THE LOCAL STAR CONDITION FOR GENERIC TRANSITIVE DIFFEOMORPHISMS

Title & Authors
THE LOCAL STAR CONDITION FOR GENERIC TRANSITIVE DIFFEOMORPHISMS
Lee, Manseob;

Abstract
Let $\small{f:M{\rightarrow}M}$ be a diffeomorphism on a closed $\small{C^{\infty}\;d({\geq}2)}$ dimensional manifold M. For $\small{C^1}$-generic f, if a diffeomorphism f satisfies the local star condition on a transitive set, then it is hyperbolic.
Keywords
transitive set;star diffeomorphisms;local star diffeomorphisms;hyperbolic;
Language
English
Cited by
References
1.
F. Abdenur, C. Bonatti, and S. Crovisier, Nonuniform hyperbolicity for $C^1$-generic diffeomorphisms, Israel J. Math. 183 (2011), 1-60.

2.
N. Aoki, The set of Axiom A diffeomorphisms with no cycles, Bol. Soc. Brasil. Mat. (N.S.) 23 (1992), no. 1-2, 21-65.

3.
S. Hayashi, Diffeomorphisms in $F^1$(M) satisfy Axiom A, Ergodic Theory Dynam. Systems 12 (1992), no. 2, 233-253.

4.
K. Lee and X. Wen, Shadowable chain transitive sets of $C^1$-generic diffeomorphisms, Bull. Korean Math. Soc. 49 (2012), no. 2, 263-270.

5.
R. Mane, Contributions to the stability conjecture, Topology 17 (1978), no. 4, 383-396.

6.
R. Mane, An ergodic closing lemma, Ann of Math. 116 (1982), no. 3, 503-540.

7.
R. Mane, A proof of the $C^1$ stability conjecture, Inst. Hautes etudes Sci. Publ. Math. No. 66 (1988), 161-210.

8.
C. Robinson, Structural stability of $C^1$-diffeomorphisms, J. Differential Equations 22 (1976), no. 1, 28-73.

9.
W. Sun and X. Tian, Diffeomorphisms with various $C^1$-stalbe properties, Acta Math. Sci. Ser. B Engl. Ed. 32 (2012), no. 2, 552-558.