Periodic orbits and chain-transitive sets of C1-diffeomorphisms
Publications Mathématiques de l'IHÉS, Tome 104 (2006), pp. 87-141.

We prove that the chain-transitive sets of C1-generic diffeomorphisms are approximated in the Hausdorff topology by periodic orbits. This implies that the homoclinic classes are dense among the chain-recurrence classes. This result is a consequence of a global connecting lemma, which allows to build by a C1-perturbation an orbit connecting several prescribed points. One deduces a weak shadowing property satisfied by C1-generic diffeomorphisms: any pseudo-orbit is approximated in the Hausdorff topology by a finite segment of a genuine orbit. As a consequence, we obtain a criterion for proving the tolerance stability conjecture in Diff1(M).

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Crovisier, Sylvain. Periodic orbits and chain-transitive sets of $C^1$-diffeomorphisms. Publications Mathématiques de l'IHÉS, Tome 104 (2006), pp. 87-141. doi : 10.1007/s10240-006-0002-4. https://www.numdam.org/articles/10.1007/s10240-006-0002-4/

1. F. Abdenur, C. Bonatti, S. Crovisier, Global dominated splittings and the C1 Newhouse phenomenon, Proc. Amer. Math. Soc., 134 (2006), 2229-2237 | MR | Zbl

2. F. Abdenur, C. Bonatti, S. Crovisier, L. Díaz, Generic diffeomorphisms on compact surfaces, Fundam. Math., 187 (2005), 127-159 | MR | Zbl

3. F. Abdenur and L. Díaz, Pseudo-orbit shadowing in the C1-topology, to appear in Discrete Cont. Dyn. Syst. | MR | Zbl

4. R. Abraham, S. Smale, Nongenericity of Ω-stability, Global analysis I, Proc. Symp. Pure Math. AMS, 14 (1970), 5-8 | Zbl

5. M.-C. Arnaud, Création de connexions en topologie C1 , Ergodic Theory Dyn. Syst., 21 (2001), 339-381 | MR | Zbl

6. M.-C. Arnaud, Approximation des ensembles ω-limites des difféomorphismes par des orbites périodiques, Ann. Sci. Éc. Norm. Supér., IV. Sér., 36 (2003), 173-190 | Numdam | Zbl

7. M.-C. Arnaud, C. Bonatti, S. Crovisier, Dynamiques symplectiques génériques, Ergodic Theory Dyn. Syst., 25 (2005), 1401-1436 | MR | Zbl

8. C. Bonatti, S. Crovisier, Récurrence et généricité, Invent. Math., 158 (2004), 33-104 | MR | Zbl

9. C. Bonatti, L. Díaz, Persistent nonhyperbolic transitive diffeomorphisms, Ann. Math., 143 (1996), 357-396 | MR | Zbl

10. C. Bonatti, L. Díaz, On maximal transitive sets of generic diffeomorphisms, Publ. Math., Inst. Hautes Étud. Sci., 96 (2003), 171-197 | EuDML | Numdam | MR | Zbl

11. C. Bonatti, L. Díaz, E. Pujals, A C1-generic dichotomy for diffeomorphisms: weak forms of hyperbolicicity or infinitely many sinks or sources, Ann. Math., 158 (2003), 355-418 | MR | Zbl

12. C. Bonatti, L. Díaz, G. Turcat, Pas de “shadowing lemma” pour des dynamiques partiellement hyperboliques, C. R. Acad. Sci. Paris, 330 (2000), 587-592 | Zbl

13. R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Springer, Berlin - New York (1975) | MR | Zbl

14. C. Conley, Isolated invariant sets and Morse index, AMS, Providence (1978) | MR | Zbl

15. C. Carballo, C. Morales, M.-J. Pacífico, Homoclinic classes for C1-generic vector fields, Ergodic Theory Dyn. Syst., 23 (2003), 1-13 | MR | Zbl

16. R. Corless, S. Pilyugin, Approximate and real trajectories for generic dynamical systems, J. Math. Anal. Appl., 189 (1995), 409-423 | MR | Zbl

17. W. Melo, Structural stability of diffeomorphisms on two-manifolds, Invent. Math., 21 (1973), 233-246 | EuDML | MR | Zbl

18. G. Gan, L. Wen, Heteroclinic cycles and homoclinic closures for generic diffeomorphisms, J. Dyn. Differ. Equations, 15 (2003), 451-471 | MR | Zbl

19. S. Gonchenko, L. Shilńikov, D. Turaev, Dynamical phenomena in systems with structurally unstable Poincaré homoclinic orbits, Chaos, 6 (1996), 15-31 | MR | Zbl

20. S. Hayashi, Connecting invariant manifolds and the solution of the C1-stability and Ω-stability conjectures for flows, Ann. Math., 145 (1997), 81-137 | Zbl

21. I. Kupka, Contribution à la théorie des champs génériques, Contrib. Differ. Equ., 2 (1963), 457-484 | MR | Zbl

22. R. Mañé, Contributions to the stability conjecture, Topology, 17 (1978), 383-396 | MR | Zbl

23. R. Mañé, An ergodic closing lemma, Ann. Math., 116 (1982), 503-540 | MR | Zbl

24. R. Mañé, A proof of the C1 stability conjecture, Publ. Math., Inst. Hautes Étud. Sci., 66 (1988), 161-210 | EuDML | Numdam | MR | Zbl

25. M. Mazur, Tolerance stability conjecture revisited, Topology Appl., 131 (2003), 33-38 | MR | Zbl

26. S. Newhouse, Hyperbolic limit sets, Trans. Amer. Math. Soc., 167 (1972), 125-150 | MR | Zbl

27. S. Newhouse, Diffeomorphisms with infinitely many sinks, Topology, 13 (1974), 9-18 | MR | Zbl

28. S. Newhouse, The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms, Publ. Math., Inst. Hautes Étud. Sci., 50 (1979), 101-151 | EuDML | Numdam | MR | Zbl

29. K. Odani, Generic homeomorphisms have the pseudo-orbit tracing property, Proc. Amer. Math. Soc., 110 (1990), 281-284 | MR | Zbl

30. J. Palis, On the C1 Ω-stability conjecture, Publ. Math., Inst. Hautes Étud. Sci., 66 (1988), 211-215 | EuDML | Numdam | Zbl

31. J. Palis, S. Smale, Structural stability theorem, Proc. Amer. Math. Soc. Symp. Pure Math., 14 (1970), 223-232 | MR | Zbl

32. J. Palis and F. Takens, Hyperbolicity & sensitive chaotic dynamics at homoclinic bifurcations, Cambridge Studies in Advanced Mathematics, 35, Cambridge University Press, Cambridge, 1993. | MR | Zbl

33. J. Palis, M. Viana, High dimension diffeomorphisms displaying infinitely many periodic attractors, Ann. Math., 140 (1994), 207-250 | MR | Zbl

34. S. Pilyugin, Shadowing in dynamical systems, Lect. Notes Math., vol. 1706, Springer, Berlin, 1999. | MR | Zbl

35. C. Pugh, The closing lemma, Amer. J. Math., 89 (1967), 956-1009 | MR | Zbl

36. C. Pugh, An improved closing lemma and a general density theorem, Amer. J. Math., 89 (1967), 1010-1021 | MR | Zbl

37. C. Pugh, C. Robinson, The C1-closing lemma, including hamiltonians, Ergodic Theory Dyn. Syst., 3 (1983), 261-314 | MR | Zbl

38. J. Robbin, A structural stability theorem, Ann. Math., 94 (1971), 447-493 | MR | Zbl

39. C. Robinson, Generic properties of conservative systems, Amer. J. Math., 92 (1970), 562-603 | MR | Zbl

40. C. Robinson, Cr - structural stability implies Kupka-Smale, Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971), pp. 443-449, Academic Press, New York, 1973. | MR | Zbl

41. C. Robinson, Structural stability of C1-diffeomorphisms, J. Differ. Equ., 22 (1976), 28-73 | MR | Zbl

42. C. Robinson, Stability theorems and hyperbolicity in dynamical systems, Rocky Mt. J. Math., 7 (1977), 425-437 | MR | Zbl

43. N. Romero, Persistence of homoclinic tangencies in higher dimensions, Ergodic Theory Dyn. Syst., 15 (1995), 735-757 | MR | Zbl

44. K. Sakai, Diffeomorphisms with weak shadowing, Fundam. Math., 168 (2001), 57-75 | EuDML | MR | Zbl

45. M. Shub, Stability and genericity for diffeomorphisms, Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971), pp. 493-514, Academic Press, New York, 1973. | MR | Zbl

46. M. Shub, Topologically transitive diffeomorphisms of T4, Lect. Notes Math., vol. 206, pp. 39-40, Springer, Berlin-New York, 1971.

47. C. Simon, A 3-dimensional Abraham-Smale example, Proc. Amer. Math. Soc., 34 (1972), 629-630 | MR | Zbl

48. S. Smale, Stable manifolds for differential equations and diffeomorphisms, Ann. Sc. Norm. Super. Pisa, 17 (1963), 97-116 | EuDML | Numdam | MR | Zbl

49. S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817 | MR | Zbl

50. F. Takens, On Zeeman's tolerance stability conjecture, Lect. Notes Math., vol. 197, 209-219, Springer, Berlin, 1971. | Zbl

51. F. Takens, Tolerance stability, Lect. Notes Math., vol. 468, 293-304, Springer, Berlin, 1975. | MR | Zbl

52. L. Wen, A uniform C1 connecting lemma, Discrete Contin. Dyn. Syst., 8 (2002), 257-265 | MR | Zbl

53. L. Wen, Z. Xia, C1 connecting lemmas, Trans. Amer. Math. Soc., 352 (2000), 5213-5230 | MR | Zbl

54. W. White, On the tolerance stability conjecture, Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971), pp. 663-665, Academic Press, New York, 1973. | MR | Zbl

55. G. Yau, J. Yorke, An open set of maps for which every point is absolutely non-shadowable, Proc. Amer. Math. Soc., 128 (2000), 909-918 | MR | Zbl

  • Li, Ming; Liu, Xingzhong A uniform C1 connecting lemma for singular flows, Journal of Differential Equations, Volume 429 (2025), p. 247 | DOI:10.1016/j.jde.2025.02.043
  • Li, Ming; Liang, Chao; Liu, Xingzhong A closing lemma for non-uniformly hyperbolic singular flows, Communications in Mathematical Physics, Volume 405 (2024) no. 8, p. 35 (Id/No 195) | DOI:10.1007/s00220-024-05045-z | Zbl:1550.37035
  • Shi, Yi; Wang, Xiaodong Cr-chain closing lemma for certain partially hyperbolic diffeomorphisms, Ergodic Theory and Dynamical Systems, Volume 44 (2024) no. 7, pp. 1923-1944 | DOI:10.1017/etds.2023.71 | Zbl:7983536
  • Bonatti, Christian; Shinohara, Katsutoshi A mechanism for ejecting a horseshoe from a partially hyperbolic chain recurrence class, Ergodic Theory and Dynamical Systems, Volume 44 (2024) no. 8, pp. 2080-2142 | DOI:10.1017/etds.2023.76 | Zbl:7932135
  • Wen, Xiao; Yang, Dawei On the partial hyperbolicity of robustly transitive sets with singularities, Journal of Dynamics and Differential Equations, Volume 35 (2023) no. 3, pp. 2035-2068 | DOI:10.1007/s10884-022-10132-7 | Zbl:1527.37034
  • Lee, Manseob Local topological stability for diffeomorphisms, Qualitative Theory of Dynamical Systems, Volume 22 (2023) no. 2, p. 8 (Id/No 51) | DOI:10.1007/s12346-023-00755-6 | Zbl:1518.37022
  • Pacifico, Maria José; Yang, Fan; Yang, Jiagang An entropy dichotomy for singular star flows, Transactions of the American Mathematical Society, Volume 376 (2023) no. 10, pp. 6845-6871 | DOI:10.1090/tran/8989 | Zbl:7735075
  • Lee, K.; Rojas, A. Generalized hyperbolic sets on Banach spaces, Acta Mathematica Hungarica, Volume 168 (2022) no. 1, pp. 63-77 | DOI:10.1007/s10474-022-01266-7 | Zbl:1524.37031
  • Gan, Shaobo; Shi, Yi Cr-closing lemma for partially hyperbolic diffeomorphisms with 1D-center bundle, Advances in Mathematics, Volume 407 (2022), p. 76 (Id/No 108553) | DOI:10.1016/j.aim.2022.108553 | Zbl:1503.37047
  • Le Calvez, Patrice; Tal, Fabio Topological horseshoes for surface homeomorphisms, Duke Mathematical Journal, Volume 171 (2022) no. 12, pp. 2519-2626 | DOI:10.1215/00127094-2022-0057 | Zbl:1518.37054
  • Xiao, Huasong Weakly shadowable vector fields on non-oriented surfaces, Dynamical Systems, Volume 37 (2022) no. 1, pp. 127-135 | DOI:10.1080/14689367.2021.2016631 | Zbl:1501.37024
  • Zheng, Ru Song Bi-Lyapunov stable homoclinic classes for C1 generic flows, Acta Mathematica Sinica. English Series, Volume 37 (2021) no. 7, pp. 1023-1040 | DOI:10.1007/s10114-021-0420-8 | Zbl:1476.37033
  • Obata, Davi Symmetries of vector fields: the diffeomorphism centralizer, Discrete and Continuous Dynamical Systems, Volume 41 (2021) no. 10, pp. 4943-4957 | DOI:10.3934/dcds.2021063 | Zbl:1477.37030
  • Bonatti, Christian; Da Luz, Adriana Star flows and multisingular hyperbolicity, Journal of the European Mathematical Society (JEMS), Volume 23 (2021) no. 8, pp. 2649-2705 | DOI:10.4171/jems/1064 | Zbl:1487.37037
  • Lee, Manseob Eventual shadowing for chain transitive sets of C1 generic dynamical systems, Journal of the Korean Mathematical Society, Volume 58 (2021) no. 5, pp. 1059-1079 | DOI:10.4134/jkms.j190083 | Zbl:1482.37026
  • Leguil, Martin; Obata, Davi; Santiago, Bruno On the centralizer of vector fields: criteria of triviality and genericity results, Mathematische Zeitschrift, Volume 297 (2021) no. 1-2, pp. 283-337 | DOI:10.1007/s00209-020-02511-x | Zbl:1459.37021
  • Lee, Manseob Orbital shadowing property on chain transitive sets for generic diffeomorphisms, Acta Universitatis Sapientiae. Mathematica, Volume 12 (2020) no. 1, pp. 146-154 | DOI:10.2478/ausm-2020-0009 | Zbl:1458.37031
  • Li, Ming Orbital shadowing and stability for vector fields, Journal of Differential Equations, Volume 269 (2020) no. 2, pp. 1360-1382 | DOI:10.1016/j.jde.2020.01.026 | Zbl:1441.37033
  • Da Luz, Adriana Singular robustly chain transitive sets are singular volume partial hyperbolic, Mathematische Zeitschrift, Volume 294 (2020) no. 1-2, pp. 687-712 | DOI:10.1007/s00209-019-02291-z | Zbl:1442.37049
  • Lee, Manseob; Park, Junmi Vector fields with the asymptotic orbital pseudo-orbit tracing property, Qualitative Theory of Dynamical Systems, Volume 19 (2020) no. 2, p. 16 (Id/No 52) | DOI:10.1007/s12346-020-00388-z | Zbl:1441.37032
  • Cheng, Cheng; Crovisier, Sylvain; Gan, Shaobo; Wang, Xiaodong; Yang, Dawei Hyperbolicity versus non-hyperbolic ergodic measures inside homoclinic classes, Ergodic Theory and Dynamical Systems, Volume 39 (2019) no. 7, pp. 1805-1823 | DOI:10.1017/etds.2017.106 | Zbl:1422.37013
  • Lee, Manseob Asymptotic orbital shadowing property for diffeomorphisms, Open Mathematics, Volume 17 (2019), pp. 191-201 | DOI:10.1515/math-2019-0002 | Zbl:1436.37033
  • Jung, Woochul The closure of periodic orbits in the Gromov-Hausdorff space, Topology and its Applications, Volume 264 (2019), pp. 493-497 | DOI:10.1016/j.topol.2019.06.048 | Zbl:1422.54046
  • Wen, Xiao; Wen, Lan A rescaled expansiveness for flows, Transactions of the American Mathematical Society, Volume 371 (2019) no. 5, pp. 3179-3207 | DOI:10.1090/tran/7382 | Zbl:1409.37025
  • Gan, Shaobo; Yang, Dawei Morse-Smale systems and horseshoes for three dimensional singular flows, Annales Scientifiques de l'École Normale Supérieure. Quatrième Série, Volume 51 (2018) no. 1, pp. 39-112 | Zbl:1466.37026
  • Lee, Manseob A type of the shadowing properties for generic view points, Axioms, Volume 7 (2018) no. 1, p. 7 (Id/No 18) | DOI:10.3390/axioms7010018 | Zbl:1432.37049
  • Xiao, Qianying; Zheng, Zuohuan C1 weak Palis conjecture for nonsingular flows, Discrete and Continuous Dynamical Systems, Volume 38 (2018) no. 4, pp. 1809-1832 | DOI:10.3934/dcds.2018074 | Zbl:1394.37035
  • Wang, Xiaodong Hyperbolicity versus weak periodic orbits inside homoclinic classes, Ergodic Theory and Dynamical Systems, Volume 38 (2018) no. 6, pp. 2345-2400 | DOI:10.1017/etds.2016.122 | Zbl:1397.37029
  • Gan, Shaobo; Li, Ming Orbital shadowing for 3-flows, Journal of Differential Equations, Volume 262 (2017) no. 10, pp. 5022-5051 | DOI:10.1016/j.jde.2017.01.015 | Zbl:1373.37078
  • Pilyugin, Sergei Yu.; Sakai, Kazuhiro Chain Transitive Sets and Shadowing, Shadowing and Hyperbolicity, Volume 2193 (2017), p. 181 | DOI:10.1007/978-3-319-65184-2_4
  • Wen, Xiao Structurally stable homoclinic classes, Discrete and Continuous Dynamical Systems, Volume 36 (2016) no. 3, pp. 1693-1707 | DOI:10.3934/dcds.2016.36.1693 | Zbl:1338.37031
  • Gourmelon, Nikolaz A Franks' lemma that preserves invariant manifolds, Ergodic Theory and Dynamical Systems, Volume 36 (2016) no. 4, pp. 1167-1203 | DOI:10.1017/etds.2014.101 | Zbl:1369.37025
  • Bonatti, Christian; Crovisier, Sylvain Center manifolds for partially hyperbolic sets without strong unstable connections, Journal of the Institute of Mathematics of Jussieu, Volume 15 (2016) no. 4, pp. 785-828 | DOI:10.1017/s1474748015000055 | Zbl:1362.37069
  • Wen, Xiao; Wen, Lan Codimension one structurally stable chain classes, Transactions of the American Mathematical Society, Volume 368 (2016) no. 6, pp. 3849-3870 | DOI:10.1090/tran/6440 | Zbl:1359.37060
  • Lee, Manseob; Lee, Seunghee Robust chain transitive vector fields, Asian-European Journal of Mathematics, Volume 8 (2015) no. 2, p. 9 (Id/No 1550026) | DOI:10.1142/s1793557115500266 | Zbl:1353.37043
  • Bessa, Mário; Ribeiro, Raquel Conservative flows with various types of shadowing, Chaos, Solitons and Fractals, Volume 75 (2015), pp. 243-252 | DOI:10.1016/j.chaos.2015.02.022 | Zbl:1352.37056
  • Wang, Xiaodong On the hyperbolicity of C1-generic homoclinic classes, Comptes Rendus. Mathématique. Académie des Sciences, Paris, Volume 353 (2015) no. 11, pp. 1047-1051 | DOI:10.1016/j.crma.2015.07.017 | Zbl:1332.37020
  • Lee, Manseob Robustly chain transitive diffeomorphisms, Journal of Inequalities and Applications, Volume 2015 (2015), p. 6 (Id/No 230) | DOI:10.1186/s13660-015-0752-y | Zbl:1353.37059
  • Lee, Keonhee; Tajbakhsh, Khosro DYNAMICAL SYSTEMS WITH SPECIFICATION, Journal of the Chungcheong Mathematical Society, Volume 28 (2015) no. 1, p. 103 | DOI:10.14403/jcms.2015.28.1.103
  • Wang, Xiaodong On the dominated splitting of Lyapunov stable aperiodic classes, Nonlinearity, Volume 28 (2015) no. 11, pp. 4209-4226 | DOI:10.1088/0951-7715/28/11/4209 | Zbl:1357.37050
  • Bessa, Mário; Lee, Manseob; Vaz, Sandra Stable weakly shadowable volume-preserving systems are volume-hyperbolic, Acta Mathematica Sinica. English Series, Volume 30 (2014) no. 6, pp. 1007-1020 | DOI:10.1007/s10114-014-3093-8 | Zbl:1325.37015
  • Fakhari, Abbas; Lee, Seunghee; Tajbakhsh, Khosro HYPERBOLICITY OF CHAIN TRANSITIVE SETS WITH LIMIT SHADOWING, Bulletin of the Korean Mathematical Society, Volume 51 (2014) no. 5, p. 1259 | DOI:10.4134/bkms.2014.51.5.1259
  • Bessa, Mario; Vaz, Sandra STABLE WEAK SHADOWABLE SYMPLECTOMORPHISMS ARE PARTIALLY HYPERBOLIC, Communications of the Korean Mathematical Society, Volume 29 (2014) no. 2, p. 285 | DOI:10.4134/ckms.2014.29.2.285
  • Ribeiro, Raquel Hyperbolicity and types of shadowing for C1 generic vector fields, Discrete and Continuous Dynamical Systems, Volume 34 (2014) no. 7, pp. 2963-2982 | DOI:10.3934/dcds.2014.34.2963 | Zbl:1341.37012
  • Kościelniak, Piotr; Mazur, Marcin; Oprocha, Piotr; Pilarczyk, Paweł Shadowing is generic – a continuous map case, Discrete and Continuous Dynamical Systems, Volume 34 (2014) no. 9, pp. 3591-3609 | DOI:10.3934/dcds.2014.34.3591 | Zbl:1351.37090
  • Bahabadi, Alireza Zamani Average shadowing property with non uniformly hyperbolicity on periodic points, Journal of Dynamical Systems and Geometric Theories, Volume 12 (2014) no. 1, pp. 11-17 | DOI:10.1080/1726037x.2014.917826 | Zbl:1365.54030
  • Shi, Yi; Gan, Shaobo; Wen, Lan On the singular-hyperbolicity of star flows, Journal of Modern Dynamics, Volume 8 (2014) no. 2, pp. 191-219 | DOI:10.3934/jmd.2014.8.191 | Zbl:1351.37141
  • Lee, Manseob Orbital shadowing property for generic divergence-free vector fields, Chaos, Solitons and Fractals, Volume 54 (2013), pp. 71-75 | DOI:10.1016/j.chaos.2013.05.013 | Zbl:1341.37011
  • Sakai, Kazuhiro Shadowable chain transitive sets, Journal of Difference Equations and Applications, Volume 19 (2013) no. 10, pp. 1601-1618 | DOI:10.1080/10236198.2013.767897 | Zbl:1364.37060
  • Mazur, Marcin; Oprocha, Piotr s-limit shadowing is C0-dense, Journal of Mathematical Analysis and Applications, Volume 408 (2013) no. 2, pp. 465-475 | DOI:10.1016/j.jmaa.2013.06.004 | Zbl:1317.37032
  • Fakhari, Abbas Uniform hyperbolicity along periodic orbits, Proceedings of the American Mathematical Society, Volume 141 (2013) no. 9, pp. 3107-3118 | DOI:10.1090/s0002-9939-2013-11553-4 | Zbl:1297.37007
  • Lee, Manseob Stably asymptotic average shadowing property and dominated splitting, Advances in Difference Equations, Volume 2012 (2012), p. 6 (Id/No 25) | DOI:10.1186/1687-1847-2012-25 | Zbl:1285.37008
  • Lee, Manseob Usual limit shadowable homoclinic classes of generic diffeomorphisms, Advances in Difference Equations, Volume 2012 (2012), p. 8 (Id/No 91) | DOI:10.1186/1687-1847-2012-91 | Zbl:1285.37007
  • Lee, Keon-Hee; Wen, Xiao SHADOWABLE CHAIN TRANSITIVE SETS OF C1-GENERIC DIFFEOMORPHISMS, Bulletin of the Korean Mathematical Society, Volume 49 (2012) no. 2, p. 263 | DOI:10.4134/bkms.2012.49.2.263
  • Lee, Manseob Robustly chain transitive sets with orbital shadowing diffeomorphisms, Dynamical Systems, Volume 27 (2012) no. 4, p. 507 | DOI:10.1080/14689367.2012.725032
  • Abdenur, Flavio; Crovisier, Sylvain Transitivity and Topological Mixing for C1 Diffeomorphisms, Essays in Mathematics and its Applications (2012), p. 1 | DOI:10.1007/978-3-642-28821-0_1
  • Lee, Manseob; Park, Junmi ASYMPTOTIC AVERAGE SHADOWING PROPERTY ON A CLOSED SET, Journal of the Chungcheong Mathematical Society, Volume 25 (2012) no. 1, p. 27 | DOI:10.14403/jcms.2012.25.1.027
  • Lee, Manseob; Kang, Bowon; Oh, Jumi GENERIC DIFFEOMORPHISM WITH SHADOWING PROPERTY ON TRANSITIVE SETS, Journal of the Chungcheong Mathematical Society, Volume 25 (2012) no. 4, p. 643 | DOI:10.14403/jcms.2012.25.4.643
  • Dai, Xiongping Dominated splitting of differentiable dynamics with C1-topological weak-star property, Journal of the Mathematical Society of Japan, Volume 64 (2012) no. 4, pp. 1249-1295 | DOI:10.2969/jmsj/06441249 | Zbl:1281.37009
  • Buzzi, Jérôme Chaos and Ergodic Theory, Mathematics of Complexity and Dynamical Systems (2012), p. 63 | DOI:10.1007/978-1-4614-1806-1_6
  • Crovisier, Sylvain Partial hyperbolicity far from homoclinic bifurcations, Advances in Mathematics, Volume 226 (2011) no. 1, pp. 673-726 | DOI:10.1016/j.aim.2010.07.013 | Zbl:1215.37018
  • Pilyugin, S. Yu. Theory of pseudo-orbit shadowing in dynamical systems, Differential Equations, Volume 47 (2011) no. 13, pp. 1929-1938 | DOI:10.1134/s0012266111130040 | Zbl:1252.37019
  • Arbieto, A.; Ribeiro, R. Flows with the (asymptotic) average shadowing property on three-dimensional closed manifolds, Dynamical Systems, Volume 26 (2011) no. 4, pp. 425-432 | DOI:10.1080/14689367.2011.604025 | Zbl:1244.37018
  • Bonatti, C. Towards a global view of dynamical systems, for the C1-topology, Ergodic Theory and Dynamical Systems, Volume 31 (2011) no. 4, pp. 959-993 | DOI:10.1017/s0143385710000891 | Zbl:1256.37006
  • Yang, Jiagang Newhouse phenomenon and homoclinic classes, Ergodic Theory and Dynamical Systems, Volume 31 (2011) no. 5, pp. 1537-1562 | DOI:10.1017/s0143385710000465 | Zbl:1276.37032
  • Yang, Dawei Stably weakly shadowing transitive sets and dominated splittings, Proceedings of the American Mathematical Society, Volume 139 (2011) no. 8, pp. 2747-2751 | DOI:10.1090/s0002-9939-2011-10699-3 | Zbl:1234.37025
  • Osipov, A. V. Nondensity of the orbital shadowing property in C1-topology, St. Petersburg Mathematical Journal, Volume 22 (2011) no. 2, pp. 267-292 | DOI:10.1090/s1061-0022-2011-01140-9 | Zbl:1221.37042
  • Crovisier, Sylvain Birth of homoclinic intersections: a model for the central dynamics of partially hyperbolic systems, Annals of Mathematics. Second Series, Volume 172 (2010) no. 3, pp. 1641-1677 | DOI:10.4007/annals.2010.172.1641 | Zbl:1233.37009
  • Buzzi, Jérôme Chaos and Ergodic Theory, Encyclopedia of Complexity and Systems Science (2009), p. 953 | DOI:10.1007/978-0-387-30440-3_64
  • Buzzi, Jérôme Chaos and Ergodic Theory, Ergodic Theory (2009), p. 633 | DOI:10.1007/978-1-0716-2388-6_64
  • Díaz, Lorenzo J.; Gorodetski, Anton Non-hyperbolic ergodic measures for non-hyperbolic homoclinic classes, Ergodic Theory and Dynamical Systems, Volume 29 (2009) no. 5, pp. 1479-1513 | DOI:10.1017/s0143385708000849 | Zbl:1184.37010
  • Bonatti, Christian; Gan, Shaobo; Wen, Lan On the existence of non-trivial homoclinic classes, Ergodic Theory and Dynamical Systems, Volume 27 (2007) no. 5, pp. 1473-1508 | DOI:10.1017/s0143385707000090 | Zbl:1128.37021

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