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

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

Raquel Ribeiro 1,

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.
Keywords: generic vector fields., chain transitivity, hyperbolicity, limit shadowing, Shadowing.
Mathematics Subject Classification: Primary: 37C20, 37D20; Secondary: 37C10, 37C5.
Citation: Raquel Ribeiro. Hyperbolicity and types of shadowing for $C^1$ generic vector fields. Discrete & Continuous Dynamical Systems - A, 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.   Google Scholar

[2]

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

[3]

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

[4]

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

[5]

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

[6]

S. Bautista and C. Morales, Lectures on sectional Anosov flows,, preprint, (2011).   Google Scholar

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms,, Lecture Notes in Mathematics, (1975).   Google Scholar

[13]

R. Bowen, On Axiom A Diffeomorphisms,, Regional Conference Series in Mathematics, (1978).   Google Scholar

[14]

C. Conley, Isolated Invariant sets and the Morse Index,, CBMS Regional Conference Series in Mathematics, (1978).   Google Scholar

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[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.  doi: 10.3934/dcds.2005.13.239.  Google Scholar

[21]

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

[22]

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

[23]

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

[24]

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

[25]

J. K. Hale, Asymptotic Behaviour of Dissipative Systems,, Math. Surveys and Monographs, 25 (1988).   Google Scholar

[26]

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

[27]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems,, Encyclopedia of Mathematics and its Applications, (1995).   Google Scholar

[28]

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

[29]

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

[30]

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

[31]

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

[32]

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

[33]

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

[34]

S. Y. Pilyugin, Shadowing in Dynamical Systems,, Lecture Notes in Mathematics, (1706).   Google Scholar

[35]

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

[36]

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

[37]

M. Shub, Global Stability of Dynamical Systems,, Springer-Verlag, (1987).   Google Scholar

[38]

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

[39]

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

[40]

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

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.   Google Scholar

[2]

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

[3]

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

[4]

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

[5]

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

[6]

S. Bautista and C. Morales, Lectures on sectional Anosov flows,, preprint, (2011).   Google Scholar

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms,, Lecture Notes in Mathematics, (1975).   Google Scholar

[13]

R. Bowen, On Axiom A Diffeomorphisms,, Regional Conference Series in Mathematics, (1978).   Google Scholar

[14]

C. Conley, Isolated Invariant sets and the Morse Index,, CBMS Regional Conference Series in Mathematics, (1978).   Google Scholar

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[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.  doi: 10.3934/dcds.2005.13.239.  Google Scholar

[21]

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

[22]

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

[23]

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

[24]

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

[25]

J. K. Hale, Asymptotic Behaviour of Dissipative Systems,, Math. Surveys and Monographs, 25 (1988).   Google Scholar

[26]

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

[27]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems,, Encyclopedia of Mathematics and its Applications, (1995).   Google Scholar

[28]

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

[29]

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

[30]

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

[31]

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

[32]

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

[33]

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

[34]

S. Y. Pilyugin, Shadowing in Dynamical Systems,, Lecture Notes in Mathematics, (1706).   Google Scholar

[35]

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

[36]

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

[37]

M. Shub, Global Stability of Dynamical Systems,, Springer-Verlag, (1987).   Google Scholar

[38]

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

[39]

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

[40]

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

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