Orbital shadowing property on chain transitive sets for generic diffeomorphisms

[1] F. Abdenur, C. Bonatti and S. Crovisier, Nonuniform hyperbolicty for C¹-generic diffeomorphisms, Isral J. Math., 183 (2011), 1–60.10.1007/s11856-011-0041-5Search in Google Scholar

[2] F. Abdenur, C. Bonatti and L. J. Díaz, Non-wandering sets with non empty interior, Noinearity, 17 (2004), 175–191.10.1088/0951-7715/17/1/011Search in Google Scholar

[3] F. Abdenur and L. J. Díaz, Pseudo-orbit shadowing in the C¹-topology, Discrete Contin. Dyn. Syst., 17 (2) (2007), 223–245.10.3934/dcds.2007.17.223Search in Google Scholar

[4] J. Ahn, K. Lee and M. Lee, Homoclinic classes with shadowing, J. Inequal. Appl., 2012:97 (2012), 1–6.10.1186/1029-242X-2012-97Search in Google Scholar

[5] C. Bonatti and S. Crovisier, Récurrence et généricité, Invent. Math., 158 (2004), 33–104.10.1007/s00222-004-0368-1Search in Google Scholar

[6] C. Bonatti and L. J. Díaz, Robust heterodimensional cycles and C¹ generic dynamics, J. Inst. Math. Jessieu, 7 (2008), 469–525.Search in Google Scholar

[7] C. Carballo, C. A. Morales and M. J. Pcaifico, Homoclinic classes for C¹ generic vector fields, Ergod. Theorey Dynam. Syst., 23 (2003), 403–415.Search in Google Scholar

[8] S. Crovisier, Periodic orbits and chain transitive sets of C¹-diffeo-morphisms, Publ. Math. Inst. Hautes Etudes. Sci., 104 (2006), 87–141.10.1007/s10240-006-0002-4Search in Google Scholar

[9] K. Lee and M. Lee, Shadowable chain recurrence classes for generic diffeomorphisms, Taiwan J. Math., 20 (2016), 399–409.Search in Google Scholar

[10] K. Lee and M. Lee, Volume preserving diffeomorphisms with orbital shadowing, J. Inequal. Appl., 2013:18, (2013), 5 pp.10.1186/1029-242X-2013-18Search in Google Scholar

[11] K. Lee and X. Wen, Shadowable chain transitive sets of C¹ generic diffeomorphisms, Bull. Korean Math. Soc.49 (2012), 263–270.10.4134/BKMS.2012.49.2.263Search in Google Scholar

[12] M. Lee, Hamiltonian systems with orbital, orbital inverse shadowing, Adv. Difference Equ., 2014:192(2014), 9 pp.10.1186/1687-1847-2014-192Search in Google Scholar

[13] M. Lee, Orbital shadowing property for generic divergence-free vector fields, Chaos Solitons & Fractals, 54 (2013), 71–75.10.1016/j.chaos.2013.05.013Search in Google Scholar

[14] M. Lee, Orbital shadowing for C¹-generic volume-preserving diffeomorphisms, Abstr. Appl. Anal., 2013, Art. ID 693032, 4 pp.10.1155/2013/693032Search in Google Scholar

[15] M. Lee, Divergence-free vector fields with orbital shadowing, Adv. Difference Equ, 2013:132, (2013), 6 pp.10.1186/1687-1847-2013-132Search in Google Scholar

[16] M. Lee, Robustly chain transitive sets with orbital shadowing diffeomorphisms, Dyn. Syst., 27 (2012), 507–514.10.1080/14689367.2012.725032Search in Google Scholar

[17] M. Lee, Chain components with C¹-stably orbital shadowing, Adv. Difference Equ., 2013:67 (2013), 12 pp.10.1186/1687-1847-2013-255Search in Google Scholar

[18] M. Lee, Chain transitive sets with dominated splitting, J. Math. Sci. Adv. Appl., 4 (2010), 201–208.Search in Google Scholar

[19] A. V. Osipov, Nondensity of the orbital shadowing property in C^1-topology, (Russian) Algebra i Analiz 22 (2010), no. 2, 127–163; translation in St. Petersburg Math. J., 22 (2) (2011), 267–292.Search in Google Scholar

[20] S. Y. Pilyugin, A. A. Rodionova and K. Sakai, Orbital and weak shadowing properties, Discrete Contin. Dyn. Syst., 9 (2) (2003), 287–308.10.3934/dcds.2003.9.287Search in Google Scholar

[21] R. Potrie, Generic bi-Lyapunov stable homoclinic classes, Nonlinearity, 23 (2010), 1631–1649.10.1088/0951-7715/23/7/006Search in Google Scholar

[22] C. Robinson, Stability theorem and hyperbolicity in dynamical systems, Rocky Mountain J. Math., 7 (1977), 425–437.10.1216/RMJ-1977-7-3-425Search in Google Scholar

[23] K. Sakai, Sahdowable chain transitive sets, J. Diff. Eqaut. Appl., 19 (2013), 1601–1618.10.1080/10236198.2013.767897Search in Google Scholar

[24] K. Sakai, Pseudo orbit tracing property and strong transversality of diffeomorphisms on closed manifolds, Osaka J. Math., 31 (1994), 373–386.Search in Google Scholar