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 and Continuous Dynamical Systems, 2014, 34 (7) : 2963-2982. doi: 10.3934/dcds.2014.34.2963
References:
[1]

F. Abdenur and L. J. Díaz, Pseudo-orbit shadowing in the $C^1$ topology, Discrete Contin. Dyn. Syst., 17 (2007), 223-245.

[2]

A. Arbieto and T. Catalan, Hyperbolicity in the volume preserving scenario, Ergodic Theory Dynam. Systems, 33 (2013), 1644-1666. doi: 10.1017/etds.2012.111.

[3]

A. Arbieto and C. Matheus, A pasting lemma and some apllications for conservative systems, Ergodic Theory Dynam. Systems, 27 (2007), 1399-1417. doi: 10.1017/S014338570700017X.

[4]

A. Arbieto and C. Morales, A dichotomy for higher-dimensional flows, Proc. Amer. Math. Soc., 141 (2013), 2817-2827 . doi: 10.1090/S0002-9939-2013-11536-4.

[5]

M.-C. Arnaud, Le "closing lemma" en topologie $C^1$, Mem. Soc. Math. Fr. (N. S.), (1998), vi+120 pp.

[6]

S. Bautista and C. Morales, Lectures on sectional Anosov flows, preprint, IMPA, 2011.

[7]

M. Bessa, A generic incompressible flow is topological mixing, C. R. Math. Acad. Sci. Paris, 346 (2008), 1169-1174. doi: 10.1016/j.crma.2008.07.012.

[8]

M. Bessa and J. Rocha, Contributions to the geometric and ergodic theory of conservative flows, Ergod. Th. & Dynam. Sys., 33 (2013), 1709-1731. doi: 10.1017/etds.2012.110.

[9]

M. Bessa and J. Rocha, On $C^1$-robust transitivity of volume-preserving flows, J. Differential Equations, 245 (2008), 3127-3143. doi: 10.1016/j.jde.2008.02.045.

[10]

M. L. Blank, Metric properties of minimal solutions of discrete periodical variational problems, Nonlinearity, 2 (1989), 1-22. doi: 10.1088/0951-7715/2/1/001.

[11]

C. Bonatti and S. Crovisier, Récurrence et généricité, Invent. Math., 158 (2004), 33-104. doi: 10.1007/s00222-004-0368-1.

[12]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York, 1975.

[13]

R. Bowen, On Axiom A Diffeomorphisms, Regional Conference Series in Mathematics, No. 35, Amer. Math. Soc., Providence, R.I., 1978.

[14]

C. Conley, Isolated Invariant sets and the Morse Index, CBMS Regional Conference Series in Mathematics, 38, American Mathematical Society, Providence, R.I., 1978.

[15]

S. Crovisier, Periodic orbits and chain-transitive sets of $C^1$-diffeomorphisms, Publ. Math. Inst. Hautes Études Sci., (2006), 87-141. doi: 10.1007/s10240-006-0002-4.

[16]

T. Eirola, O. Nevalinna and S. Pilyugin, Limit shadowing property, Numer. Funct. Anal. Optim., 18 (1997), 75-92. doi: 10.1080/01630569708816748.

[17]

C. Ferreira, Stability properties of divergence-free vector fields, Dyn. Syst., 27 (2012), 223-238. doi: 10.1080/14689367.2012.655710.

[18]

J. Franks, Necessary conditions for the stability of diffeomorphisms, Trans. Amer. Math. Soc., 158 (1971), 301-308. doi: 10.1090/S0002-9947-1971-0283812-3.

[19]

S. Gan and L. Wen, Nonsingular star flows satisfy Axiom A and the no-cycle condition, Invent. Math., 164 (2006), 279-315. doi: 10.1007/s00222-005-0479-3.

[20]

S. Gan, M. Li and L. Wen, Robustly transitive singular sets via approach of an extended linear Poincaré flow, Discrete Contin. Dyn. Syst., 13 (2005), 239-269. doi: 10.3934/dcds.2005.13.239.

[21]

S. Gan, L. Wen and S. Zhu, Indices of singularities of robustly transitive sets, Discrete Contin. Dyn. Syst., 21 (2008), 945-957. doi: 10.3934/dcds.2008.21.945.

[22]

R. Gu, The asymptotic average shadowing property and transitivity, Nonlinear Anal., 67 (2007), 1680-1689. doi: 10.1016/j.na.2006.07.040.

[23]

R. Gu, The asymptotic average-shadowing property and transitivity for flows, Chaos Solitons Fractals, 41 (2009), 2234-2240. doi: 10.1016/j.chaos.2008.08.029.

[24]

R. Gu, Y. Sheng and Z. Xia, The average-shadowing property and transitivity for continuous flows, Chaos Solitons Fractals, 23 (2005), 989-995. doi: 10.1016/j.chaos.2004.06.059.

[25]

J. K. Hale, Asymptotic Behaviour of Dissipative Systems, Math. Surveys and Monographs, 25, Amer. Math. Soc., Providence, RI, 1988.

[26]

M. Hirsh, C. Pugh and M. Shub, Invariant manifolds, Bull. Amer. Math. Soc., 76 (1970), 1015-1019. doi: 10.1090/S0002-9904-1970-12537-X.

[27]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995.

[28]

M. Komuro, Lorenz attractors do not have the pseudo-orbit tracing property, J. Math. Soc. Japan, 37 (1985), 489-514. doi: 10.2969/jmsj/03730489.

[29]

I. Kupka, Contribution à la théorie des champs génériques, Contributions to Differential Equations, 2 (1963), 457-484.

[30]

M. Lee, Usual limit shadowable homoclinic classes of generic diffeomorphisms, Adv. Difference Equ., 2012 (2012), 8 pp. doi: 10.1186/1687-1847-2012-91.

[31]

K. Lee and X. Wen, Shadowable chain transitive sets of C1-generic diffeomorphisms, Bull. Korean Math. Soc., 49 (2012), 263-270. doi: 10.4134/BKMS.2012.49.2.263.

[32]

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

[33]

J. Lewowicz, Lyapunov functions and topological stability, J. Differential Equations, 38 (1980), 192-209. doi: 10.1016/0022-0396(80)90004-2.

[34]

S. Y. Pilyugin, Shadowing in Dynamical Systems, Lecture Notes in Mathematics, 1706, Springer-Verlag, Berlin, 1999.

[35]

C. Pugh and C. Robinson, The $C^1$ closing lemma, including Hamiltonians, Ergodic Theory Dynam. Systems, 3 (1983), 261-313. doi: 10.1017/S0143385700001978.

[36]

C. Robinson, Generic properties of conservative systems, Amer. J. Math., 92 (1970), 562-603. doi: 10.2307/2373361.

[37]

M. Shub, Global Stability of Dynamical Systems, Springer-Verlag, New York, 1987.

[38]

S. Smale, Stable manifolds for differential equations and diffeomorphisms, Ann. Sc. Norm. Super. Pisa (3), 17 (1963), 97-116.

[39]

L. Wen and Z. Xia, $C^1$-connecting lemmas, Trans. Amer, Math. Soc., 352 (2000), 5213-5230. doi: 10.1090/S0002-9947-00-02553-8.

[40]

L. Wen, On the preperiodic set, Discrete Contin. Dynam. Systems, 6 (2000), 237-241. doi: 10.3934/dcds.2000.6.237.

show all references

References:
[1]

F. Abdenur and L. J. Díaz, Pseudo-orbit shadowing in the $C^1$ topology, Discrete Contin. Dyn. Syst., 17 (2007), 223-245.

[2]

A. Arbieto and T. Catalan, Hyperbolicity in the volume preserving scenario, Ergodic Theory Dynam. Systems, 33 (2013), 1644-1666. doi: 10.1017/etds.2012.111.

[3]

A. Arbieto and C. Matheus, A pasting lemma and some apllications for conservative systems, Ergodic Theory Dynam. Systems, 27 (2007), 1399-1417. doi: 10.1017/S014338570700017X.

[4]

A. Arbieto and C. Morales, A dichotomy for higher-dimensional flows, Proc. Amer. Math. Soc., 141 (2013), 2817-2827 . doi: 10.1090/S0002-9939-2013-11536-4.

[5]

M.-C. Arnaud, Le "closing lemma" en topologie $C^1$, Mem. Soc. Math. Fr. (N. S.), (1998), vi+120 pp.

[6]

S. Bautista and C. Morales, Lectures on sectional Anosov flows, preprint, IMPA, 2011.

[7]

M. Bessa, A generic incompressible flow is topological mixing, C. R. Math. Acad. Sci. Paris, 346 (2008), 1169-1174. doi: 10.1016/j.crma.2008.07.012.

[8]

M. Bessa and J. Rocha, Contributions to the geometric and ergodic theory of conservative flows, Ergod. Th. & Dynam. Sys., 33 (2013), 1709-1731. doi: 10.1017/etds.2012.110.

[9]

M. Bessa and J. Rocha, On $C^1$-robust transitivity of volume-preserving flows, J. Differential Equations, 245 (2008), 3127-3143. doi: 10.1016/j.jde.2008.02.045.

[10]

M. L. Blank, Metric properties of minimal solutions of discrete periodical variational problems, Nonlinearity, 2 (1989), 1-22. doi: 10.1088/0951-7715/2/1/001.

[11]

C. Bonatti and S. Crovisier, Récurrence et généricité, Invent. Math., 158 (2004), 33-104. doi: 10.1007/s00222-004-0368-1.

[12]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York, 1975.

[13]

R. Bowen, On Axiom A Diffeomorphisms, Regional Conference Series in Mathematics, No. 35, Amer. Math. Soc., Providence, R.I., 1978.

[14]

C. Conley, Isolated Invariant sets and the Morse Index, CBMS Regional Conference Series in Mathematics, 38, American Mathematical Society, Providence, R.I., 1978.

[15]

S. Crovisier, Periodic orbits and chain-transitive sets of $C^1$-diffeomorphisms, Publ. Math. Inst. Hautes Études Sci., (2006), 87-141. doi: 10.1007/s10240-006-0002-4.

[16]

T. Eirola, O. Nevalinna and S. Pilyugin, Limit shadowing property, Numer. Funct. Anal. Optim., 18 (1997), 75-92. doi: 10.1080/01630569708816748.

[17]

C. Ferreira, Stability properties of divergence-free vector fields, Dyn. Syst., 27 (2012), 223-238. doi: 10.1080/14689367.2012.655710.

[18]

J. Franks, Necessary conditions for the stability of diffeomorphisms, Trans. Amer. Math. Soc., 158 (1971), 301-308. doi: 10.1090/S0002-9947-1971-0283812-3.

[19]

S. Gan and L. Wen, Nonsingular star flows satisfy Axiom A and the no-cycle condition, Invent. Math., 164 (2006), 279-315. doi: 10.1007/s00222-005-0479-3.

[20]

S. Gan, M. Li and L. Wen, Robustly transitive singular sets via approach of an extended linear Poincaré flow, Discrete Contin. Dyn. Syst., 13 (2005), 239-269. doi: 10.3934/dcds.2005.13.239.

[21]

S. Gan, L. Wen and S. Zhu, Indices of singularities of robustly transitive sets, Discrete Contin. Dyn. Syst., 21 (2008), 945-957. doi: 10.3934/dcds.2008.21.945.

[22]

R. Gu, The asymptotic average shadowing property and transitivity, Nonlinear Anal., 67 (2007), 1680-1689. doi: 10.1016/j.na.2006.07.040.

[23]

R. Gu, The asymptotic average-shadowing property and transitivity for flows, Chaos Solitons Fractals, 41 (2009), 2234-2240. doi: 10.1016/j.chaos.2008.08.029.

[24]

R. Gu, Y. Sheng and Z. Xia, The average-shadowing property and transitivity for continuous flows, Chaos Solitons Fractals, 23 (2005), 989-995. doi: 10.1016/j.chaos.2004.06.059.

[25]

J. K. Hale, Asymptotic Behaviour of Dissipative Systems, Math. Surveys and Monographs, 25, Amer. Math. Soc., Providence, RI, 1988.

[26]

M. Hirsh, C. Pugh and M. Shub, Invariant manifolds, Bull. Amer. Math. Soc., 76 (1970), 1015-1019. doi: 10.1090/S0002-9904-1970-12537-X.

[27]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995.

[28]

M. Komuro, Lorenz attractors do not have the pseudo-orbit tracing property, J. Math. Soc. Japan, 37 (1985), 489-514. doi: 10.2969/jmsj/03730489.

[29]

I. Kupka, Contribution à la théorie des champs génériques, Contributions to Differential Equations, 2 (1963), 457-484.

[30]

M. Lee, Usual limit shadowable homoclinic classes of generic diffeomorphisms, Adv. Difference Equ., 2012 (2012), 8 pp. doi: 10.1186/1687-1847-2012-91.

[31]

K. Lee and X. Wen, Shadowable chain transitive sets of C1-generic diffeomorphisms, Bull. Korean Math. Soc., 49 (2012), 263-270. doi: 10.4134/BKMS.2012.49.2.263.

[32]

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

[33]

J. Lewowicz, Lyapunov functions and topological stability, J. Differential Equations, 38 (1980), 192-209. doi: 10.1016/0022-0396(80)90004-2.

[34]

S. Y. Pilyugin, Shadowing in Dynamical Systems, Lecture Notes in Mathematics, 1706, Springer-Verlag, Berlin, 1999.

[35]

C. Pugh and C. Robinson, The $C^1$ closing lemma, including Hamiltonians, Ergodic Theory Dynam. Systems, 3 (1983), 261-313. doi: 10.1017/S0143385700001978.

[36]

C. Robinson, Generic properties of conservative systems, Amer. J. Math., 92 (1970), 562-603. doi: 10.2307/2373361.

[37]

M. Shub, Global Stability of Dynamical Systems, Springer-Verlag, New York, 1987.

[38]

S. Smale, Stable manifolds for differential equations and diffeomorphisms, Ann. Sc. Norm. Super. Pisa (3), 17 (1963), 97-116.

[39]

L. Wen and Z. Xia, $C^1$-connecting lemmas, Trans. Amer, Math. Soc., 352 (2000), 5213-5230. doi: 10.1090/S0002-9947-00-02553-8.

[40]

L. Wen, On the preperiodic set, Discrete Contin. Dynam. Systems, 6 (2000), 237-241. doi: 10.3934/dcds.2000.6.237.

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