Browse > Article
http://dx.doi.org/10.4134/BKMS.2014.51.1.067

SHADOWING, EXPANSIVENESS AND STABILITY OF DIVERGENCE-FREE VECTOR FIELDS  

Ferreira, Celia (Departamento de Matematica Faculdade de Ciencias Universidade do Porto)
Publication Information
Bulletin of the Korean Mathematical Society / v.51, no.1, 2014 , pp. 67-76 More about this Journal
Abstract
Let X be a divergence-free vector field defined on a closed, connected Riemannian manifold. In this paper, we show the equivalence between the following conditions: ${\bullet}$ X is a divergence-free vector field satisfying the shadowing property. ${\bullet}$ X is a divergence-free vector field satisfying the Lipschitz shadowing property. ${\bullet}$ X is an expansive divergence-free vector field. ${\bullet}$ X has no singularities and is Anosov.
Keywords
shadowing; Lipschitz shadowing; expansiveness; Anosov vector fields;
Citations & Related Records
연도 인용수 순위
  • Reference
1 D. V. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Proc. Steklov Math. Inst. 90 (1967), 1-235.
2 M. Bessa, C. Ferreira, and J. Rocha, On the stability of the set of hyperbolic closed orbits of a Hamiltonian, Math. Proc. Cambridge Philos. Soc. 149 (2010), no. 2, 373-383.   DOI   ScienceOn
3 M. Bessa and J. Rocha, On $C^1$-robust transitivity of volume-preserving flows, J. Differential Equations 245 (2008), no. 11, 3127-3143.   DOI   ScienceOn
4 M. Bessa and J. Rocha, Three-dimensional conservative star flows are Anosov, Discrete Contin. Dyn. Syst. 26 (2010), no. 3, 839-846.
5 R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, Vol. 470. Springer-Verlag, Berlin-New York, 1975.
6 R. Bowen and P. Walters, Expansive one-parameter flows, J. Differential Equations 12 (1972), 180-193.   DOI
7 C. 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.
8 C. Ferreira, Stability properties of divergence-free vector fields, Dyn. Syst. 27 (2012), no. 2, 223-238.   DOI
9 K. Lee and K. Sakai, Structural stability of vector fields with shadowing, J. Differential Equations 232 (2007), no. 1, 303-313.   DOI   ScienceOn
10 R. Mane, Quasi-Anosov diffeomorphisms and hyperbolic manifolds, Trans. Amer. Math. Soc. 229 (1977), 351-370.   DOI
11 K. Moriyasu, K. Sakai, and W. Sun, $C^1$-stably expansive flows, J. Differential Equations 213 (2005), no. 2, 352-367.   DOI   ScienceOn
12 J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965), 286-294.   DOI   ScienceOn
13 K. Palmer, S. Piyugin, and S. Tikhomirov, Lipschitz shadowing and structural stability of flows, J. Differential Equations 252 (2012), no. 2, 1723-1747.   DOI   ScienceOn
14 S. Pilyugin, Shadowing in Dynamical Systems, Lecture Notes Math., Springer, Berlin, 1999.
15 S. Pilyugin and S. Tikhomirov, Lipschitz shadowing implies structural stability, Nonlinearity 23 (2010), no. 10, 2509-2515.   DOI   ScienceOn
16 C. Zuppa, Regularisation $C^{\infty}$ des champs vectoriels qui preservent l'element de volume, Bol. Soc. Bras. Mat. 10 (1979), no. 2, 51-56.   DOI
17 S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747-817.   DOI
18 S. Tikhomirov, Interiors of sets of vector fields with shadowing corresponding to certain classes of reparameterizations, Vestnik St. Petersburg Univ. Math. 41 (2008), no. 4, 360-366.   DOI
19 T. Vivier, Projective hyperbolicity and fixed points, Ergodic Theory Dynam. Systems 26 (2006), no. 3, 923-936.   DOI   ScienceOn