Geodesic flows with positive topological entropy, twist maps and hyperbolicity

Abstract

We prove a perturbation lemma for the derivative of geodesic flows in high dimension. This implies that a $C^2$ generic riemannian metric has a nontrivial hyperbolic basic set in its geodesic flow.

  • [Ab-Rob] R. Abraham and J. Robbin, Transversal Mappings and Flows, New York-Amsterdam: W. A. Benjamin, 1967.
    @book {Ab-Rob, MRKEY = {0240836},
      AUTHOR = {Abraham, Ralph and Robbin, Joel},
      TITLE = {Transversal Mappings and Flows},
      PUBLISHER = {W. A. Benjamin},
      ADDRESS={New York-Amsterdam},
      YEAR = {1967},
      PAGES = {x+161},
      MRCLASS = {57.55},
      MRNUMBER = {39 \#2181},
      MRREVIEWER = {J. Palis},
      ZBLNUMBER = {0171.44404},
      }
  • [An1] D. V. Anosov, "Generic properties of closed geodesics," Math. USSR Izv., vol. 21, pp. 1-29, 1983.
    @article {An1, MRKEY = {670163},
      AUTHOR = {Anosov, D. V.},
      TITLE = {Generic properties of closed geodesics},
      JOURNAL = {Math. {\rm USSR} Izv.},
      FJOURNAL = {Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya},
      VOLUME = {21},
      YEAR = {1983},
      PAGES = {1--29},
      ISSN = {0373-2436},
      MRCLASS = {58E10 (53C22)},
      MRNUMBER = {84b:58029},
      MRREVIEWER = {Wolfgang Ziller},
      ZBLNUMBER={0554.58043},
      }
  • [Arnaud2] M. Arnaud, Type des Points Fixes des Difféomorphismes Symplectiques de ${\bf T}^n\times {\bf R}^n$, , 1992, vol. 48.
    @book {Arnaud2, MRKEY = {1146837},
      AUTHOR = {Arnaud, Marie-Claude},
      TITLE = {Type des Points Fixes des Difféomorphismes Symplectiques de {${\bf T}\sp n\times {\bf R}\sp n$}},
      SERIES = {Mém. Soc. Math. France},
      FJOURNAL = {Mémoires de la Société Mathématique de France. Nouvelle Série},
      VOLUME = {48},
      YEAR = {1992},
      PAGES = {63 pp.},
      ISSN = {0037-9484},
      MRCLASS = {58F05 (58F30 70H05)},
      MRNUMBER = {93c:58065},
      MRREVIEWER = {Helmut R{ü}ssmann},
      ZBLNUMBER={0758.58009},
      }
  • [BH] V. Bangert and N. Hingston, "Closed geodesics on manifolds with infinite abelian fundamental group," J. Differential Geom., vol. 19, pp. 277-282, 1984.
    @article{BH,
      author={Bangert, V. and Hingston, N.},
      TITLE={Closed geodesics on manifolds with infinite abelian fundamental group},
      JOURNAL={J. Differential Geom.},
      VOLUME={19},
      PAGES={277--282},
      YEAR={1984},
      ZBLNUMBER={0545.53036},
      MRNUMBER={86e:53032},
      }
  • [BTZ1] W. Ballmann, G. Thorbergsson, and W. Ziller, "Existence of closed geodesics on positively curved manifolds," J. Differential Geom., vol. 18, pp. 221-252, 1983.
    @article{BTZ1,
      author={Ballmann, W. and Thorbergsson, G. and Ziller, W.},
      TITLE={Existence of closed geodesics on positively curved manifolds},
      JOURNAL={J. Differential Geom.},
      VOLUME={18},
      PAGES={221--252},
      YEAR={1983},
      MRNUMBER={84i:58032},
      ZBLNUMBER={0514.53044},
      }
  • [BTZ2] W. Ballmann, G. Thorbergsson, and W. Ziller, "Closed geodesics on positively curved manifolds," Ann. of Math., vol. 116, pp. 213-247, 1982.
    @article{BTZ2,
      author={Ballmann, W. and Thorbergsson, G. and Ziller, W.},
      TITLE={Closed geodesics on positively curved manifolds},
      JOURNAL={Ann. of Math.},
      VOLUME={116},
      PAGES={213--247},
      YEAR={1982},
      ZBLNUMBER={0495.58010},
      MRNUMBER={83k:58022},
      }
  • [CO] Go to document G. Contreras and F. Oliveira, "$C^2$ densely the 2-sphere has an elliptic closed geodesic," Ergodic Theory Dynam. Systems, vol. 24, iss. 5, pp. 1395-1423, 2004.
    @article {CO, MRKEY = {2104590},
      AUTHOR = {Contreras, Gonzalo and Oliveira, Fernando},
      TITLE = {{$C\sp 2$} densely the 2-sphere has an elliptic closed geodesic},
      JOURNAL = {Ergodic Theory Dynam. Systems},
      FJOURNAL = {Ergodic Theory and Dynamical Systems},
      VOLUME = {24},
      YEAR = {2004},
      NUMBER = {5},
      PAGES = {1395--1423},
      ISSN = {0143-3857},
      MRCLASS = {53D25 (37C27 37D40 37E30 37J25 53C22)},
      MRNUMBER = {2006a:53096},
      MRREVIEWER = {Victor Bangert},
      DOI = {10.1017/S0143385704000422},
      ZBLNUMBER = {1093.53035},
      }
  • [gpe] Go to document G. Contreras-Barandiarán and G. P. Paternain, "Genericity of geodesic flows with positive topological entropy on $S^2$," J. Differential Geom., vol. 61, iss. 1, pp. 1-49, 2002.
    @article {gpe, MRKEY = {1949783},
      AUTHOR = {Contreras-Barandiar{á}n, Gonzalo and Paternain, Gabriel P.},
      TITLE = {Genericity of geodesic flows with positive topological entropy on {$S\sp 2$}},
      JOURNAL = {J. Differential Geom.},
      FJOURNAL = {Journal of Differential Geometry},
      VOLUME = {61},
      YEAR = {2002},
      NUMBER = {1},
      PAGES = {1--49},
      ISSN = {0022-040X},
      CODEN = {JDGEAS},
      MRCLASS = {53D25 (37B40 37C20 37D40)},
      MRNUMBER = {2003j:53134},
      MRREVIEWER = {Rafael Oswaldo Ruggiero},
      URL = {http://projecteuclid.org/getRecord?id=euclid.jdg/1090351319},
      ZBLNUMBER = {1048.53061},
      }
  • [DLS2] Go to document A. Delshams, R. de la Llave, and T. M. Seara, "Orbits of unbounded energy in quasi-periodic perturbations of geodesic flows," Adv. Math., vol. 202, iss. 1, pp. 64-188, 2006.
    @article {DLS2, MRKEY = {2218821},
      AUTHOR = {Delshams, Amadeu and de la Llave, Rafael and Seara, Tere M.},
      TITLE = {Orbits of unbounded energy in quasi-periodic perturbations of geodesic flows},
      JOURNAL = {Adv. Math.},
      FJOURNAL = {Advances in Mathematics},
      VOLUME = {202},
      YEAR = {2006},
      NUMBER = {1},
      PAGES = {64--188},
      ISSN = {0001-8708},
      CODEN = {ADMTA4},
      MRCLASS = {37J45 (37D10 37D40 37J40 53D25 70H08 70H09)},
      MRNUMBER = {2007a:37070},
      MRREVIEWER = {Mikhail B. Sevryuk},
      DOI = {10.1016/j.aim.2005.03.005},
      ZBLNUMBER = {1091.37018},
      }
  • [DL3] Go to document J. Denef and F. Loeser, "Germs of arcs on singular algebraic varieties and motivic integration," Invent. Math., vol. 135, iss. 1, pp. 201-232, 1999.
    @article {DL3, MRKEY = {1664700},
      AUTHOR = {Denef, Jan and Loeser, Fran{ç}ois},
      TITLE = {Germs of arcs on singular algebraic varieties and motivic integration},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {135},
      YEAR = {1999},
      NUMBER = {1},
      PAGES = {201--232},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {14A20 (14D10 14G99)},
      MRNUMBER = {99k:14002},
      MRREVIEWER = {Jos{é} Ignacio Burgos Gil},
      DOI = {10.1007/s002220050284},
      ZBLNUMBER = {0928.14004},
      }
  • [Din] E. I. Dinaburg, "On the relations among various entropy characteristics of dynamical systems," Math. USSR Izv., vol. 5, pp. 337-378, 1971.
    @article{Din,
      author={Dinaburg, E. I.},
      TITLE={On the relations among various entropy characteristics of dynamical systems},
      JOURNAL={Math. {\rm USSR} Izv.},
      VOLUME={5},
      YEAR={1971},
      PAGES={337--378},
      }
  • [EH] D. Eisenbud and J. Harris, The Geometry of Schemes, New York: Springer-Verlag, 2000, vol. 197.
    @book {EH, MRKEY = {1730819},
      AUTHOR = {Eisenbud, David and Harris, Joe},
      TITLE = {The Geometry of Schemes},
      SERIES = {Grad. Texts in Math.},
      VOLUME = {197},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {2000},
      PAGES = {x+294},
      ISBN = {0-387-98638-3; 0-387-98637-5},
      MRCLASS = {14A15 (14-01 14C05 14H50 14N05)},
      MRNUMBER = {2001d:14002},
      MRREVIEWER = {Allen B. Altman},
      ZBLNUMBER = {0960.14002},
      }
  • [Fra1] Go to document J. Franks, "Necessary conditions for stability of diffeomorphisms," Trans. Amer. Math. Soc., vol. 158, pp. 301-308, 1971.
    @article {Fra1, MRKEY = {0283812},
      AUTHOR = {Franks, John},
      TITLE = {Necessary conditions for stability of diffeomorphisms},
      JOURNAL = {Trans. Amer. Math. Soc.},
      FJOURNAL = {Transactions of the American Mathematical Society},
      VOLUME = {158},
      YEAR = {1971},
      PAGES = {301--308},
      ISSN = {0002-9947},
      MRCLASS = {57.20},
      MRNUMBER = {44 \#1042},
      MRREVIEWER = {R. F. Williams},
      DOI = {10.2307/1995906},
      ZBLNUMBER = {0219.58005},
      }
  • [FLC] Go to document J. Franks and P. Le Calvez, "Regions of instability for non-twist maps," Ergodic Theory Dynam. Systems, vol. 23, iss. 1, pp. 111-141, 2003.
    @article {FLC, MRKEY = {1971199},
      AUTHOR = {Franks, John and Le Calvez, Patrice},
      TITLE = {Regions of instability for non-twist maps},
      JOURNAL = {Ergodic Theory Dynam. Systems},
      FJOURNAL = {Ergodic Theory and Dynamical Systems},
      VOLUME = {23},
      YEAR = {2003},
      NUMBER = {1},
      PAGES = {111--141},
      ISSN = {0143-3857},
      MRCLASS = {37E30 (37C20 37J10)},
      MRNUMBER = {2003m:37053},
      MRREVIEWER = {Christophe Gol{é}},
      DOI = {10.1017/S0143385702000858},
      ZBLNUMBER = {1140.37331},
      }
  • [GKM] D. Gromoll, W. Klingenberg, and W. Meyer, Riemannsche Geometrie im Gro\ss en, New York: Springer-Verlag, 1975, vol. 55.
    @book {GKM, MRKEY = {0365399},
      AUTHOR = {Gromoll, D. and Klingenberg, W. and Meyer, W.},
      TITLE = {Riemannsche {G}eometrie im {G}ro\ss en},
      SERIES = {Lecture Notes in Math.},
      VOLUME={55},
      NOTE = {Zweite Auflage},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1975},
      PAGES = {vi+287},
      MRCLASS = {53CXX (58-02)},
      MRNUMBER = {51 \#1651},
      ZBLNUMBER={0293.53001},
      }
  • [Harris] J. Harris, Algebraic Geometry, A First Course, New York: Springer-Verlag, 1992, vol. 133.
    @book {Harris, MRKEY = {1182558},
      AUTHOR = {Harris, Joe},
      TITLE = {Algebraic Geometry, A First Course},
      SERIES = {Grad. Texts in Math.},
      VOLUME = {133},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1992},
      PAGES = {xx+328},
      ISBN = {0-387-97716-3},
      MRCLASS = {14-01},
      MRNUMBER = {93j:14001},
      MRREVIEWER = {Liam O'Carroll},
      ZBLNUMBER = {0779.14001},
      }
  • [Hartshorne] R. Hartshorne, Algebraic Geometry, New York: Springer-Verlag, 1977, vol. 52.
    @book {Hartshorne, MRKEY = {0463157},
      AUTHOR = {Hartshorne, Robin},
      TITLE = {Algebraic Geometry},
      SERIES = {Grad. Texts in Math.},
      VOLUME={52},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1977},
      PAGES = {xvi+496},
      ISBN = {0-387-90244-9},
      MRCLASS = {14-01},
      MRNUMBER = {57 \#3116},
      MRREVIEWER = {Robert Speiser},
      ZBLNUMBER = {0367.14001},
      }
  • [HK] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge: Cambridge Univ. Press, 1995, vol. 54.
    @book {HK, MRKEY = {1326374},
      AUTHOR = {Katok, Anatole and Hasselblatt, Boris},
      TITLE = {Introduction to the Modern Theory of Dynamical Systems},
      SERIES = {Encyclopedia Math. Appl.},
      VOLUME = {54},
      NOTE = {with a supplementary chapter by Katok and Leonardo Mendoza},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {1995},
      PAGES = {xviii+802},
      ISBN = {0-521-34187-6},
      MRCLASS = {58Fxx (34Cxx 34Dxx 58-01 58F11 58F15)},
      MRNUMBER = {96c:58055},
      MRREVIEWER = {Edoh Amiran},
      ZBLNUMBER = {0878.58020},
      }
  • [HPS] M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant Manifolds, New York: Springer-Verlag, 1977, vol. 583.
    @book {HPS, MRKEY = {0501173},
      AUTHOR = {Hirsch, M. W. and Pugh, C. C. and Shub, M.},
      TITLE = {Invariant Manifolds},
      SERIES = {Lecture Notes in Math.},
      VOLUME={583},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1977},
      PAGES = {ii+149},
      MRCLASS = {58F15 (58F10)},
      MRNUMBER = {58 \#18595},
      MRREVIEWER = {M. C. Irwin},
      ZBLNUMBER = {0355.58009},
      }
  • [Kli] W. Klingenberg, Lectures on Closed Geodesics, New York: Springer-Verlag, 1978, vol. 230.
    @book {Kli, MRKEY = {0478069},
      AUTHOR = {Klingenberg, Wilhelm},
      TITLE = {Lectures on Closed Geodesics},
      SERIES = {Grundleheren Math. Wiss.},
      VOLUME={230},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1978},
      PAGES = {x+227},
      ISBN = {3-540-08393-6},
      MRCLASS = {53C20 (58E10)},
      MRNUMBER = {57 \#17563},
      MRREVIEWER = {Y. Muto},
      ZBLNUMBER = {0397.58018},
      }
  • [KT] Go to document W. Klingenberg and F. Takens, "Generic properties of geodesic flows," Math. Ann., vol. 197, pp. 323-334, 1972.
    @article {KT, MRKEY = {0307282},
      AUTHOR = {Klingenberg, Wilhelm and Takens, Floris},
      TITLE = {Generic properties of geodesic flows},
      JOURNAL = {Math. Ann.},
      FJOURNAL = {Mathematische Annalen},
      VOLUME = {197},
      YEAR = {1972},
      PAGES = {323--334},
      ISSN = {0025-5831},
      MRCLASS = {58F99},
      MRNUMBER = {46 \#6402},
      MRREVIEWER = {L. A. Santalo},
      DOI = {10.1007/BF01428204},
      ZBLNUMBER = {0225.58006},
      }
  • [KW2] Go to document G. Knieper and H. Weiss, "$C^\infty$ genericity of positive topological entropy for geodesic flows on $S^2$," J. Differential Geom., vol. 62, iss. 1, pp. 127-141, 2002.
    @article {KW2, MRKEY = {1987379},
      AUTHOR = {Knieper, Gerhard and Weiss, Howard},
      TITLE = {{$C\sp \infty$} genericity of positive topological entropy for geodesic flows on {$S\sp 2$}},
      JOURNAL = {J. Differential Geom.},
      FJOURNAL = {Journal of Differential Geometry},
      VOLUME = {62},
      YEAR = {2002},
      NUMBER = {1},
      PAGES = {127--141},
      ISSN = {0022-040X},
      CODEN = {JDGEAS},
      MRCLASS = {37D40 (37B40 37C20 37D05 37J45 53D25)},
      MRNUMBER = {2004c:37051},
      MRREVIEWER = {Gabriel P. Paternain},
      URL = {http://projecteuclid.org/getRecord?id=euclid.jdg/1090425531},
      ZBLNUMBER = {1067.37038},
      }
  • [LC1] P. Le Calvez, Propriétés Dynamiques des Difféomorphismes de l’Anneau et du Tore, Paris: Soc. Math. France, 1991, vol. 204.
    @book {LC1, MRKEY = {1183304},
      AUTHOR = {Le Calvez, Patrice},
      TITLE = {Propriétés Dynamiques des Difféomorphismes de l'Anneau et du Tore},
      SERIES = {Astérisque},
      VOLUME= {204},
      FJOURNAL = {Astérisque},
      PUBLISHER={Soc. Math. France},
      ADDRESS={Paris},
      YEAR = {1991},
      PAGES = {131},
      ISSN = {0303-1179},
      MRCLASS = {58F12 (58F27)},
      MRNUMBER = {94d:58092},
      MRREVIEWER = {Jaroslav Stark},
      ZBLNUMBER = {0784.58033},
      }
  • [Ma8] Go to document R. Mañé, "An ergodic closing lemma," Ann. of Math., vol. 116, iss. 3, pp. 503-540, 1982.
    @article {Ma8, MRKEY = {678479},
      AUTHOR = {Ma{ñ}{é},
      Ricardo},
      TITLE = {An ergodic closing lemma},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {116},
      YEAR = {1982},
      NUMBER = {3},
      PAGES = {503--540},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {58F10 (28D10 58F15)},
      MRNUMBER = {84f:58070},
      MRREVIEWER = {Carmen Chicone},
      DOI = {10.2307/2007021},
      ZBLNUMBER = {0511.58029},
      }
  • [Ma9] Go to document R. Mañé, "On the topological entropy of geodesic flows," J. Differential Geom., vol. 45, iss. 1, pp. 74-93, 1997.
    @article {Ma9, MRKEY = {1443332},
      AUTHOR = {Ma{ñ}{é},
      Ricardo},
      TITLE = {On the topological entropy of geodesic flows},
      JOURNAL = {J. Differential Geom.},
      FJOURNAL = {Journal of Differential Geometry},
      VOLUME = {45},
      YEAR = {1997},
      NUMBER = {1},
      PAGES = {74--93},
      ISSN = {0022-040X},
      CODEN = {JDGEAS},
      MRCLASS = {58F17 (53C22)},
      MRNUMBER = {98d:58141},
      MRREVIEWER = {Boris Hasselblatt},
      URL = {http://projecteuclid.org/getRecord?id=euclid.jdg/1214459755},
      ZBLNUMBER = {0896.58052},
      }
  • [Manning] Go to document A. Manning, "Topological entropy for geodesic flows," Ann. of Math., vol. 110, iss. 3, pp. 567-573, 1979.
    @article {Manning, MRKEY = {554385},
      AUTHOR = {Manning, Anthony},
      TITLE = {Topological entropy for geodesic flows},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {110},
      YEAR = {1979},
      NUMBER = {3},
      PAGES = {567--573},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {58F17 (28D20 58F11)},
      MRNUMBER = {81e:58044},
      MRREVIEWER = {William Perrizo},
      DOI = {10.2307/1971239},
      ZBLNUMBER = {0426.58016},
      }
  • [Mat9] J. N. Mather, "Invariant subsets for area preserving homeomorphisms of surfaces," in Mathematical Analysis and Applications, Part B, New York: Academic Press, 1981, vol. 7, pp. 531-562.
    @incollection {Mat9, MRKEY = {634258},
      AUTHOR = {Mather, John N.},
      TITLE = {Invariant subsets for area preserving homeomorphisms of surfaces},
      BOOKTITLE = {Mathematical Analysis and Applications, {P}art {\rm B}},
      SERIES = {Adv. in Math. Suppl. Stud.},
      VOLUME = {7},
      PAGES = {531--562},
      PUBLISHER = {Academic Press},
      ADDRESS = {New York},
      YEAR = {1981},
      MRCLASS = {58F08 (54H20 58F99)},
      MRNUMBER = {84j:58069},
      MRREVIEWER = {B. Brechner},
      ZBLNUMBER = {0505.58027},
      }
  • [Moser1] J. Moser, "Proof of a generalized form of a fixed point theorem due to G. D. Birkhoff," in Geometry and Topology, New York: Springer-Verlag, 1977, vol. 597, pp. 464-494.
    @incollection {Moser1, MRKEY = {0494305},
      AUTHOR = {Moser, J.},
      TITLE = {Proof of a generalized form of a fixed point theorem due to {G}. {D}. {B}irkhoff},
      BOOKTITLE = {Geometry and Topology},
      VENUE={{P}roc. {III} {L}atin {A}mer. {S}chool of {M}ath., {I}nst. {M}at. {P}ura {A}plicada {CNP}q, {R}io de {J}aneiro, 1976},
      PAGES = {464--494},
      SERIES={Lecture Notes in Math.},
      VOLUME={597},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1977},
      MRCLASS = {58F20},
      MRNUMBER = {58 \#13205},
      MRREVIEWER = {Julian I. Palmore},
      ZBLNUMBER = {0358.58009},
      }
  • [N1] Go to document S. E. Newhouse, "Quasi-elliptic periodic points in conservative dynamical systems," Amer. J. Math., vol. 99, iss. 5, pp. 1061-1087, 1977.
    @article {N1, MRKEY = {0455049},
      AUTHOR = {Newhouse, Sheldon E.},
      TITLE = {Quasi-elliptic periodic points in conservative dynamical systems},
      JOURNAL = {Amer. J. Math.},
      FJOURNAL = {American Journal of Mathematics},
      VOLUME = {99},
      YEAR = {1977},
      NUMBER = {5},
      PAGES = {1061--1087},
      ISSN = {0002-9327},
      MRCLASS = {58F20 (58F10)},
      MRNUMBER = {56 \#13290},
      MRREVIEWER = {Zbigniew Nitecki},
      DOI = {10.2307/2374000},
      ZBLNUMBER = {0379.58011},
      }
  • [PatPe] Go to document G. P. Paternain and J. Petean, "Zero entropy and bounded topology," Comment. Math. Helv., vol. 81, iss. 2, pp. 287-304, 2006.
    @article {PatPe, MRKEY = {2225629},
      AUTHOR = {Paternain, Gabriel P. and Petean, Jimmy},
      TITLE = {Zero entropy and bounded topology},
      JOURNAL = {Comment. Math. Helv.},
      FJOURNAL = {Commentarii Mathematici Helvetici. A Journal of the Swiss Mathematical Society},
      VOLUME = {81},
      YEAR = {2006},
      NUMBER = {2},
      PAGES = {287--304},
      ISSN = {0010-2571},
      MRCLASS = {37D40 (37B40 53D25 55P99)},
      MRNUMBER = {2007g:37023},
      MRREVIEWER = {Andrea Sambusetti},
      DOI = {10.4171/CMH/53},
      ZBLNUMBER = {1093.53090},
      }
  • [PS] Go to document C. Pugh and M. Shub, "The $\Omega $-stability theorem for flows," Invent. Math., vol. 11, pp. 150-158, 1970.
    @article {PS, MRKEY = {0287579},
      AUTHOR = {Pugh, Charles and Shub, Michael},
      TITLE = {The {$\Omega $}-stability theorem for flows},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {11},
      YEAR = {1970},
      PAGES = {150--158},
      ISSN = {0020-9910},
      MRCLASS = {57.48},
      MRNUMBER = {44 \#4782},
      MRREVIEWER = {R. Ellis},
      DOI = {10.1007/BF01404608},
      ZBLNUMBER = {0212.29102},
      }
  • [PR] Go to document C. C. Pugh and C. Robinson, "The $C^{1}$ closing lemma, including Hamiltonians," Ergodic Theory Dynam. Systems, vol. 3, iss. 2, pp. 261-313, 1983.
    @article {PR, MRKEY = {742228},
      AUTHOR = {Pugh, Charles C. and Robinson, Clark},
      TITLE = {The {$C\sp{1}$} closing lemma, including {H}amiltonians},
      JOURNAL = {Ergodic Theory Dynam. Systems},
      FJOURNAL = {Ergodic Theory and Dynamical Systems},
      VOLUME = {3},
      YEAR = {1983},
      NUMBER = {2},
      PAGES = {261--313},
      ISSN = {0143-3857},
      MRCLASS = {58F10 (58F30)},
      MRNUMBER = {85m:58106},
      MRREVIEWER = {Richard Cushman},
      DOI = {10.1017/S0143385700001978},
      ZBLNUMBER = {0548.58012},
      }
  • [Rad1] Go to document H. Rademacher, "On a generic property of geodesic flows," Math. Ann., vol. 298, iss. 1, pp. 101-116, 1994.
    @article {Rad1, MRKEY = {1252820},
      AUTHOR = {Rademacher, Hans-Bert},
      TITLE = {On a generic property of geodesic flows},
      JOURNAL = {Math. Ann.},
      FJOURNAL = {Mathematische Annalen},
      VOLUME = {298},
      YEAR = {1994},
      NUMBER = {1},
      PAGES = {101--116},
      ISSN = {0025-5831},
      CODEN = {MAANA},
      MRCLASS = {58E10 (58F17)},
      MRNUMBER = {94j:58040},
      MRREVIEWER = {Gabriel P. Paternain},
      DOI = {10.1007/BF01459728},
      ZBLNUMBER = {0813.53029},
      }
  • [Ro0] C. Robinson, Lectures on Hamiltonian Systems, Rio de Janeiro: IMPA, 1971, vol. 7.
    @book{Ro0,
      author={Robinson, C.},
      TITLE={Lectures on Hamiltonian Systems},
      SERIES={Monografias de Matemática},
      VOLUME={7},
      PUBLISHER={IMPA},
      ADDRESS={Rio de Janeiro},
      YEAR={1971},
      }
  • [Smale] Go to document S. Smale, "Differentiable dynamical systems," Bull. Amer. Math. Soc., vol. 73, pp. 747-817, 1967.
    @article {Smale, MRKEY = {0228014},
      AUTHOR = {Smale, S.},
      TITLE = {Differentiable dynamical systems},
      JOURNAL = {Bull. Amer. Math. Soc.},
      FJOURNAL = {Bulletin of the American Mathematical Society},
      VOLUME = {73},
      YEAR = {1967},
      PAGES = {747--817},
      ISSN = {0002-9904},
      MRCLASS = {57.47 (34.00)},
      MRNUMBER = {37 \#3598},
      MRREVIEWER = {M. M. Peixoto},
      DOI = {10.1090/S0002-9904-1967-11798-1},
      ZBLNUMBER = {0202.55202},
      }
  • [Veys] W. Veys, "Arc spaces, motivic integration and stringy invariants," in Singularity Theory and its Applications, Tokyo: Math. Soc. Japan, 2006, vol. 43, pp. 529-572.
    @incollection {Veys, MRKEY = {2325153},
      AUTHOR = {Veys, Willem},
      TITLE = {Arc spaces, motivic integration and stringy invariants},
      BOOKTITLE = {Singularity Theory and its Applications},
      SERIES = {Adv. Stud. Pure Math.},
      VOLUME = {43},
      PAGES = {529--572},
      PUBLISHER = {Math. Soc. Japan},
      ADDRESS = {Tokyo},
      YEAR = {2006},
      MRCLASS = {14E99 (14B05)},
      MRNUMBER = {2008g:14023},
      MRREVIEWER = {Tommaso De Fernex},
      ZBLNUMBER = {1127.14004},
      }

Authors

Gonzalo Contreras

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