {Valuations and local uniformization},
      booktitle = {Singularity Theory and its Applications},
      series = {Adv. Stud. Pure Math.},
      volume = {43},
      pages = {477--527},
      publisher = {Math. Soc. Japan, Tokyo},
      year = {2006},
      mrclass = {12J20 (13A18 14E15 14J15)},
      mrnumber = {2325152},
      mrreviewer = {K. Kiyek},
      doi = {10.2969/aspm/04310477},
      url = {https://doi.org/10.2969/aspm/04310477},
      zblnumber = {},
      }
  • [MR1155288] Go to document A. P. Veselov, "Growth and integrability in the dynamics of mappings," Comm. Math. Phys., vol. 145, iss. 1, pp. 181-193, 1992.
    @ARTICLE{MR1155288,
      author = {Veselov, Alexander P.},
      title = {Growth and integrability in the dynamics of mappings},
      journal = {Comm. Math. Phys.},
      fjournal = {Communications in Mathematical Physics},
      volume = {145},
      year = {1992},
      number = {1},
      pages = {181--193},
      issn = {0010-3616},
      mrclass = {58F23 (14H99)},
      mrnumber = {1155288},
      mrreviewer = {Qi Keith Chen},
      doi = {10.1007/BF02099285},
      url = {https://doi.org/10.1007/BF02099285},
      zblnumber = {0751.58034},
      }
  • [MR3335696] Go to document C. -M. Viallet, "On the algebraic structure of rational discrete dynamical systems," J. Phys. A, vol. 48, iss. 16, p. 16, 2015.
    @ARTICLE{MR3335696,
      author = {Viallet, C.-M.},
      title = {On the algebraic structure of rational discrete dynamical systems},
      journal = {J. Phys. A},
      fjournal = {Journal of Physics. A. Mathematical and Theoretical},
      volume = {48},
      year = {2015},
      number = {16},
      pages = {16FT01, 21},
      issn = {1751-8113},
      mrclass = {37J35 (37A35 65Q30)},
      mrnumber = {3335696},
      mrreviewer = {Guizhang Tu},
      doi = {10.1088/1751-8113/48/16/16FT01},
      url = {https://doi.org/10.1088/1751-8113/48/16/16FT01},
      zblnumber = {1320.26014},
      }
  • [xiao] Go to document J. Xiao, "Weak transcendental holomorphic Morse inequalities on compact Kähler manifolds," Ann. Inst. Fourier (Grenoble), vol. 65, iss. 3, pp. 1367-1379, 2015.
    @ARTICLE{xiao,
      author = {Xiao, Jian},
      title = {Weak transcendental holomorphic {M}orse inequalities on compact {K}ähler manifolds},
      journal = {Ann. Inst. Fourier (Grenoble)},
      fjournal = {Université de Grenoble. Annales de l'Institut Fourier},
      volume = {65},
      year = {2015},
      number = {3},
      pages = {1367--1379},
      issn = {0373-0956},
      mrclass = {32Q15 (32C30)},
      mrnumber = {3449182},
      mrreviewer = {Severin Barmeier},
      doi = {10.5802/aif.2959},
      url = {https://doi.org/10.5802/aif.2959},
      zblnumber = {1333.32025},
      }
  • [xie_periodic] Go to document J. Xie, "Periodic points of birational transformations on projective surfaces," Duke Math. J., vol. 164, iss. 5, pp. 903-932, 2015.
    @ARTICLE{xie_periodic,
      author = {Xie, Junyi},
      title = {Periodic points of birational transformations on projective surfaces},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {164},
      year = {2015},
      number = {5},
      pages = {903--932},
      issn = {0012-7094},
      mrclass = {37P55 (14E05 32H50 37C25)},
      mrnumber = {3332894},
      mrreviewer = {Yu Yasufuku},
      doi = {10.1215/00127094-2877402},
      url = {https://doi.org/10.1215/00127094-2877402},
      zblnumber = {1392.37120},
      }
  • [yosida] K. Yosida, Functional Analysis, Sixth ed., Springer-Verlag, New York, 1980, vol. 123.
    @BOOK{yosida,
      author = {Yosida, Kôsaku},
      title = {Functional Analysis},
      series = {Grundlehren Math. Wissen.},
      volume = {123},
      edition = {Sixth},
      publisher = {Springer-Verlag, New York},
      year = {1980},
      pages = {xii+501},
      isbn = {3-540-10210-8},
      mrclass = {46-01 (47-01)},
      mrnumber = {0617913},
      zblnumber = {0435.46002},
      }

Authors

Nguyen-Bac Dang

Institute for Mathematical Sciences, Stony Brook University, Stony Brook, NY, U.S.A.

Charles Favre

CMLS, CNRS, École polytechnique, Institut Polytechnique de Paris, 91128 Palaiseau Cedex, France