Propriétés dynamiques des difféomorphismes de l'anneau et du tore

Le Calvez, Patrice

Astérisque, no. 204 (1991) , 133 p.

MR: 1183304 | Zbl: 0784.58033 | 4 citations in Numdam

@book{AST_1991__204__1_0,
     author = {Le Calvez, Patrice},
     title = {Propri\'et\'es dynamiques des diff\'eomorphismes de l'anneau et du tore},
     series = {Ast\'erisque},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {204},
     year = {1991},
     zbl = {0784.58033},
     mrnumber = {1183304},
     language = {fr},
     url = {http://www.numdam.org/item/AST_1991__204__1_0/}
}

TY  - BOOK
AU  - Le Calvez, Patrice
TI  - Propriétés dynamiques des difféomorphismes de l'anneau et du tore
T3  - Astérisque
PY  - 1991
IS  - 204
PB  - Société mathématique de France
UR  - http://www.numdam.org/item/AST_1991__204__1_0/
LA  - fr
ID  - AST_1991__204__1_0
ER  -

%0 Book
%A Le Calvez, Patrice
%T Propriétés dynamiques des difféomorphismes de l'anneau et du tore
%S Astérisque
%D 1991
%N 204
%I Société mathématique de France
%U http://www.numdam.org/item/AST_1991__204__1_0/
%G fr
%F AST_1991__204__1_0

Le Calvez, Patrice. Propriétés dynamiques des difféomorphismes de l'anneau et du tore. Astérisque, no. 204 (1991), 133 p. http://numdam.org/item/AST_1991__204__1_0/

References
Cited by

[AY] K. Alligood, J. Yorke. Cascades of period-doubling bifurcations: a prerequisite for horseshoes, Bull. Amer. Math. Soc., 9 (1983), 319-322. | DOI | Zbl | MR

[An] S. B. Angenent. Monotone recurrence relations, their Birkhoff orbits and topological entropy, Ergod. Th. Dynam. Sys., 10 (1990), 15-41. | DOI | Zbl | MR

[AG] S. B. Angenent, C. Gole. Lamination by ghost circles, preprint E.T.H. Zurich (1991).

[AL] S. Aubry, P. Y. Le Daeron. The discrete Frenkel-Kontorova model and its generalizations, Physica 8D(1983), 381-422. | Zbl

[Ba1] V. Bangert. Mather sets for twist maps and geodesics on tori, Dynamics reported, Vol I, 1988, John Wiley and Sons. | DOI | Zbl | MR

[Ba2] V. Bangert. Minimal geodesics, Ergod. Th. Dynam. Sys., 10 (1989), 263-286. | Zbl | MR

[BS] M. Barge, R. Swanson. Rotation shadowing properties of circle and annulus maps, Ergod. Th. Dynam. Sys., 8 (1988), 509-521. | DOI | Zbl | MR

[BG] M. Barge, R. M. Gillette. Rotation and periodicity in plane separating continua, preprint, Montana State University. | DOI | Zbl | MR

[BC] M. Benedicks, L. Carleson. The dynamics of the Hénon map, Ann. of Math., 133 (1991), 73-169. | DOI | Zbl | MR

[Be] D. Bernstein. Birkhoff periodic orbits for twist maps with the circle intersection property, Ergod. Th. Dynam. Sys., 5 (1985), 531-537. | DOI | Zbl | MR

[BK] D. Bernstein, A. Katok. Birkhoff periodic orbits for small perturbations of completely integrable Hamiltonian systems with convex Hamiltonians, Invent. Math., 88 (1987), 225-241. | DOI | EuDML | Zbl | MR

[Bi1] G. D. Birkhoff. Proof of Poincaré's last geometric theorem, Trans. Amer. Math. Soc., 14 (1913), 14-22 | JFM | MR

G. D. Birkhoff. Proof of Poincaré's last geometric theorem, Collected Math. Papers, vol I, 673-681, Dover, New-York, (1978).

[Bi2] G. D. Birkhoff. Surface transformation and their dynamical applications. Acta Math., 43 (1920), 1-119 ; et Collected Math. Papers, vol II, 111-229. | DOI | MR | JFM

[Bi3] G. D. Birkhoff. An extension of Poincaré's last geometric theorem, Acta Math., 47 (1926), 297-311 ; et Collected. Math. Papers vol II, 252-260. | DOI | JFM | MR

[Bi4] G. D. Birkhoff. On the periodic motions of dynamical systems, Acta. Math., 50 (1927), 359-379 ; et Collected Math. Papers, vol II, 333-353. | DOI | JFM | MR

[Bi5] G. D. Birkhoff. Sur quelques courbes fermées remarquables, Bull. Soc. Math. France, 80 (1932), 1-26 ; et Collected Math. Papers, vol II, 444-461. | EuDML | Numdam | JFM | Zbl | MR

[Bi6] G. D. Birkhoff. Sur l'existence de régions d'instabilité en dynamique, Ann. Inst. Henri Poincaré, 8, (1932) ; et Collected Math. Papers, vol II, 418-443. | Numdam | EuDML | JFM | MR | Zbl

[Bi7] G. D. Birkhoff. Nouvelles recherches sur les systèmes dynamiques, Memoriae Pont. Acad. Sci. Novi Lyncaei, 1 (1935), 85-216 ; et Collected Math. Papers, vol II, 530-661. | JFM | Zbl

[BGMY] L. Block, J. Guckenheimer, J. Misiurewicz, L. S. Young. Periodic points and topological entropy of one dimensional maps, L. N. in Math., 819, Springer Verlag, (1980), 18-34. | Zbl | MR

[Bo1] P. L. Boyland. Braid Types and a topological method of proving positive entropy, preprint.

[Bo2] P. L. Boyland. Rotations sets and Morse decomposition in twist maps, Ergod. Th. Dynam. Sys., 8 (1988), 33-61. | DOI | Zbl | MR

[Bo3] P. L. Boyland. Rotation sets and topologically monotone orbits for annulus homeo-morphisms, preprint, Univ. of Minessota.

[BH] P. L. Boyland, G. R. Hall. Invariant circles and the order structure of periodic orbits in monotone twist maps, Topology, 26 (1987), 21-35. | DOI | Zbl | MR

[Bw] R. Bowen. Entropy and the fundamental group, L. N. in Math. 668, Springer Verlag, (1978), 21-29. | Zbl | MR

[BF] R. Bowen, J. Franks. The periodic points of maps of the disk and the interval, Topology, 15 (1976), 337-342. | DOI | Zbl | MR

[Bwr] L. E. J. Brouwer. Beweis des ebenen Translationssatzes, Math. Ann., 72 (1912), 37-54. | DOI | EuDML | JFM | MR

[Br] M. Brown. A new proof of Brouwer's lemma on translation arcs, Houston J. of Math., 10 (1984), 35-41. | Zbl | MR

[BN] M. Brown, W. D. Neumann. Proof of the Poincaré-Birkhoff fixed point theorem, Mich. Math. J., 24 (1977), 21-31. | DOI | Zbl | MR

[By] M.-L. Byalyi. Aubry-Mather sets and Birkhoff s theorem for geodesic flows on a two-dimensional torus, preprint, Weizmann Inst., (1988). | Zbl | MR

[Cr] P. H. Carter. An improvement of the Poincaré-Birkhoff theorem, Trans. Amer. Math. Soc., 269 (1982), 285-299. | Zbl | MR

[CL] M. L. Cartwright, J. E. Littlewood. On non-linear differential equations of the second order : I. The equation $ÿ - k (1 - y^{2}) ẏ + y = b k cos (λ t + α), k$ large, J. London Math. Soc., 20 (1945), 180-189. | DOI | MR | Zbl

[Ci1] M. Casdagli. Periodic orbits for dissipative twist maps, Ergod. Th. Dynam. Sys., 7 (1987), 165-173. | DOI | Zbl | MR

[Ci2] M. Casdagli. Rotational chaos in dissipative systems, Physica 29D (1988), 365-386. | Zbl | MR

[CB] A. J. Casson, S. A. Bleiler, Automorphisms of surfaces after Nielsen and Thurston, London Math. Soc. Students Texts, 9 (1988). | Zbl | MR

[Chp1] M. Chaperon, Une idée de "géodésiques brisées" pour les systèmes hamiltoniens, C. R. Acad. Sc. Paris, 298 (1984), 293-296. | Zbl | MR

[Chp2] M. Chaperon. An elementary proof of the Conley-Zehnder theorem, L. N. in Math., 1125 (1985), 1-8. | Zbl | MR

[Chr1] M. Charpentier. Sur quelques propriétés des courbes de M. Birkhoff, Bull. Soc. Math. France, 62 (1934), 193-224. | DOI | Numdam | EuDML | Zbl | MR

[Chr2] M. Charpentier. Sur des courbes fermées analogues aux courbes de M. Birkhoff, J. Math. Pures et Appl., 14 (1935), 1-48. | Numdam | JFM | Zbl

[Che1] A. Chenciner. Sur un énoncé dissipatif du théorème géométrique de Poincaré-Birkhoff, C. R. Acad. Sc. Paris, 294 (1982), 243-246. | Zbl | MR

[Che2] A. Chenciner. La dynamique au voisinage d'un point fixe elliptique conservatif : de Poincaré et Birkhoff à Aubry et Mather, Séminaire Bourbaki, 622, Astérisque, Soc. Math. France, (1985), 165-173. | Numdam | EuDML | MR | Zbl

[CGT] A. Chenciner, J. M. Gambaudo, C. Tresser. Une remarque sur la structure des endomorphismes de degré $1$ du cercle, C. R. Acad. Sc. Paris, 299 (1984), 145-147. | Zbl | MR

[Co1] C. Conley. Isolated invariant sets and the Morse index, CBMS Regional Conference, 38, Providence RI, Amer. Math. Soc., 1978. | DOI | Zbl | MR

[Co2] C. Conley. The gradient structure of aflow (I), Ergod. Th. Dynam. Sys., 8* (1988), 11-31. | DOI | Zbl | MR

[CE] C. Conley, R. Easton. Isolated invariant sets and isolating blocks, Trans. Amer. Math. Soc., 158 (1971), 35-61. | DOI | Zbl | MR

[CZ] C. Conley, E. Zehnder. The Birkhoff-Lewis fixed point theorem and a conjecture of Arnold, Invent. Math., 73 (1983), 33-49. | DOI | EuDML | Zbl | MR

[D] J. Denzler. Mather sets for plane Hamiltonian systems. J. Appl. Math. Phys. (ZAMP), 38 (1987), 791-812. | DOI | Zbl | MR

[Do] R. Douady. Application du théorème des tores invariants, Thèse de troisième cycle, Univ. Paris VII, 1982.

[EE] C. Earle, J. Eells. A fibrebundle description of Teichmuller theory, J. Differential Geometry, 4 (1970), 169-185.

[Fa1] A. Fathi. Une interprétation plus topologique de la démonstration du théorème de Birkhoff, Appendice du Chap. I de [Hm2].

[Fa2] A. Fathi. An orbit closing proof of Brouwer's lemma on translation arcs, L'enseignement Math., 33 (1987), 315-322. | Zbl | MR

[Fa3] A. Fathi. Expansiveness, hyperbolicity and Hausdorff dimension. Comm. in Math. Phys., 126 (1989), 249-262. | DOI | Zbl | MR

[FLP] A. Fathi, F. Laudenbach, V. Poenaru. Travaux de Thurston sur les surfaces, Astérisque, 66-67, Soc. Math. France, (1979). | Zbl | Numdam | MR

[F1] A. Floer. A refinement of Conley index and an application to the stability of hyperbolic invariant sets, Ergod. Th. Dynam. Sys., 7 (1987), 93-103. | DOI | Zbl | MR

[Fr1] J. Franks. Generalizations of the Poincaré-Birkhoff theorem, Ann. of Math., 128 (1988), 139-151. | DOI | Zbl | MR

[Fr2] J. Franks. Recurrence and fixed points of surface homeomorphisms, Ergod. Th. Dynam. Sys., 8* (1988), 99-107. | DOI | Zbl | MR

[Fr3] J. Franks. A variation on the Poincaré-Birkhoff theorem, Contemporary Mathematics, 81 (1988), 111-117. | DOI | Zbl | MR

[Fr4] J. Franks. Realizing rotation vectors for torus homeomorphisms, Trans. Amer. Math. Soc., 311 (1989), 107-115. | DOI | Zbl | MR

[Gol] C. Gole. Ghost circle for twist maps, preprint, University of Minnesota, (1990). | MR | Zbl

[Gor] D. L. Goroff. Hyperbolic sets for twist maps, Ergod. Th. Dynam. Sys., 5 (1985), 337-354. | DOI | Zbl | MR

[Gu] L. Guillou. Théorème de translation plane de Brouwer et généralisations du théorème de Poincaré-Birkhoff, preprint, Univ. Paris-Sud, (1990). | Zbl | MR

[Ha1] G. R. Hall. A topological version of a theorem of Mather on twist maps. Ergod. Th. Dynam. Sys., 4 (1984), 585-603. | Zbl | MR

[Ha2] G. R. Hall. Some problems on dynamics of annulus maps, Contemporary Mathematics, 81 (1988), 135-151. | DOI | Zbl | MR

[Hd1] M. Handel. A pathological area preserving $C^{\infty}$ diffeomorphism of the plane, Proc. Amer. Math. Soc., 86 (1982), 163-168. | Zbl | MR

[Hd2] M. Handel. Zero entropy surfaces homeomorphisms, preprint, CUNY, (1986).

[Hd3] M. Handel. The rotation set of a homeomorphism is closed, Comm. Math. Phys., 127 (1990), 339-349. | DOI | Zbl | MR

[H1] G. A. Hedlund. Geodesics on a two-dimensional Riemannian manifold with periodic coefficients. Ann. of Math., 33 (1932), 719-739. | DOI | JFM | MR

[Hn] M. Henon. A two-dimensional mapping with a strange attractor, Comm. Math. Phys., 50 (1976), 69-77. | DOI | Zbl | MR

[Hm1] M. R. Herman. Sur la conjugaison différentiate des difféomorphismes du cercle à une rotation. Publ. Math. I.H.E.S., 49 (1979), 5-234. | DOI | Numdam | EuDML | Zbl | MR

[Hm2] M. R. Herman. Sur les courbes invariantes par les difféomorphismes de l'anneau, Astérisque, 103-104, Soc. Math. France, (1983). | Numdam | Zbl | MR

[Hm3] M. R. Herman. Inégalités à priori pour des tores lagrangiens invariants par des difféomorphismes symplectiques, Publ. Math. I.H.E.S., 70 (1989), 47-101. | DOI | Numdam | EuDML | Zbl | MR

[Hm4] M. Herman. On the dynamics on Lagrangian tori invariant by symplectic diffeomorphisms, preprint, Ecole Polytechnique, (1990). | Zbl | MR

[HH] K. Hockett, P. Holmes. Josephson's junction, annulus maps, Birkhoff attractors, horseshoes and rotation sets, Ergod. Th. Dynam. Sys., 6 (1986), 205-239. | DOI | Zbl | MR

[Hu] M. Hurley. Attractors : persistence, and density of their basins, Trans. Amer. Math. Soc., 269 (1982), 247-271. | DOI | Zbl | MR

[Ka1] A. Katok. Lyapounov exponents, entropy and periodic points for diffeomorphisms, Publ. Math. I.H.E.S., 51 (1980). | DOI | Numdam | Zbl | MR

[Ka2] A. Katok. Some remarks on Birkhoff and Mather twist map theorem, Ergod. Th. Dynam. Sys., 2 (1982), 185-194. | DOI | Zbl | MR

[Ka3] A. Katok. Minimal orbits for small perturbations of completely integrable Hamiltonian systems, preprint, Cal. Tech. | MR | Zbl

[Ke] B. De Kerekjarto. The plane translation theorem of Brouwer and the last geometric theorem of Poincaré, Acta Sci. Math. Szeged, 4 (1928-29), 86-102. | JFM

[Ku] I. Kupka. Contribution à la théorie des champs génériques, Contributions to Diff. Equations, 2 (1963), 457-484. | Zbl | MR

[L1] P. Le Calvez. Existence d'orbites quasi-périodiques dans les attracteurs de Birkhoff, Commun. Math. Phys., 106 (1986), 383-394. | DOI | Zbl | MR

[L2] P. Le Calvez. Propriétés dynamiques des régions d'instabilité, Ann. Scient. Ec. Norm. Sup., 20 (1987), 443-464. | DOI | Numdam | EuDML | Zbl | MR

[L3] P. Le Calvez. Propriétés des attracteurs de Birkhoff, Ergod. Th. Dynam. Sys. 8 (1987), 241-310. | Zbl | MR

[L4] P. Le Calvez. Les ensembles d'Aubry-Mather d'un difféomorphisme conservatif de l'anneau déviant la verticale sont en général hyperboliques. C. R. Acad. Sci. Paris. 306 (1988), 51-54. | Zbl | MR

[L5] P. Le Calvez. Propriétés générales des applications déviant la verticale. Bull. Soc. Math. France, 117 (1989), 69-102. | DOI | Numdam | EuDML | Zbl | MR

[L6] P. Le Calvez. Etude topologique des applications déviant la verticale, Ensaios Matematicos, Soc. Bras. Math., vol. 2 (1990). | Zbl | EuDML

[L7] P. Le Calvez. Existence d' orbites de Birkhoff généralisées pour les difféomorphismes conservatifs de l'anneau, preprint, Univ. Paris-Sud, (1989).

[L8] P. Le Calvez. Construction d'orbites périodiques par perturbation d'un difféomorphisme déviant la verticale, preprint, Univ. Paris-Sud, (1991). | Zbl | MR

[Lv] M. Levi. Qualitative analysis of the periodically forced relaxation oscillations, Mem. Amer. Math. Soc., 24 (1981), 1-147. | Zbl | MR

[Ln] N. Levinson. A second order differential equation with singular solutions, Ann. of Math., 50 (1949), 127-153. | DOI | Zbl | MR

[Li1] J. E. Littlewood. On non linear differential equations of the second order : III, Acta Math., 97 (1957), 267-308. | Zbl | MR

[Li2] J. E. Littlewood. On non linear differential equations of the second order : IV, Acta Math., 98 (1957), 1-110. | Zbl | MR

[LM] J. Llibre, R. S. Mackay. Rotation vectors and entropy for diffeomorphisms of the torus isotopic to the identity, Ergod. Th and Dynam. Sys., 11 (1991), 115-128. | DOI | MR

[Ma1] J. Mather. Invariant subsets of area-preserving homeomorphisms of surfaces, Advances in Math. Suppl. Studies, 7B. | Zbl

[Ma2] J. Mather. Existence of quasi-periodic orbits for twist homeomorphisms of the annulus, Topology, 21 (1982), 457-467. | DOI | Zbl | MR

[Ma3] J. Mather. Glancing billards, Ergod. Th. Dynam. Sys. 2 (1982), 397-403. | DOI | Zbl | MR

[Ma4] J. Mather. Non-uniqueness of solutions of Percival's Euler-Lagrange equation, Comm. Math. Phys., 86 (1982), 465-473. | DOI | Zbl | MR

[Ma5] J. Mather. A criterion for the non-existence of invariant circles, Publ. IHES, 63 (1986), 153-204. | DOI | Numdam | EuDML | Zbl | MR

[Ma6] J. Mather. Destruction of invariant circles Ergod. Th. Dynam. Sys., 8* (1988), 199-214. | DOI | Zbl | MR

[Ma7] J. Mather. Minimal measures, Comment. Math. Helv., 64 (1989), 375-394. | DOI | EuDML | Zbl | MR

[Ma8] J. Mather. Minimal action measures for positive definite Lagrangian systems, a paraître dans Proc. IXth Int. Conf. Math. Phys. | Zbl

[Ma9] J. Mather. Variational construction of orbits of twist diffeomorphisms, (1990). | Zbl

[MS] R. S. Mac Kay, J. Stark. Lectures on orbits of minimal actions for area preserving maps, preprint, Univ. of Warwick.

[Mi] J. Milnor, On the concept of attactor, Comm. Math. Phys., Commun. Math. Phys., 99 (1985), 177-195. | DOI | Zbl | MR

[MZ1] M. Misiurewicz, K. Zieman. Rotation sets for maps of tori, preprint, Univ. of Warwick, 1988. | MR | Zbl

[MZ2] M. Misiurewicz, K. Zieman. Rotation sets and ergodic measures for torus homeomorphism, Fund. Math., 137 (1991), 45-52. | DOI | EuDML | Zbl | MR

[MV] L. Mora, M. Viana. Abundance of strange attractors, preprint, IMPA, Rio de Janeiro, (1989). | Zbl | MR

[Mo1] J. Moser. On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss., Gottingen, Math. Phys., KI (1962), 1-20. | Zbl | MR

[Mo2] J. Moser. Monotone twist mappings and the calculus of variation, Ergod. Th. Dynam. Sys., 6 (1986), 401-413. | DOI | Zbl | MR

[Mo3] J. Moser. Recent developments in the theory of Hamiltonian systems, SIAM Review, 8 (1986), 459-485. | Zbl | MR

[N1] S. Newhouse. Diffeomorphisms with infinitely many sinks Topology, 13 (1974), 9-18. | DOI | Zbl | MR

[N2] S. Newhouse. The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms, Publ. Math. I.H.E.S., 50 (1979), 101-151. | DOI | Numdam | EuDML | Zbl | MR

[NPT] S. E. Newhouse, J. Palis, F. Takens. Bifurcations and stability of families of diffeomorphisms, Publ. Math. I.H.E.S., 57 (1983), 5-72. | DOI | Numdam | EuDML | Zbl | MR

[PT] J. Palis, F. Takens. Homoclinic bifurcations : hyperbolicity, fractional dimension and infinitely many attractors, Cambridge Univers. Press, à paraître. | Zbl

[P1] P. Plykin. Sources and sinks for $A$ -diffeomorphisms, USSR Math. Sb. 23 (1978), 233-253. | DOI | Zbl

[Po] H. Poincare. Sur un théorème de géométrie, Rediconti del Circolo Matematico di Palermo, 33 (1912), 375-407. | DOI | JFM

[Ro] R. C. Robinson. Generic properties of conservative systems : I, II, Amer. J. Math., 92 (1970), 562-603 et 897-906. | DOI | MR | Zbl

[Ru] D. Ruelle. Small random perturbations of dynamical systems and the definition of attractors, Commun. Math. Phys., 82 (1981), 137-151. | DOI | Zbl | MR

[Sc] S. Schwartzman. Asymptotic cycles, Ann. of Maths, 68 (1957), 270-284. | DOI | Zbl | MR

[SM] C. L. Siegel, J. K. Moser. Lectures on celestial mechanics, Springer-Verlag, Berlin, (1971). | Zbl | MR

[Sm1] S. Smale, Stable manifolds for differential equations and diffeomorphisms, Ann. Scuola Norm. Sup. Pisa, 17 (1963), 97-116. | Zbl | Numdam | EuDML | MR

[Sm2] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747-817. | DOI | Zbl | MR

[Sm3] S. Smale, Diffeomorphisms of the $2$ -sphere, Proc. Amer. Math. Soc., 10 (1959), 621-626. | Zbl | MR

[T] W. P. Thurston. On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc., 19 (1988), 417-431. | DOI | Zbl | MR