On the stability conjecture for geodesic flows of manifolds without conjugate points

Rifford, Ludovic; Ruggiero, Rafael

Annales Henri Lebesgue, Tome 4 (2021), pp. 759-784.

Nous étudions la conjecture de stabilité en topologie $C^2$ du point de vue de Mañé pour le flots géodésiques sans points conjugués sur les variétés compactes. La conjecture de stabilité en topologie $C^1$ pour les flots géodésiques est un problème ouvert car le $C^1$-Closing Lemma n’est pas connu dans ce contexte. Sans Closing Lemma, nous démontrons que la théorie des variétés sans points conjugués et une version récente du Lemme de Franks du point de vue de Mañé permettent d’obtenir une réponse positive à la conjecture dans le cas des variétés compactes sans points conjugués de dimension 2 et 3 ayant un revêtement universel quasi-convexe avec des rayons géodésiques divergents, ainsi que pour les variétés de dimension $n$ de rank un généralisées.

We study the $C^2$-structural stability conjecture from Mañé’s viewpoint for geodesics flows of compact manifolds without conjugate points. The structural stability conjecture is an open problem in the category of geodesic flows because the $C^1$ closing lemma is not known in this context. Without the $C^1$ closing lemma, we combine the geometry of manifolds without conjugate points and a recent version of Franks’ Lemma from Mañé’s viewpoint to prove the conjecture for compact surfaces, for compact three dimensional manifolds with quasi-convex universal coverings where geodesic rays diverge, and for $n$-dimensional, generalized rank one manifolds.

Classification : 37C20, 37D40, 53C22, 20F65
Mots clés : Geodesic flow, structural stability, closing lemma, conjugate points, quasi-convex space, Gromov hyperbolic space

[Ano69] Anosov, Dmitriĭ V. Geodesic flows on closed Riemannian manifolds of negative curvature, Proceedings of the Steklov Institute of Mathematics, 90, American Mathematical Society, 1969 (Translated from the Russian by S. Feder)

[Arn78] Arnold, Vladimir I. Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, 60, Springer, 1978 | MR | Zbl

[Bal95] Ballmann, Werner Lectures on spaces of nonpositive curvature, DMW Seminar, 25, Birkhäuser, 1995 | MR | Zbl

[BBB87] Ballmann, Werner; Brin, Michael; Burns, Keith On surfaces with no conjugate points, J. Differ. Geom., Volume 25 (1987), pp. 249-273 | MR | Zbl

[BGS85] Ballmann, Werner; Gromov, Michael L.; Schroeder, Viktor Manifolds of nonpositive curvature, Progress in Mathematics, 61, Birkhäuser, 1985 | MR | Zbl

[Bur92] Burns, Keith The flat strip theorem fails for surfaces with no conjugate points, Proc. Am. Math. Soc., Volume 115 (1992), pp. 199-206 | DOI | MR | Zbl

[CS86] Croke, Christopher B.; Schroeder, Viktor The fundamental group of compact manifolds without conjugate points, Comment. Math. Helv., Volume 61 (1986) no. 1, pp. 161-175 | DOI | MR | Zbl

[Ebe72] Eberlein, Patrick B. Geodesic flows in certain manifolds without conjugate points, Trans. Am. Math. Soc., Volume 167 (1972), pp. 151-170 | DOI | MR | Zbl

[Ebe96] Eberlein, Patrick B. Geometry of nonpositively curved manifolds, Chicago Lectures in Mathematics, University of Chicago Press, 1996 | MR | Zbl

[Esc77] Eschenburg, Jost-Hinrich Horospheres and the stable part of the geodesic flow, Math. Z., Volume 153 (1977), pp. 237-251 | DOI | MR | Zbl

[Gre54] Green, Leon W. Surfaces without conjugate points, Trans. Am. Math. Soc., Volume 76 (1954), pp. 529-546 | DOI | MR | Zbl

[Gro87] Gromov, Mikhael L. Hyperbolic Groups, Essays in Group Theory (Mathematical Sciences Research Institute Publications), Volume 8, Springer, 1987, pp. 75-264 | DOI | MR | Zbl

[Hop48] Hopf, Eberhard Closed surfaces without conjugate points, Proc. Natl. Acad. Sci. USA, Volume 34 (1948), pp. 47-51 | DOI | MR | Zbl

[HPS77] Hirsch, Morris; Pugh, Charles C.; Shub, Michael Invariant Manifolds, Lecture Notes in Mathematics, 583, Springer, 1977 | Zbl

[Kli74] Klingenberg, Wilhelm Riemannian manifolds with geodesic flows of Anosov type, Ann. Math., Volume 99 (1974), pp. 1-13 | DOI | MR | Zbl

[LRR16] Lazrag, Ayadi; Rifford, Ludovic; Ruggiero, Rafael Franks’ Lemma for $C^2$ Mañé perturbations of Riemannian metrics and applications to persistence, J. Mod. Dyn., Volume 10 (2016), pp. 379-411 | Zbl

[Mañ82] Mañé, Ricardo An ergodic closing lemma, Ann. Math., Volume 116 (3) (1982), pp. 503-540 | DOI | MR | Zbl

[Mañ87] Mañé, Ricardo On a theorem of Klingenberg, Dynamical systems and bifurcation theory (Rio de Janeiro, 1985) (M. Camacho, F. Takens Editors M. Pacífico, ed.) (Pitman Research Notes in Mathematics Series), Volume 160, Longman Scientific & Technical, Harlow (1987), pp. 319-345 | MR | Zbl

[Mañ88] Mañé, Ricardo A proof of the $C^1$ stability conjecture, Publ. Math., Inst. Hautes Étud. Sci., Volume 66 (1988), pp. 161-210 | DOI | Numdam | Zbl

[Mor24] Morse, Harold Marston A fundamental class of geodesics on any closed surface of genus greater than one, Trans. Am. Math. Soc., Volume 26 (1924), pp. 25-60 | DOI | MR | Zbl

[New77] Newhouse, Sheldon E. Quasi-elliptic periodic points in conservative dynamical systems, Am. J. Math., Volume 99 (1977) no. 5, pp. 1061-1087 | DOI | MR | Zbl

[Pes77] Pesin, Ja B. Geodesic flows on closed Riemannian manifolds without focal points, Math. USSR, Izv., Volume 11 (1977) no. 6, pp. 1195-1228 | DOI | MR | Zbl

[PR83] Pugh, C. Charles; Robinson, Clark The $C^1$ closing lemma, including Hamiltonians, Ergodic Theory Dyn. Syst., Volume 3 (1983), pp. 261-313 | DOI | MR | Zbl

[Rif12] Rifford, Ludovic Closing geodesics in $C^1$ topology, J. Differ. Geom., Volume 91 (2012) no. 3, pp. 361-382 | MR | Zbl

[Rob70a] Robinson, Clark Generic properties of conservative systems, Am. J. Math., Volume 92 (1970), pp. 562-603 | DOI | MR | Zbl

[Rob70b] Robinson, Clark Generic properties of conservative systems. II., Am. J. Math., Volume 92 (1970), pp. 897-906 | DOI | MR | Zbl

[Rug94] Ruggiero, Rafael Expansive dynamics and hyperbolic geometry, Bol. Soc. Bras. Mat., Nova Sér., Volume 25 (1994) no. 2, pp. 139-172 | DOI | MR | Zbl

[Rug97] Ruggiero, Rafael Expansive geodesic flows in manifolds without conjugate points, Ergodic Theory Dyn. Syst., Volume 17 (1997), pp. 211-225 | DOI | MR | Zbl

[Rug03] Ruggiero, Rafael On the divergence of geodesic rays in manifolds without conjugate points, dynamics of the geodesic flow and global geometry, Geometric methods in dynamics (II). Volume in honor of Jacob Palis. In part papers presented at the international conference on dynamical systems held at IMPA, Rio de Janeiro, Brazil, July 2000, to celebrate Jacob Palis’ 60th birthday (Astérisque), Volume 287, Société Mathématique de France, 2003, pp. 231-249 | Numdam | Zbl

[Rug07] Ruggiero, Rafael Dynamics and global geometry of manifolds without conjugate points, Ensaios Matemáticos, 12, Sociedade Brasileira de Matemática, 2007 | MR | Zbl

[Rug08] Ruggiero, Rafael A note on the divergence of geodesic rays in manifolds without conjugate points, Geom. Dedicata, Volume 134 (2008), pp. 131-138 | DOI | MR | Zbl