A dichotomy for higher-dimensional flows

Arbieto, A.; Morales, C.

doi:10.1090/S0002-9939-2013-11536-4

A dichotomy for higher-dimensional flows
HTML articles powered by AMS MathViewer

by A. Arbieto and C. A. Morales PDF

Proc. Amer. Math. Soc. 141 (2013), 2817-2827 Request permission

Abstract:

We analyze the dichotomy between sectional-Axiom A flows and flows with points accumulated by periodic orbits of different indices. Indeed, this is proved for $C^1$ generic flows whose singularities accumulated by periodic orbits have codimension one. Our result improves the work of M. J. Pacifico and the second author.

References

Flavio Abdenur, Generic robustness of spectral decompositions, Ann. Sci. École Norm. Sup. (4) 36 (2003), no. 2, 213–224 (English, with English and French summaries). MR 1980311, DOI 10.1016/S0012-9593(03)00008-9
V. S. Afraĭmovich, V. V. Bykov, and L. P. Shil′nikov, On attracting structurally unstable limit sets of Lorenz attractor type, Trudy Moskov. Mat. Obshch. 44 (1982), 150–212 (Russian). MR 656286
Christian Bonatti and Marcelo Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting, Israel J. Math. 115 (2000), 157–193. MR 1749677, DOI 10.1007/BF02810585
C. M. Carballo, C. A. Morales, and M. J. Pacifico, Homoclinic classes for generic $C^1$ vector fields, Ergodic Theory Dynam. Systems 23 (2003), no. 2, 403–415. MR 1972228, DOI 10.1017/S0143385702001116
C. M. Carballo, C. A. Morales, and M. J. Pacifico, Maximal transitive sets with singularities for generic $C^1$ vector fields, Bol. Soc. Brasil. Mat. (N.S.) 31 (2000), no. 3, 287–303. MR 1817090, DOI 10.1007/BF01241631
Shaobo Gan and Lan Wen, Nonsingular star flows satisfy Axiom A and the no-cycle condition, Invent. Math. 164 (2006), no. 2, 279–315. MR 2218778, DOI 10.1007/s00222-005-0479-3
Shaobo Gan and Lan Wen, Heteroclinic cycles and homoclinic closures for generic diffeomorphisms, J. Dynam. Differential Equations 15 (2003), no. 2-3, 451–471. Special issue dedicated to Victor A. Pliss on the occasion of his 70th birthday. MR 2046726, DOI 10.1023/B:JODY.0000009743.10365.9d
Ming Li, Shaobo Gan, and Lan Wen, Robustly transitive singular sets via approach of an extended linear Poincaré flow, Discrete Contin. Dyn. Syst. 13 (2005), no. 2, 239–269. MR 2152388, DOI 10.3934/dcds.2005.13.239
Shengzhi Zhu, Shaobo Gan, and Lan Wen, Indices of singularities of robustly transitive sets, Discrete Contin. Dyn. Syst. 21 (2008), no. 3, 945–957. MR 2399444, DOI 10.3934/dcds.2008.21.945
J. E. Marsden and M. McCracken, The Hopf bifurcation and its applications, Applied Mathematical Sciences, Vol. 19, Springer-Verlag, New York, 1976. With contributions by P. Chernoff, G. Childs, S. Chow, J. R. Dorroh, J. Guckenheimer, L. Howard, N. Kopell, O. Lanford, J. Mallet-Paret, G. Oster, O. Ruiz, S. Schecter, D. Schmidt and S. Smale. MR 0494309
John Guckenheimer and R. F. Williams, Structural stability of Lorenz attractors, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 59–72. MR 556582
Shuhei Hayashi, Diffeomorphisms in $\scr F^1(M)$ satisfy Axiom A, Ergodic Theory Dynam. Systems 12 (1992), no. 2, 233–253. MR 1176621, DOI 10.1017/S0143385700006726
Shuhei 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. MR 1432037, DOI 10.2307/2951824
Anatole Katok and Boris Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza. MR 1326374, DOI 10.1017/CBO9780511809187
M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. MR 0501173
Shantao Liao, Qualitative theory of differentiable dynamical systems, Science Press Beijing, Beijing; distributed by American Mathematical Society, Providence, RI, 1996. Translated from the Chinese; With a preface by Min-de Cheng. MR 1449640
Ricardo Mañé, A proof of the $C^1$ stability conjecture, Inst. Hautes Études Sci. Publ. Math. 66 (1988), 161–210. MR 932138
Ricardo Mañé, An ergodic closing lemma, Ann. of Math. (2) 116 (1982), no. 3, 503–540. MR 678479, DOI 10.2307/2007021
Ricardo Mañé, Contributions to the stability conjecture, Topology 17 (1978), no. 4, 383–396. MR 516217, DOI 10.1016/0040-9383(78)90005-8
R. Metzger and C. Morales, Sectional-hyperbolic systems, Ergodic Theory Dynam. Systems 28 (2008), no. 5, 1587–1597. MR 2449545, DOI 10.1017/S0143385707000995
C. A. Morales, Strong stable manifolds for sectional-hyperbolic sets, Discrete Contin. Dyn. Syst. 17 (2007), no. 3, 553–560. MR 2276427, DOI 10.3934/dcds.2007.17.553
C. A. Morales and M. J. Pacifico, A dichotomy for three-dimensional vector fields, Ergodic Theory Dynam. Systems 23 (2003), no. 5, 1575–1600. MR 2018613, DOI 10.1017/S0143385702001621
C. A. Morales, M. J. Pacifico, and E. R. Pujals, Singular hyperbolic systems, Proc. Amer. Math. Soc. 127 (1999), no. 11, 3393–3401. MR 1610761, DOI 10.1090/S0002-9939-99-04936-9
V. A. Pliss, On a conjecture of Smale, Differencial′nye Uravnenija 8 (1972), 268–282 (Russian). MR 0299909
Charles C. Pugh, An improved closing lemma and a general density theorem, Amer. J. Math. 89 (1967), 1010–1021. MR 226670, DOI 10.2307/2373414
D. V. Turaev and L. P. Shil′nikov, An example of a wild strange attractor, Mat. Sb. 189 (1998), no. 2, 137–160 (Russian, with Russian summary); English transl., Sb. Math. 189 (1998), no. 1-2, 291–314. MR 1622321, DOI 10.1070/SM1998v189n02ABEH000300
Leonid P. Shilnikov, Andrey L. Shilnikov, Dmitry Turaev, and Leon O. Chua, Methods of qualitative theory in nonlinear dynamics. Part II, World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, vol. 5, World Scientific Publishing Co., Inc., River Edge, NJ, 2001. MR 1884710, DOI 10.1142/9789812798558_{0}001
Lan Wen, On the preperiodic set, Discrete Contin. Dynam. Systems 6 (2000), no. 1, 237–241. MR 1739926, DOI 10.3934/dcds.2000.6.237