July  2014, 34(7): 2963-2982. doi: 10.3934/dcds.2014.34.2963

Hyperbolicity and types of shadowing for $C^1$ generic vector fields

1. 

Instituto de Matemática, Universidade Federal do Rio de Janeiro, P. O. Box 68530, 21945-970 Rio de Janeiro, Brazil

Received  July 2013 Revised  September 2013 Published  December 2013

We study various types of shadowing properties and their implication for $C^1$ generic vector fields. We show that, generically, any of the following three hypotheses implies that an isolated set is topologically transitive and hyperbolic: (i) the set is chain transitive and satisfies the (classical) shadowing property, (ii) the set satisfies the limit shadowing property, or (iii) the set satisfies the (asymptotic) shadowing property with the additional hypothesis that stable and unstable manifolds of any pair of critical orbits intersect each other. In our proof we essentially rely on the property of chain transitivity and, in particular, show that it is implied by the limit shadowing property. We also apply our results to divergence-free vector fields.
Citation: Raquel Ribeiro. Hyperbolicity and types of shadowing for $C^1$ generic vector fields. Discrete & Continuous Dynamical Systems, 2014, 34 (7) : 2963-2982. doi: 10.3934/dcds.2014.34.2963
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show all references

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Discrete Contin. Dyn. Syst., 17 (2007), 223-245.  Google Scholar

[2]

Ergodic Theory Dynam. Systems, 33 (2013), 1644-1666. doi: 10.1017/etds.2012.111.  Google Scholar

[3]

Ergodic Theory Dynam. Systems, 27 (2007), 1399-1417. doi: 10.1017/S014338570700017X.  Google Scholar

[4]

Proc. Amer. Math. Soc., 141 (2013), 2817-2827 . doi: 10.1090/S0002-9939-2013-11536-4.  Google Scholar

[5]

Mem. Soc. Math. Fr. (N. S.), (1998), vi+120 pp.  Google Scholar

[6]

preprint, IMPA, 2011. Google Scholar

[7]

C. R. Math. Acad. Sci. Paris, 346 (2008), 1169-1174. doi: 10.1016/j.crma.2008.07.012.  Google Scholar

[8]

Ergod. Th. & Dynam. Sys., 33 (2013), 1709-1731. doi: 10.1017/etds.2012.110.  Google Scholar

[9]

J. Differential Equations, 245 (2008), 3127-3143. doi: 10.1016/j.jde.2008.02.045.  Google Scholar

[10]

Nonlinearity, 2 (1989), 1-22. doi: 10.1088/0951-7715/2/1/001.  Google Scholar

[11]

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[12]

Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York, 1975.  Google Scholar

[13]

Regional Conference Series in Mathematics, No. 35, Amer. Math. Soc., Providence, R.I., 1978.  Google Scholar

[14]

CBMS Regional Conference Series in Mathematics, 38, American Mathematical Society, Providence, R.I., 1978.  Google Scholar

[15]

Publ. Math. Inst. Hautes Études Sci., (2006), 87-141. doi: 10.1007/s10240-006-0002-4.  Google Scholar

[16]

Numer. Funct. Anal. Optim., 18 (1997), 75-92. doi: 10.1080/01630569708816748.  Google Scholar

[17]

Dyn. Syst., 27 (2012), 223-238. doi: 10.1080/14689367.2012.655710.  Google Scholar

[18]

Trans. Amer. Math. Soc., 158 (1971), 301-308. doi: 10.1090/S0002-9947-1971-0283812-3.  Google Scholar

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[22]

Nonlinear Anal., 67 (2007), 1680-1689. doi: 10.1016/j.na.2006.07.040.  Google Scholar

[23]

Chaos Solitons Fractals, 41 (2009), 2234-2240. doi: 10.1016/j.chaos.2008.08.029.  Google Scholar

[24]

Chaos Solitons Fractals, 23 (2005), 989-995. doi: 10.1016/j.chaos.2004.06.059.  Google Scholar

[25]

Math. Surveys and Monographs, 25, Amer. Math. Soc., Providence, RI, 1988.  Google Scholar

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Bull. Amer. Math. Soc., 76 (1970), 1015-1019. doi: 10.1090/S0002-9904-1970-12537-X.  Google Scholar

[27]

Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995.  Google Scholar

[28]

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[29]

Contributions to Differential Equations, 2 (1963), 457-484.  Google Scholar

[30]

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[31]

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[32]

Ergodic Theory Dynam. Systems, 28 (2008), 1587-1597. doi: 10.1017/S0143385707000995.  Google Scholar

[33]

J. Differential Equations, 38 (1980), 192-209. doi: 10.1016/0022-0396(80)90004-2.  Google Scholar

[34]

Lecture Notes in Mathematics, 1706, Springer-Verlag, Berlin, 1999.  Google Scholar

[35]

Ergodic Theory Dynam. Systems, 3 (1983), 261-313. doi: 10.1017/S0143385700001978.  Google Scholar

[36]

Amer. J. Math., 92 (1970), 562-603. doi: 10.2307/2373361.  Google Scholar

[37]

Springer-Verlag, New York, 1987.  Google Scholar

[38]

Ann. Sc. Norm. Super. Pisa (3), 17 (1963), 97-116.  Google Scholar

[39]

Trans. Amer, Math. Soc., 352 (2000), 5213-5230. doi: 10.1090/S0002-9947-00-02553-8.  Google Scholar

[40]

Discrete Contin. Dynam. Systems, 6 (2000), 237-241. doi: 10.3934/dcds.2000.6.237.  Google Scholar

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