대한수학회지 (Journal of the Korean Mathematical Society)

  • 제51권1호
  • /
  • Pages.17-53
  • /
  • 2014
  • /
  • 0304-9914(pISSN)
  • /
  • 2234-3008(eISSN)
대한수학회 (Korean Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

ROBUSTLY SHADOWABLE CHAIN COMPONENTS OF C1 VECTOR FIELDS

  • Lee, Keonhee (Department of Mathematics Chungnam National University) ;
  • Le, Huy Tien (Department of Mathematics Vietnam National University) ;
  • Wen, Xiao (The School of Mathematics and System Science Beihang University)
  • 투고 : 2012.08.13
  • 발행 : 2014.01.01
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

Let ${\gamma}$ be a hyperbolic closed orbit of a $C^1$ vector field X on a compact boundaryless Riemannian manifold M, and let $C_X({\gamma})$ be the chain component of X which contains ${\gamma}$. We say that $C_X({\gamma})$ is $C^1$ robustly shadowable if there is a $C^1$ neighborhood $\mathcal{U}$ of X such that for any $Y{\in}\mathcal{U}$, $C_Y({\gamma}_Y)$ is shadowable for $Y_t$, where ${\gamma}_Y$ denotes the continuation of ${\gamma}$ with respect to Y. In this paper, we prove that any $C^1$ robustly shadowable chain component $C_X({\gamma})$ does not contain a hyperbolic singularity, and it is hyperbolic if $C_X({\gamma})$ has no non-hyperbolic singularity.

키워드

과제정보

연구 과제 주관 기관 : National Research Foundation (NRF)

참고문헌

  1. A. Arroyo and F. R. Hertz, Homoclinic bifurcations and uniform hyperbolicity for three-dimensional flows, Ann. Inst. H. Poincare Anal. Non Lineaire 20 (2003), no. 5, 805-841. https://doi.org/10.1016/S0294-1449(03)00016-7
  2. C. Bonatti, N. Gourmelon, and T. Vivier, Perturbations of the derivative along periodic orbits, Ergodic Theory Dynam. Systems 26 (2006), no. 5, 1307-1337. https://doi.org/10.1017/S0143385706000253
  3. C. I. Doering, Persistently transitive vector fields on three-dimensional manifolds, Dynamical systems and bifurcation theory (Rio de Janeiro, 1985), 59-89, Pitman Res. Notes Math. Ser., 160, Longman Sci. Tech., Harlow, 1987.
  4. S. Gan, K. Sakai, and L. Wen, $C^1$-stably weakly shadowing homoclinic classes admit dominated splitting, Discrete Contin. Dyn. Syst. 27 (2010), no. 1, 205-216. https://doi.org/10.3934/dcds.2010.27.205
  5. S. Gan and L. Wen, Nonsingular star flows satisfy Axiom A and the no-cycle condition, Invent. Math. 164 (2006), no. 2, 279-315. https://doi.org/10.1007/s00222-005-0479-3
  6. N. Gourmelon, A Franks' lemma that preserves invariant manifolds, preprint at http://www.preprint.impa.br/.
  7. S. Hayashi, Connecting invariant manifolds and the solution of the $C^1$-stability and ${\Omega}$-stability conjectures for flows, Ann. of Math. (2) 145 (1997), no. 1, 81-137. https://doi.org/10.2307/2951824
  8. M. Komuro, One-parameter flows with the pseudo orbit tracing property, Monatsh. Math. 98 (1984), no. 3, 219-253. https://doi.org/10.1007/BF01507750
  9. M. Li, S. Gan, and L. Wen, Robustly transitive singular sets via approach of an extended linear Poincare flow, Discrete Contin. Dyn. Syst. 13 (2005), no. 2, 239-269. https://doi.org/10.3934/dcds.2005.13.239
  10. S. Liao, An existence theorem for periodic orbits, Acta. Sci. Nat. Univ. Pekin. 1 (1979), 1-20.
  11. K. Lee and M. Lee, Hyperbolicity of $C^1$-stably expansive homoclinic classes, Discrete Contin. Dyn. Syst. 27 (2010), no. 3, 1133-1145. https://doi.org/10.3934/dcds.2010.27.1133
  12. K. Lee, K. Moriyasu, and K. Sakai, $C^1$-stable shadowing diffeomorphisms, Discrete Contin. Dyn. Syst. 23 (2008), no. 3, 683-697.
  13. K. Lee and K. Sakai, Structural stability of vector fields with shadowing, J. Differential Equations 232 (2007), no. 1, 303-313. https://doi.org/10.1016/j.jde.2006.08.012
  14. R. Mane, An ergodic closing lemma, Ann. of Math. (2) 116 (1982), no. 3, 503-540. https://doi.org/10.2307/2007021
  15. J. Palis and W. de Melo, Geometric Theory of Dynamical Systems: An Introduction, Springer-Verlag, 1982.
  16. C. Pugh and C. Robinson, The $C^r$ closing lemma, including Hamiltonians, Ergodic Theory Dynam. Systems 3 (1983), no. 2, 261-313.
  17. S. Yu. Pilyugin and S. B. Tikhomirov, Vector fields with the oriented shadowing property, J. Differential Equations 248 (2010), no. 6, 1345-1375. https://doi.org/10.1016/j.jde.2009.09.024
  18. C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, Second edition, Studies in Advanced Mathematics, CRC Press, 1999.
  19. K. Sakai, Pseudo-orbit tracing property and strong transversality of diffeomorphisms on closed manifolds, Osaka J. Math. 31 (1994), no. 2, 373-386.
  20. K. Sakai, $C^1$-stably shadowable chain components, Ergodic Theory Dynam. Systems 28 (2008), no. 3, 987-1029.
  21. M. Sambarino and J. L. Vieitez, On $C^1$-persistently expansive homoclinic classes, Discrete Contin. Dyn. Syst. 14 (2006), no. 3, 465-481.
  22. X. Wen, S. Gan, and L. Wen, $C^1$-stably shadowable chain components are hyperbolic, J. Differential Equations 246 (2009), no. 1, 340-357. https://doi.org/10.1016/j.jde.2008.03.032

피인용 문헌

  1. SHADOWABLE CHAIN COMPONENTS AND HYPERBOLICITY vol.52, pp.1, 2015, https://doi.org/10.4134/BKMS.2015.52.1.149
  2. Weak measure expansive flows vol.260, pp.2, 2016, https://doi.org/10.1016/j.jde.2015.09.017
  3. HYPERBOLICITY OF HOMOCLINIC CLASSES OF VECTOR FIELDS vol.98, pp.03, 2015, https://doi.org/10.1017/S1446788714000640
  4. Robustly shadowable chain transitive sets and hyperbolicity vol.33, pp.4, 2018, https://doi.org/10.1080/14689367.2017.1417355