Abstract
We give an affirmative answer to a problem of Liao and Mañé which asks whether, for a nonsingular flow to loose the Ω-stability, it must go through a critical-element-bifurcation. More precisely, a vector field S on a compact boundaryless manifold is called a star system if S has a C 1 neighborhood \(\mathcal{U}\) in the set of C 1 vector fields such that every singularity and every periodic orbit of every \(X\in\mathcal{U}\) is hyperbolic. We prove that any nonsingular star flow satisfies Axiom A and the no cycle condition.
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Aoki, N.: The set of Axiom A diffeomorphisms with no cycles. Bol. Soc. Bras. Mat. 23, 21–65 (1992)
Bowen, R.: Periodic orbits for hyperbolic flows. Am. J. Math. 94, 1–30 (1972)
Diaz, L., Rocha, J.: Partially hyperbolic and transitive dynamics generated by heteroclinic cycles. Ergodic Theory Dyn. Syst. 21, 25–76 (2001)
Ding, H.: Disturbance of the homoclinic trajectory and applications. Acta Sci. Nat. Univ. Pekin., no. 1, 53–63 (1986)
Franks, J.: Necessary conditions for stability of diffeomorphisms. Trans. Am. Math. Soc. 158, 301–308 (1971)
Gan, S.: Another proof for C 1 stability conjecture for flows. Sci. China, Ser. A 41, 1076–1082 (1998)
Gan, S.: The star systems \(\mathcal{X}^*\) and a proof of the C 1 Ω-stability conjecture for flows. J. Differ. Equations 163, 1–17 (2000)
Gan, S.: A generalized shadowing lemma. Discrete Contin. Dyn. Syst. 8, 627–632 (2002)
Guchenheimer, J.: A strange, strange attractor. The Hopf bifurcation theorems and its applications. Applied Mathematical Series, vol. 19, pp. 368–381. Springer 1976
Hayashi, S.: Diffeomorphisms in \(\mathcal{F}^1(M)\) satisfy Axiom A. Ergodic Theory Dyn. Syst. 12, 233–253 (1992)
Hayashi, S.: Connecting invariant manifolds and the solution of the C 1 stability conjecture and Ω-stability conjecture for flows. Ann. Math. 145, 81–137 (1997)
Hu, S.: A proof of C 1 stability conjecture for 3-dimensional flows. Trans. Am. Math. Soc. 342, 753–772 (1994)
Ito, R., Toyoshiba, H.: On vector fields without singularity in \(\mathcal{G}^1(M)\). Gakujutu Kenkyu (Academic Studies), Mathematics 47, 9–12 (1999)
Li, C., Wen, L.: \(\mathcal{X}^*\) plus Axiom A does not imply no-cycle. J. Differ. Equations 119, 395–400 (1995)
Liao, S.T.: A basic property of a certain class of differential systems (in Chinese). Acta Math. Sin. 22, 316–343 (1979)
Liao, S.T.: Obstruction sets (I) (in chinese). Acta Math. Sin. 23, 411–453 (1980)
Liao, S.T.: Obstruction sets (II). Acta Sci. Nat. Univ. Pekin., no. 2, 1–36 (1981)
Liao, S.T.: An existence theorem for periodic orbits. Acta Sci. Nat. Univ. Pekin., no. 1, 1–20 (1979)
Liao, S.T.: The qualitative theory of differential dynamical systems. Science Press 1996
Mañé, R.: An ergodic closing lemma. Ann. Math. 116, 503–540 (1982)
Mañé, R.: A proof of the C 1 stability conjecture. Publ. Math., Inst. Hautes Étud. Sci. 66, 161–210 (1988)
Mañé, R.: Quasi-Anosov diffeomorphisims and hyperbolic manifolds. Trans. Am. Math. Soc. 229, 351–370 (1977)
Morales, C.A., Pacifico, M.J., Pujals, E.R.: Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers. Ann. Math. (2) 160, 375–432 (2004)
Moriyasu, K., Sakai, K.: A note on Mañé’s proof of the stability conjecture. Far East J. Dyn. Syst. 4, 97–106 (2002)
Palis, J.: On the C 1 Ω-stability conjecture. Publ. Math., Inst. Hautes Étud. Sci. 66, 211–215 (1988)
Palis, J., Smale, S.: Structural Stability Theorems, Global Analysis. Proc. Symp. Pure Math., vol. 14, pp. 223–231. Am. Math. Soc. 1970
Peixoto, M.: Structural stability on two-dimensional manifolds. Topology 1, 101–120 (1962)
Pugh, C., Robinson, C.: The closing lemma, including Hamiltonians. Ergodic Theory Dyn. Syst. 3, 261–313 (1983)
Pujals, E., Sambarino, M.: Homoclinic tangencies and hyperbolicity for surface diffeomorphisms. Ann. Math., 151, 961–1023 (2000)
Selgrade, J.: Isolated invariant sets for flows on vector bundles. Trans. Am. Math. Soc., 203, 359–390 (1975)
Shub, M.: Global Stability of Dynamical Systems. Springer 1987
Smale, S.: Differentiable dynamical systems. Bull. Am. Math. Sci. 73, 747–817 (1967)
Sacker, R., Sell, G.: Existence of dichotomies and invariant splittings for linear differential systems. J. Differ. Equations 22, 478–496 (1976)
Toyoshiba, H.: A property of vector fields without singularity in \(\mathcal{G}^1(M)\). Ergodic Theory Dyn. Syst. 21, 303–314 (2001)
Wen, L.: On the C 1 stability conjecture for flows. J. Differ. Equations 129, 334–357 (1996)
Wen, L.: On the preperiodic set. Discrete Contin. Dyn. Syst. 6, 237–241 (2000)
Wen, L.: Generic diffeomorphisms away from homoclinic tangencies and heterodimensional cycles. Bull. Braz. Math. Soc., New Ser. 35, 419–452 (2004)
Wen, L., Xia, Z.: C 1 connecting lemmas. Trans. Am. Math. Soc. 352, 5213–5230 (2000)
Zhang, Y., Gan, S.: On Mañé’s proof of the C 1 stability conjecture. Acta Math. Sin., Engl. Ser. 21, 533–540 (2005)
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Dedicated to Shaotao Liao and Ricardo Mañé
Mathematics Subject Classification (2000)
37D30
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Gan, S., Wen, L. Nonsingular star flows satisfy Axiom A and the no-cycle condition. Invent. math. 164, 279–315 (2006). https://doi.org/10.1007/s00222-005-0479-3
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DOI: https://doi.org/10.1007/s00222-005-0479-3