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Discrete admissibility and exponential trichotomy of dynamical systems
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 |
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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