Abstract
It seems that in Mañé’s proof of the C 1Ω-stability conjecture containing in the famous paper which published in I. H. E. S. (1988), there exists a deficiency in the main lemma which says that for f∈ \({\fancyscript F}\) 1(M) there exists a dominated splitting \( TM_{{\left| {\overline{P} _{i} {\left( f \right)}} \right.}} = \widetilde{E}^{s}_{i} \oplus \widetilde{F}^{u}_{i} \) (0 < i < dim M) such that if \( \widetilde{E}^{s}_{i} \) is contracting, then \( \widetilde{F}^{u}_{i} \) is expanding. In the first part of the paper, we give a proof to fill up this deficiency. In the last part of the paper, we, under a weak assumption, prove a result that seems to be useful in the study of dynamics in some other stability context.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Mañé, R.: A proof of the C 1 Ω–stability conjecture. Publ. Math. I.H.E.S., 66, 160–210 (1988)
Liao, S. T.: A basic property of a certain class of differential systems. Acta Mathematica Sinica, Chinese Series, 22, 316–343 (1979)
Aoki, N.: The set of Axiom A diffeomorphisms with no cycle. Bol. Soc. Brasil. Mat. (N. S.), 23, 21–65 (1992)
Gan, S.: The star system χ* and a proof of the C 1 Ω–stability conjecture for flows. J. Diff. Eqs., 163, 1–17 (2000)
Hayashi, S.: Connecting invariant manifolds and the solution of the C 1 stability and Ω–stability conjectures for flows. Annals of Math., 145, 81–137 (1997)
Hayashi, S.: Diffeomorphisms in F 1(M) satisfy Axiom A. Ergodic Theory Dynam. Systems, 12, 233–253 (1992)
Palis, J.: On the C 1 Ω–Stability Conjecture. Publ. Math. I.H.E.S., 66, 211–215 (1988)
Wen, L.: On the C 1 stabililty conjecture for flows. J. Diff. Eqs., 129, 334–357 (1996)
Liao, S. T.: An existence theorem for periodic orbits. Acta Sci. Natur. Univ. Pekinensis, 1, 1–20 (1979)
Liao, S. T.: Certain uniformity properties of differential systems and a generalization of an existence theorem for periodic orbits. Acta Sci. Natur. Univ. Pekinensis, 2, 1–19 (1985)
Gan, S.: A generalized shadowing lemma. Discrete and Continuous Dynamical Systems, 8(9), 627–632 (2002)
Pliss, V. A.: On a conjecture due to Smale. Diff. Uravnenija., 8, 268–282 (1972)
Author information
Authors and Affiliations
Corresponding author
Additional information
The first author is supported by NSFC (No. 10171004) and Ministry of Education Special Funds for Excellent Doctoral Thesis. The second author is supported by Ministry of Education Special Funds for Excellent Doctoral Thesis
Rights and permissions
About this article
Cite this article
Zhang, Y., Gan, S.B. On Mañé’s Proof of the C 1 Stability Conjecture. Acta Math Sinica 21, 533–540 (2005). https://doi.org/10.1007/s10114-005-0522-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-005-0522-0