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  2017, 24: 1-9. doi: 10.3934/era.2017.24.001

Desingularization of surface maps

Erica Clay , Boris Hasselblatt and  Enrique Pujals

1. 

Department of Mathematics, Tufts University, Medford, MA 02155, USA
2. 

IMPA, Estrada Dona Castorina 110, Rio de Janeiro, Brasil 22460-320

Received  November 15, 2016 Revised  January 11, 2017 Published  February 2017

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.

Keywords: Singularization, hyperbolicity.
Mathematics Subject Classification: Primary: 37D20; Secondary: 57N10.
Citation: Erica Clay, Boris Hasselblatt, Enrique Pujals. Desingularization of surface maps. Electronic Research Announcements, 2017, 24: 1-9. doi: 10.3934/era.2017.24.001
References:
[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.  Google Scholar

[2]

C. Bonatti, A. 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.  Google Scholar

[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.  Google Scholar

[4]

B. Hasselblatt and A. Katok, A First Course in Dynamics. With a Panorama of Recent Developments, Cambridge University Press, New York, 2003.  Google Scholar

[5]

C. A. Morales and M. J. Pacifico, Strange attractors arising from hyperbolic flows, preprint. Google Scholar

[6]

C. A. Morales, M. 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.  Google Scholar

[7]

C. A. Morales, M. 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.  Google Scholar

[8]

R. Metzger and C. Morales, Sectional-Hyperbolic Systems, Ergodic Theory Dynam. Systems, 28 (2008), 1587-1597.  doi: 10.1017/S0143385707000995.  Google Scholar

[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.  Google Scholar

[10]

S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817.  doi: 10.1090/S0002-9904-1967-11798-1.  Google Scholar

show all references

References:
[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.  Google Scholar

[2]

C. Bonatti, A. 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.  Google Scholar

[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.  Google Scholar

[4]

B. Hasselblatt and A. Katok, A First Course in Dynamics. With a Panorama of Recent Developments, Cambridge University Press, New York, 2003.  Google Scholar

[5]

C. A. Morales and M. J. Pacifico, Strange attractors arising from hyperbolic flows, preprint. Google Scholar

[6]

C. A. Morales, M. 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.  Google Scholar

[7]

C. A. Morales, M. 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.  Google Scholar

[8]

R. Metzger and C. Morales, Sectional-Hyperbolic Systems, Ergodic Theory Dynam. Systems, 28 (2008), 1587-1597.  doi: 10.1017/S0143385707000995.  Google Scholar

[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.  Google Scholar

[10]

S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817.  doi: 10.1090/S0002-9904-1967-11798-1.  Google Scholar

Figure 1.  The Geometric Lorenz Attractor (by Mattias Lindkvist, from [4])
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Figure 2.  Singularization (from [1])
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