One-dimensional contracting singular horseshoe

Carrasco-Olivera, D.; Morales, C.; San Martín, B.

doi:10.1090/S0002-9939-2010-10392-1

One-dimensional contracting singular horseshoe
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by D. Carrasco-Olivera, C. A. Morales and B. San Martín

Proc. Amer. Math. Soc. 138 (2010), 4009-4023

DOI: https://doi.org/10.1090/S0002-9939-2010-10392-1

Published electronically: June 16, 2010

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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.

References

A. M. Blokh and M. Yu. Lyubich, Nonexistence of wandering intervals and structure of topological attractors of one-dimensional dynamical systems. II. The smooth case, Ergodic Theory Dynam. Systems 9 (1989), no. 4, 751–758. MR 1036906, DOI 10.1017/S0143385700005319
A. M. Blokh and M. Yu. Lyubich, The absence of wandering intervals in one-dimensional smooth dynamical systems, Dokl. Akad. Nauk SSSR 304 (1989), no. 5, 1033–1036 (Russian); English transl., Soviet Math. Dokl. 39 (1989), no. 1, 169–172. MR 997178
D. Berry and B. D. Mestel, Wandering intervals for Lorenz maps with bounded nonlinearity, Bull. London Math. Soc. 23 (1991), no. 2, 183–189. MR 1122907, DOI 10.1112/blms/23.2.183
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.
Milton Cobo, Piece-wise affine maps conjugate to interval exchanges, Ergodic Theory Dynam. Systems 22 (2002), no. 2, 375–407. MR 1898797, DOI 10.1017/S0143385702000196
Ethan M. Coven and Zbigniew Nitecki, Nonwandering sets of the powers of maps of the interval, Ergodic Theory Dynam. Systems 1 (1981), no. 1, 9–31. MR 627784, DOI 10.1017/s0143385700001139
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.
John Guckenheimer, Sensitive dependence to initial conditions for one-dimensional maps, Comm. Math. Phys. 70 (1979), no. 2, 133–160. MR 553966
R. F. Williams, The structure of Lorenz attractors, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 73–99. MR 556583
Hall, C.R., A $C^{\infty }$ Denjoy counterexample. Ergod. Th. and Dynam. Sys. 7 (1981), 509–530.
J. Harrison, Wandering intervals, Dynamical systems and turbulence, Warwick 1980 (Coventry, 1979/1980), Lecture Notes in Math., vol. 898, Springer, Berlin-New York, 1981, pp. 154–163. MR 654888
A. F. Ivanov, An example of infinitely many sinks for smooth interval maps, Acta Math. Univ. Comenian. (N.S.) 61 (1992), no. 1, 3–9. MR 1205854
M. Yu. Lyubich, 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), no. 4, 737–749. MR 1036905, DOI 10.1017/S0143385700005307
R. Labarca and M. J. Pacífico, Stability of singularity horseshoes, Topology 25 (1986), no. 3, 337–352. MR 842429, DOI 10.1016/0040-9383(86)90048-0
W. de Melo, A finiteness problem for one-dimensional maps, Proc. Amer. Math. Soc. 101 (1987), no. 4, 721–727. MR 911040, DOI 10.1090/S0002-9939-1987-0911040-1
M. Martens, W. de Melo, and S. van Strien, Julia-Fatou-Sullivan theory for real one-dimensional dynamics, Acta Math. 168 (1992), no. 3-4, 273–318. MR 1161268, DOI 10.1007/BF02392981
W. de Melo and S. van Strien, One-dimensional dynamics: the Schwarzian derivative and beyond, Bull. Amer. Math. Soc. (N.S.) 18 (1988), no. 2, 159–162. MR 929092, DOI 10.1090/S0273-0979-1988-15633-9
W. de Melo and S. van Strien, A structure theorem in one-dimensional dynamics, Ann. of Math. (2) 129 (1989), no. 3, 519–546. MR 997312, DOI 10.2307/1971516
Welington de Melo and Sebastian van Strien, One-dimensional dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 25, Springer-Verlag, Berlin, 1993. MR 1239171, DOI 10.1007/978-3-642-78043-1
Eduardo M. Muñoz Morales, Bernardo San Martín Rebolledo, and Jaime A. Vera Valenzuela, Nonhyperbolic persistent attractors near the Morse-Smale boundary, Ann. Inst. H. Poincaré C Anal. Non Linéaire 20 (2003), no. 5, 867–888 (English, with English and French summaries). MR 1995505, DOI 10.1016/S0294-1449(03)00015-5
John Milnor and William Thurston, On iterated maps of the interval, Dynamical systems (College Park, MD, 1986–87) Lecture Notes in Math., vol. 1342, Springer, Berlin, 1988, pp. 465–563. MR 970571, DOI 10.1007/BFb0082847
Zbigniew Nitecki, Differentiable dynamics. An introduction to the orbit structure of diffeomorphisms, The M.I.T. Press, Cambridge, Mass.-London, 1971. MR 0649788
Tomasz Nowicki and Sebastian van Strien, Hyperbolicity properties of $C^2$ multi-modal Collet-Eckmann maps without Schwarzian derivative assumptions, Trans. Amer. Math. Soc. 321 (1990), no. 2, 793–810. MR 994169, DOI 10.1090/S0002-9947-1990-0994169-6
William Parry, Symbolic dynamics and transformations of the unit interval, Trans. Amer. Math. Soc. 122 (1966), 368–378. MR 197683, DOI 10.1090/S0002-9947-1966-0197683-5
Alvaro Rovella, The dynamics of perturbations of the contracting Lorenz attractor, Bol. Soc. Brasil. Mat. (N.S.) 24 (1993), no. 2, 233–259. MR 1254985, DOI 10.1007/BF01237679
Arthur J. Schwartz, A generalization of a Poincaré-Bendixson theorem to closed two-dimensional manifolds, Amer. J. Math. 85 (1963), 453–458; errata: 85 (1963), 753. MR 0155061
A. N. Sharkovskiĭ and A. F. Ivanov, $C^{\infty }$-mappings of an interval with attracting cycles with arbitrarily large periods, Ukrain. Mat. Zh. 35 (1983), no. 4, 537–539 (Russian). MR 712483
David Singer, Stable orbits and bifurcation of maps of the interval, SIAM J. Appl. Math. 35 (1978), no. 2, 260–267. MR 494306, DOI 10.1137/0135020
Sebastian van Strien and Edson Vargas, Real bounds, ergodicity and negative Schwarzian for multimodal maps, J. Amer. Math. Soc. 17 (2004), no. 4, 749–782. MR 2083467, DOI 10.1090/S0894-0347-04-00463-1
Jean-Christophe Yoccoz, 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 (French, with English summary). MR 741080