충청수학회지 (Journal of the Chungcheong Mathematical Society)

  • 제30권2호
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  • Pages.177-181
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  • 2017
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  • 1226-3524(pISSN)
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  • 2383-6245(eISSN)
충청수학회 (The Chungcheong Mathematical Society)
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CHAIN TRANSITIVE SETS AND DOMINATED SPLITTING FOR GENERIC DIFFEOMORPHISMS

  • Lee, Manseob (Department of Mathematics, Mokwon University)
  • 투고 : 2016.11.04
  • 심사 : 2017.02.01
  • 발행 : 2017.05.15
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초록

Let $f:M{\rightarrow}M$ be a diffeomorphism of a compact smooth manifold M. In this paper, we show that $C^1$ generically, if a chain transitive set ${\Lambda}$ is locally maximal then it admits a dominated splitting. Moreover, $C^1$ generically if a chain transitive set ${\Lambda}$ of f is locally maximal then it has zero entropy.

키워드

참고문헌

  1. F. Abdenur, C. Bonatti, and S. Croviser, Global dominated splitting and the $C^1$ Newhouse phenomenon, Proc. Amer. Math. Soc. 134 (2006), 2229-2237. https://doi.org/10.1090/S0002-9939-06-08445-0
  2. F. Abdenur, C. Bonatti, and S. Croviser, Nonuniform hyperbolicity for generic diffeomorphisms, Israel J. Math. 183 (2011), 1-60. https://doi.org/10.1007/s11856-011-0041-5
  3. C. Bonatti, L. J. Diaz, and E. R. Pujals, A $C^1$-generic dichotomy for diffeomor-phisms: weak forms of hyperbolicity or infinitely many sinks or sources, Ann. of Math. 158 (2003), no. 2, 355-418. https://doi.org/10.4007/annals.2003.158.355
  4. S. Crovisier, Periodic orbits and chain transitive sets of $C^1$ diffeomorphisms, Publ. Math. de L'ihes 104 (2006), 87-141. https://doi.org/10.1007/s10240-006-0002-4
  5. K. Lee and M. Lee, Stably inverse shadowable transitive sets and dominated splitting, Proc. Amer. Math. Soc. 140 (2012), no. 1, 217-226. https://doi.org/10.1090/S0002-9939-2011-10882-7
  6. M. Lee, Chain transitive sets with dominated splitting, J. Math. Sci. Adv. Appl. 4 (2010), 201-208.
  7. M. Lee, Dominated splitting with stably expansive, J. Korean Soc. Math. Educ. Ser. B Pure Appl. Math. 18 (2011), no. 4, 285-291.
  8. M. Lee, Stably asymptotic average shadowing property and dominated splitting, Adv. Difference Equ. 2012, 2012:25, 6 pages.
  9. M. Lee, Stably ergodic shadowing and dominated splitting, Far East J. Math. Sci. 62 (2012), no. 2, 275-284.
  10. M. Lee, Limit weak shadowing property and dominated splitting, Far East J. Math. Sci. 66 (2012), no. 2, 171-180.
  11. M. Lee, Continuum-wise expansive and dominated splitting, Int. J. Math. Anal. 7 (2013), no. 23, 1149-1154. https://doi.org/10.12988/ijma.2013.13113
  12. M. Lee, Continuum-wise fully expansive diffeomorphisms and dominated splitting, Int. J. Math. Anal. 8 (2014), no. 7, 329-335. https://doi.org/10.12988/ijma.2014.4113
  13. M. Lee, Robustly chain transitive diffeomorphisms, J. Inequal. Appl. (2015), 2015:230, 6 pages.
  14. M. Lee and X. Wen, Diffeomorphisms with $C^1$-stably average shadowing, Acta Math. Sin. (Engl. Ser.) 29 (2013), no. 1, 85-92. https://doi.org/10.1007/s10114-012-1162-4
  15. E. R. Pujals and M. Sambarino, Homoclinic tangencies and hyperbolicity for surface diffeomorphisms, Ann. of Math. 151 (2000), no. 3, 961-1023. https://doi.org/10.2307/121127

피인용 문헌

  1. EVENTUAL SHADOWING FOR CHAIN TRANSITIVE SETS OF C1 GENERIC DYNAMICAL SYSTEMS vol.58, pp.5, 2021, https://doi.org/10.4134/jkms.j190083