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A pasting lemma and some applications for conservative systems

Published online by Cambridge University Press:  01 October 2007

ALEXANDER ARBIETO  and
CARLOS MATHEUS
ALEXANDER ARBIETO
Affiliation:
IMPA, Estrada D. Castorina 110, Jardim Botânico, 22460-320 Rio de Janeiro, Brazil (email: alexande@impa.br, cmateus@impa.br)
CARLOS MATHEUS
Affiliation:
IMPA, Estrada D. Castorina 110, Jardim Botânico, 22460-320 Rio de Janeiro, Brazil (email: alexande@impa.br, cmateus@impa.br)
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Abstract

We prove that in a compact manifold of dimension n≥2, C1+α volume-preserving diffeomorphisms that are robustly transitive in the C1-topology have a dominated splitting. Also we prove that for three-dimensional compact manifolds, an isolated robustly transitive invariant set for a divergence-free vector field cannot have a singularity. In particular, we prove that robustly transitive divergence-free vector fields in three-dimensional manifolds are Anosov. For this, we prove a ‘pasting’ lemma, which allows us to make perturbations in conservative systems.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 27 , Issue 5 , October 2007 , pp. 1399 - 1417
Copyright
Copyright © Cambridge University Press 2007

References

[1]Aubin, T.. Nonlinear Analysis on Manifolds. Monge-Ampère Equations. Springer, Berlin, 1982.CrossRefGoogle Scholar
[2]Bourgain, J. and Brezis, H.. On the equation div Y =f and applications to control of phases. J. Amer. Math. Soc. 16(2) (2002), 393–426.CrossRefGoogle Scholar
[3]Burago, D. and Kleiner, B.. Separated nets in Euclidean spaces and Jacobians of biLipschitz maps. Geom. Funct. Anal. 8 (1998), 273–282.Google Scholar
[4]Bochi, J., Fayad, B. and Pujals, E.. A remark on conservative diffeomorphisms. C. R. Math. Acad. Sci. Paris 10 (2006), 763–766.CrossRefGoogle Scholar
[5]Bonatti, C., Diaz, L. and Pujals, E.. A C 1-generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources. Ann. of Math. (2) 158 (2003), 355–418.Google Scholar
[6]Bonatti, C., Diaz, L. and Viana, M.. Dynamics Beyond Uniform Hyperbolicity. Springer, Berlin, 2005.Google Scholar
[7]Biragov, V. and Shilnikov, L.. On the bifurcation of a saddle-focus separatrix loop in a three-dimensional conservative dynamical system. Selecta Math. Sov. 11(4) (1992), 333–340.Google Scholar
[8]Dacorogna, B. and Moser, J.. On a partial differential equation involving the jacobian determinant. Ann. Inst. Poincaré 7 (1990), 1–26.CrossRefGoogle Scholar
[9]Diaz, L., Pujals, E. and Ures, R.. Partial hyperbolic and robust transitivity. Acta Math. 183 (1999), 1–42.Google Scholar
[10]Doering, C.. Persistently transitive vector fields on three-dimensional manifolds. Proc. on Dynamical Systems and Bifurcation Theory (Pitman Research Notes Mathematics Series, 160). Pitman, Boston, MA, 1987, pp. 59–89.Google Scholar
[11]Gilbarg, D. and Trudinger, N.. Elliptic Partial Differential Equations of Second Order. Springer, Berlin, 1998.Google Scholar
[12]Hayashi, S.. Connecting invariant manifolds and the solution of the C 1 stability and Ω-stability conjectures for flows. Ann. of Math. (2) 145 (1997), 81–137.CrossRefGoogle Scholar
[13]Hörmander, L.. Linear Partial Differential Operators. Springer, Berlin, 1976.Google Scholar
[14]Katok, A. and Hasselblat, B.. Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar
[15]Liao, S. T.. Obstruction sets II. Beijin Daxue Xuebao 2 (1981), 1–36.Google Scholar
[16]Ladyzenskaya, O. and Uralsteva, N.. Linear and Quasilinear Elliptic Partial Differential Equations. Academic Press, New York, 1968.Google Scholar
[17]Mañé, R.. An ergodic closing lemma. Ann. of Math. (2) 116 (1982), 503–540.CrossRefGoogle Scholar
[18]Morales, C. A., Pacífico, M. J. and Pujals, E.. Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers. Ann. of Math. (2) 160 (2004), 375–432.CrossRefGoogle Scholar
[19]Moser, J.. On the volume element of a manifold. Trans. Amer. Math. Soc. 120 (1965), 286–294.CrossRefGoogle Scholar
[20]Ornstein, D.. A non-inequality for differential operators in the L 1 norm. Arch. Ration Mech. Anal. 11 (1962), 40–49.CrossRefGoogle Scholar
[21]Pugh, C. and Robinson, C.. The C 1 Closing Lemma, including Hamiltonians. Ergod. Th. & Dynam. Sys. 3(2) (1983), 261–313.CrossRefGoogle Scholar
[22]Robinson, R. C.. Generic properties of conservative systems, I and II. Amer. J. Math. 92 (1970), 562–603 and 897–906Google Scholar
[23]Riviere, T. and Ye, D.. Resolutions of the prescribed volume form equation. Nonlinear Diff. Eq. Appl. 3 (1996), 323–369.CrossRefGoogle Scholar
[24]Tahzibi, A.. Stably ergodic systems which are not partially hyperbolic. Israel J. Math. 142 (2004), 315–344.CrossRefGoogle Scholar
[25]Toyoshiba, H.. Nonsingular vector fields in $\mathcal {G}^1(M^3)$ satisfy Axiom A and no cycle: a new proof of Liao’s theorem. Hokkaido Math. J. 29(1) (2000), 45–58.CrossRefGoogle Scholar
[26]Viana, M.. What’s new on Lorenz strange attractors. Math. Intelligencer 22(3) (2000), 6–19.Google Scholar
[27]Wen, L. and Xia, Z.. C 1 connecting lemmas. Trans. Amer. Math. Soc. 352(11) (2000), 5213–5230.CrossRefGoogle Scholar
[28]Xia, Z.. Homoclinic points in symplectic and volume-preserving diffeomorphisms. Comm. Math. Phys. 177 (1996), 435–449.CrossRefGoogle Scholar
[29]Zehnder, E.. Note on Smoothing Symplectic and Volume-preserving Diffeomorphisms (Lecture Notes in Mathematics, 597). Springer, Berlin, 1977, pp. 828–854.Google Scholar
[30]Zuppa, C.. Regularisation $C^{\infty }$ des champs vectoriels qui préservent l’elément de volume. Bol. Soc. Brasil Mate. 10(2) (1979), 51–56.CrossRefGoogle Scholar