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One-dimensional contracting singular horseshoe

Authors: D. Carrasco-Olivera, C. A. Morales and B. San Martín
Journal: Proc. Amer. Math. Soc. 138 (2010), 4009-4023
MSC (2010): Primary 37E05, 37D25; Secondary 37D30, 37F15
Published electronically: June 16, 2010
MathSciNet review: 2679622
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Abstract: In this paper we prove some kind of structural stability defined as usual but restricted to a certain subset of one-dimensional maps coming from first return maps associated to singular cycles for vector fields in manifolds with boundary. The motivation is the stability of the Singular Horseshoes introduced by Labarca and Pacifico where an expanding condition on the singularity holds. Here we obtain analogous result but under a contracting condition.

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  • 1. Blokh, A.M. and Lyubich, M.Yu.,
    Nonexistence of wandering intervals and structure of topological attractors of one-dimensional dynamical systems. II. The smooth case.
    Ergodic Theory Dynam. Systems, 9 (1989), 751-758. MR 1036906 (91e:58101)
  • 2. Blokh, A.M. and Lyubich, M.Yu.,
    The absence of wandering intervals in one-dimensional smooth dynamical systems.
    Soviet Math. Dokl. 39 (1989), 169-172. MR 997178 (90f:58110)
  • 3. Berry, D. and Mestel, B.D.,
    Wandering intervals for Lorenz maps with bounded nonlinearity.
    Bull. London Math. Soc. 23 (1991), 183-189. MR 1122907 (93c:58119)
  • 4. Cedervall, S.,
    Invariant Measures and Correlation Decay for S-multimodal Interval Maps.
    Doctor of Philosophy Thesis, Imperial College of Science, Technology and Medicine, University of London, 2006.
  • 5. Cobo, Milton,
    Piece-wise affine maps conjugate to interval exchanges.
    Ergodic Theory Dynam. Systems 22 (2002), no. 2, 375-407. MR 1898797 (2003h:37003)
  • 6. Coven, E. M. and Nitecki, Z.,
    Nonwandering sets of the powers of maps of the interval.
    Ergodic Theory Dynam. Systems 1 (1981), no. 1, 9-31. MR 627784 (82m:58043)
  • 7. Denjoy, A.,
    Sur les courbes définies par les équations différentielles à la surface du tore.
    Journal de Mathématiques Pures et Appliquées 11 (1932), 333-375.
  • 8. Guckenheimer, J.,
    Sensitive dependence to initial conditions for one-dimensional maps.
    Commun. Math. Phys. 70 (1979), 133-160. MR 553966 (82c:58037)
  • 9. Guckenheimer, J. and Williams, R.,
    The structure of Lorenz attractors.
    Publ. Math. IHES, 50 (1979), 73-99. MR 556583 (82b:58055b)
  • 10. Hall, C.R.,
    A $ C^{\infty}$ Denjoy counterexample.
    Ergod. Th. and Dynam. Sys. 7 (1981), 509-530.
  • 11. Harrison, J.,
    Wandering intervals. Dynamical systems and turbulence.
    Lecture Notes in Math., 898, Springer, Berlin-New York, 1981, 154-163. MR 654888 (83h:58059)
  • 12. Ivanov, A.F.,
    An example of infinitely many sinks for smooth interval maps.
    Acta Math. Univ. Comenian. (N.S.) 61 (1992), no. 1, 3-9. MR 1205854 (93m:58060)
  • 13. Lyubich, M.Yu.,
    Nonexistence of wandering intervals and structure of topological attractors of one-dimensional dynamical systems. I. The case of negative Schwarzian derivative.
    Ergodic Theory Dynam. Systems 9 (1989), 737-749. MR 1036905 (91e:58100)
  • 14. Labarca, L. and Pacifico, M.J.,
    Stability of singularity horseshoes.
    Topology 25 (1986), no. 3, 337-352. MR 842429 (87h:58106)
  • 15. de Melo, W.,
    A finiteness problem for one-dimensional maps.
    Proc. Amer. Math. Soc. 101 (1987), 721-727. MR 911040 (89a:58063)
  • 16. Martens, M., de Melo, W., and van Strien, S.,
    Julia-Fatou-Sullivan theory for real one-dimensional dynamics.
    Acta Math. 168 (1992), no. 3-4, 273-318. MR 1161268 (93d:58137)
  • 17. de Melo, W. and van Strien, S.,
    One-dimensional dynamics: the Schwarzian derivative and beyond.
    Bull. Amer. Math. Soc. (N.S.) 18 (1988), 159-162. MR 929092 (89e:58079)
  • 18. de Melo, W. and van Strien, S.,
    A structure theorem in one-dimensional dynamics.
    Ann. of Math. (2) 129 (1989), 519-546. MR 997312 (90m:58106)
  • 19. de Melo, W. and van Strien, S.,
    One-dimensional dynamics.
    Springer-Verlag, Berlin, 1993. MR 1239171 (95a:58035)
  • 20. Muñoz, E.M., San Martín, B. and Vera, J.A., Nonhyperbolic persistent attractors near the Morse-Smale boundary. Ann. Inst. Henri Poincaré Anal. Non Linéaire 20 (2003), 867-888. MR 1995505 (2004e:37033)
  • 21. Milnor, J. and Thurston, W.,
    On iterated maps of the interval.
    Lecture Notes in Math., 1342, Springer, Berlin, 1988, 465-563. MR 970571 (90a:58083)
  • 22. Nitecki, Z.,
    Differentiable dynamics.
    MIT Press, Cambridge, Mass.-London, 1971. MR 0649788 (58:31210)
  • 23. Nowicki, T. and van Strien, S.,
    Hyperbolicity properties of $ C\sp 2$ multi-modal Collet-Eckmann maps without Schwarzian derivative assumptions.
    Trans. Amer. Math. Soc. 321 (1990), no. 2, 793-810. MR 994169 (91a:58097)
  • 24. Parry, W.,
    Symbolic dynamics and transformations of the unit interval.
    Trans. Amer. Math. Soc. 122 (1966), 368-378. MR 0197683 (33:5846)
  • 25. Rovella, A.,
    The dynamics of perturbations of the contracting Lorenz attractor.
    Bol. Soc. Brasil. Mat. (N.S.) 24 (1993), 233-259. MR 1254985 (95a:58097)
  • 26. Schwartz, A.,
    A generalization of a Poincaré-Bendixson theorem to closed two-dimensional manifolds.
    Amer. J. Math. 85 (1963), 453-458. MR 0155061 (27:5003)
  • 27. Sharkovskiĭ, A.N. and Ivanov, A.F.,
    $ C\sp{\infty }$-mappings of an interval with attracting cycles with arbitrarily large periods. (Russian)
    Ukrain. Mat. Zh. 35 (1983), no. 4, 537-539. MR 712483 (84m:58066)
  • 28. Singer, D., Stable orbits and bifurcations of maps of the interval.
    SIAM J. Appl. Math. 35 (1978), 260-267. MR 0494306 (58:13206)
  • 29. van Strien, S. and Vargas, E.,
    Real bounds, ergodicity and negative Schwarzian for multimodal maps.
    J. Amer. Math. Soc. 17 (2004), no. 4, 749-782. MR 2083467 (2005i:37043)
  • 30. Yoccoz, J.C.,
    Il n'y a pas de contre-exemple de Denjoy analytique.
    C. R. Acad. Sci. Paris Sér. I Math. 298 (1984), no. 7, 141-144. MR 741080 (85j:58134)

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Additional Information

D. Carrasco-Olivera
Affiliation: Departamento de Matemática, Facultad de Ciencias, Universidad del Bío Bío, Av. Collao #1202, Casilla 5-C, Región de Concepción, Chile

C. A. Morales
Affiliation: Instituto de Matemáticas, Universidad Federal de Rio de Janeiro, P.O. Box 68530, 21945-970, Rio de Janeiro, Brasil

B. San Martín
Affiliation: Departamento de Matematicas, Universidad Católica del Norte, Av. Angamos 0610, Casilla 1280, Antofagasta, Chile

Keywords: One-dimensional maps, conjugation, symbolic dynamic, nonwandering set, transitive set
Received by editor(s): June 11, 2009
Received by editor(s) in revised form: November 26, 2009, January 13, 2010, and January 18, 2010
Published electronically: June 16, 2010
Additional Notes: The first author was supported in part by Project Mecesup 0202-UCN; Project Fondecyt No. 1040682; Project ADI 17 Anillo en Sistemas Dinámicos de Baja Dimensión, Chile; CONICYT Proyecto Inserción de Nuevos Invertigadores en la Academia, 2009, Folio 79090039.
The second author was partially supported by CNPq, FAPERJ and PRONEX-Brazil.
The third author was partially supported by Project Fondecyt No. 1040682 and Project ADI 17, Anillo en Sistemas Dinámicos de Baja Dimensión, CONICYT - Chile and PRONEX-Brazil.
Communicated by: Bryna Kra
Article copyright: © Copyright 2010 American Mathematical Society

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