Journal of the Mathematical Society of Japan

Dominated splitting of differentiable dynamics with $\mathrm{C}^1$-topological weak-star property

Xiongping DAI

Full-text: Open access

Abstract

We study weak hyperbolicity of a differentiable dynamical system which is robustly free of non-hyperbolic periodic orbits of Markus type. Let S be a $\mathrm{C}^1$-class vector field on a closed manifold $M^n$, which is free of any singularities. It is of $\mathrm{C}^1$-weak-star in case there exists a $\mathrm{C}^1$-neighborhood $\mathscr{U}$ of S such that for any X$\in\mathscr{U}$, if $P$ is a common periodic orbit of X and S with S$_{\upharpoonright P}=$X$_{\upharpoonright P}$, then $P$ is hyperbolic with respect to X. We show, in the framework of Liao theory, that S possesses the $\mathrm{C}^1$-weak-star property if and only if it has a natural and nonuniformly hyperbolic dominated splitting on the set of periodic points $\mathrm{Per}$(S), for the case $n=3$.

Article information

Source
J. Math. Soc. Japan, Volume 64, Number 4 (2012), 1249-1295.

Dates
First available in Project Euclid: 29 October 2012

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1351516775

Digital Object Identifier
doi:10.2969/jmsj/06441249

Mathematical Reviews number (MathSciNet)
MR2998923

Zentralblatt MATH identifier
1281.37009

Subjects
Primary: 37C10: Vector fields, flows, ordinary differential equations
Secondary: 37D30: Partially hyperbolic systems and dominated splittings 37D05: Hyperbolic orbits and sets 37C27: Periodic orbits of vector fields and flows 34D30: Structural stability and analogous concepts [See also 37C20]

Keywords
dominated splitting weak-star property Liao theory

Citation

DAI, Xiongping. Dominated splitting of differentiable dynamics with $\mathrm{C}^1$-topological weak-star property. J. Math. Soc. Japan 64 (2012), no. 4, 1249--1295. doi:10.2969/jmsj/06441249. https://projecteuclid.org/euclid.jmsj/1351516775


Export citation

References

  • F. Abdenur, C. Bonatti, S. Crovisier and L. J. Díaz, Generic diffeomorphisms on compact surfaces, Fund. Math., 187 (2005), 127–159.
  • N. Aoki, The set of Axiom A diffeomorphisms with no cycles, Bol. Soc. Brasil. Mat. (N.S.), 23 (1992), 21–65.
  • C. Bonatti and S. Crovisier, Récurrence et généricité, Invent. Math., 158 (2004), 33–104.
  • C. Bonatti, L. J. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity, Encyclopaedia Math. Sci., 102, Mathematical Physics, III, Springer-Verlag, Berlin, 2005.
  • C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conf. Ser. in Math., 38, Amer. Math. Soc., Providence, RI, 1978.
  • S. Crovisier, Periodic orbits and chain-transitive sets of $C^1$-diffeomorphisms, Publ. Math. Inst. Hautes Études Sci., 104 (2006), 87–141.
  • X. Dai, On the continuity of Liao qualitative functions of differential systems and applications, Commun. Contemp. Math., 7 (2005), 747–768.
  • X. Dai, Integral expressions of Lyapunov exponents for autonomous ordinary differential systems, Sci. China Ser. A, 52 (2009), 195–216.
  • X. Dai, On the approximation of Lyapunov exponents and a question suggested by Anatole Katok, Nonlinearity, 23 (2010), 513–528.
  • X. Dai, Hyperbolicity of $C^1$-star invariant sets for $C^1$-class dynamical systems, Sci. China Math., 54 (2011), 269–280.
  • X. Dai, On sifting-shadowing combinations and hyperbolicity of nonsingular weak-star flows, preprint, 2009.
  • J. Franks, Necessary conditions for stability of diffeomorphisms, Trans. Amer. Math. Soc., 158 (1971), 301–308.
  • S. Gan, A generalized shadowing lemma, Discrete Contin. Dyn. Syst., 8 (2002), 627–632.
  • S. Gan and L. Wen, Nonsingular star flows satisfy Axiom A and the no-cycle condition, Invent. Math., 164 (2006), 279–315.
  • A. Gogolev, Diffeomorphisms Hölder conjugate to Anosov diffeomorphisms, Ergodic Theory Dynam. Systems, 30 (2010), 441–456.
  • S. Hayashi, Diffeomorphisms in $\mathscr{F} \sp 1(M)$ satisfy Axiom A, Ergodic Theory Dynam. Systems, 12 (1992), 233–253.
  • S. Hayashi, Connecting invariant manifolds and the solution of the $C^1$ stability and $\Omega$-stability conjectures for flows, Ann. of Math., 145 (1997), 81–137.
  • Y. Jiang, On a question of Katok in one-dimensional case, Discrete Contin. Dyn. Syst., 24 (2009), 1209–1213.
  • S.-T. Liao, Certain ergodic properties of a differential systems on a compact differentiable manifold, Acta Sci. Natur. Univ. Pekinensis, 9 (1963), 241–265; 309–326.
  • S.-T. Liao, Applications to phase-space structure of ergodic properties of the one-parameter transformation group induced on the tangent bundle by a differential systems on a manifold. I, Acta Sci. Natur. Univ. Pekinensis, 12 (1966), 1–43.
  • S.-T. Liao, Standard systems of differential equations, Acta Math. Sinica, 17 (1974), 100–109; 175–196; 270–295.
  • S.-T. Liao, A basic property of a certain class of differential systems, Acta Math. Sinica, 22 (1979), 316–343.
  • S.-T. Liao, An extended $C^{1}$ closing lemma, Beijing Daxue Xuebao, 3 (1979), 1–41.
  • S.-T. Liao, On the stability conjecture, Chinese Ann. Math., 1 (1980), 9–30.
  • S.-T. Liao, Obstruction sets. II, Beijing Daxue Xuebao, 2 (1981), 1–36.
  • S.-T. Liao, On characteristic exponents: construction of a new Borel set for the multiplicative ergodic theorem for vector fields, Beijing Daxue Xuebao Ziran Kexue Ban, 29 (1993), 277–302.
  • R. Mañé, Expansive diffeomorphisms, Proc. Sympos. Dynam. Systems (University of Warwick, 1974), Lecture Notes in Math., 468, Springer-Verlag, Berlin, 1975, pp.,162–174.
  • R. Mañé, Contributions to the stability conjecture, Topology, 17 (1978), 383–396.
  • R. Mañé, An ergodic closing lemma, Ann. of Math. (2), 116 (1982), 503–540.
  • R. Mañé, A proof of the $C^1$ stability conjecture, Inst. Hautes Études Sci. Publ. Math., 66 (1988), 161–210.
  • L. Markus, The behavior of the solutions of a differential systems near a periodic solution, Ann. of Math. (2), 72 (1960), 245–266.
  • K. Moriyasu, K. Sakai and W. Sun, $C^1$-stably expansive flows, J. Differential Equations, 213 (2005), 352–367.
  • S. E. Newhouse, The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 101–151.
  • M. J. Pacifico, E. R. Pujals and J. L. Vieitez, Robustly expansive homoclinic classes, Ergodic Theory Dynam. Systems, 25 (2005), 271–300.
  • J. Palis and W. de Melo, Geometric Theory of Dynamical Systems: An Introduction, Springer-Verlag, New York, 1982.
  • J. Palis and S. Smale, Structural stability theorems, In: Global Analysis, Berkeley, 1968, Proc. Sympos. Pure Math., 14, Amer. Math. Soc., Providence, RI, 1970, pp.,223–231.
  • M. M. Peixoto, Structural stability on two-dimensional manifolds, Topology, 1 (1962), 101–120.
  • V. A. Pliss, On a conjecture of Smale, Differencial'nye Uravnenija, 8 (1972), 268–282.
  • C. C. Pugh, The closing lemma, Amer. J. Math., 89 (1967), 956–1009.
  • K. Sakai, $C^1$-stably shadowable chain components, Ergodic Theory Dynam. Systems, 28 (2008), 987–1029.
  • M. Sambarino and J. L. Vieitez, On $C^1$-persistently expansive homoclinic classes, Discrete Contin. Dyn. Syst., 14 (2006), 465–481.
  • L. Wen, On the $C\sp 1$ stability conjecture for flows, J. Differential Equations, 129 (1996), 334–357.
  • X. Wen, S. Gan and L. Wen, $C^1$-stably shadowable chain components are hyperbolic, J. Differential Equations, 246 (2009), 340–357.
  • S.-Z. Zhu, Vector fields with stable shadowable property on closed surfaces, Acta Math. Sin. (Engl. Ser.), 26 (2010), 1449–1456.