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Desingularization of surface maps

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  • We prove a result for maps of surfaces that illustrates how singularhyperbolic flows can be desingularized if a global section can be collapsed to a surface along stable leaves.

    Mathematics Subject Classification: Primary: 37D20; Secondary: 57N10.


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  • Figure 1.  The Geometric Lorenz Attractor (by Mattias Lindkvist, from [4])

    Figure 2.  Singularization (from [1])

  • [1] C. Bonatti, L. J. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective, Encyclopaedia of Mathematical Sciences, 102, Mathematical Physics, Ⅲ, Springer-Verlag, Berlin, 2005.
    [2] C. BonattiA. Pumariño and M. Viana, Lorenz attractors with arbitrary expanding dimension, C. R. Acad. Paris Sér. I Math., 325 (1997), 883-888.  doi: 10.1016/S0764-4442(97)80131-0.
    [3] R. J. Daverman and G. A. Venema, Embeddings in Manifolds, Graduate Studies in Mathematics, 106, American Mathematical Society, Providence, RI, 2009. doi: 10.1090/gsm/106.
    [4] B. Hasselblatt and A. Katok, A First Course in Dynamics. With a Panorama of Recent Developments, Cambridge University Press, New York, 2003.
    [5] C. A. Morales and M. J. Pacifico, Strange attractors arising from hyperbolic flows, preprint.
    [6] C. A. MoralesM. J. Pacifico and E. R. Pujals, Global attractors from the explosion of singular cycles, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 1317-1322.  doi: 10.1016/S0764-4442(97)82362-2.
    [7] C. A. MoralesM. J. Pacifico and E. R. Pujals, Singular Hyperbolic Systems, Proc. Amer. Math. Soc., 127 (1999), 3393-3401.  doi: 10.1090/S0002-9939-99-04936-9.
    [8] R. Metzger and C. Morales, Sectional-Hyperbolic Systems, Ergodic Theory Dynam. Systems, 28 (2008), 1587-1597.  doi: 10.1017/S0143385707000995.
    [9] S. Newhouse, On simple arcs between structurally stable flows, in Dynamical SystemsWarwick 1974 (Proc. Sympos. Appl. Topology and Dynamical Systems, Univ. Warwick, Coventry, 1973/1974; presented to E. C. Zeeman on his fiftieth birthday), Lecture Notes in Math., Vol. 468, Springer, Berlin, 1975,209-233.
    [10] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817.  doi: 10.1090/S0002-9904-1967-11798-1.
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