In this paper, we prove that for any f∈Diff1 (M) and Λ ⊂ M be a nontrivial topologically transitive proper subset with a splitting EsF (without hypothesis of domination), where Es is uniformly contracting, there is no arc of the stable manifold whole contained in Λ.

1.
S.
Gan
,
A Necessary and sufficient condition for the existence of dominated splitting with a given index
,
Trends in Mathematics
7
(
2004
),
143
168
.
2.
M. W.
Hirsch
,
C. C.
Pugh
,
Stable manifolds for hyperbolic sets
,
Bull. Amer. Math. Soc.
75
No.
1
(
1969
),
149
152
.
3.
M. W.
Hirsch
,
C. C.
Pugh
,
M.
Shub
, Invariant manifolds,
Lecture Notes in Math.
No
583
,
Springer-Verlag
,
New York
, (
1976
).
4.
C.
Liang
,
G.
Liu
,
Dominated splitting versus Small angles
,
Acta Math. Sinica
,
24
, No.
7
(
2008
),
1163
1174
.
5.
C.
Liang
,
G.
Liu
,
Conditions for dominated splitting
,
Acta Math. Sinica
,
25
, No.
8
(
2009
),
1389
1398
.
6.
C.
Liang
,
G.
Liu
,
W.
Sun
,
Equivalent conditions for dominated splitting in volume-preserving diffeomorphisms
,
Acta Math. Sinica
23
, No.
9
(
2007
),
1563
1576
.
7.
S. T.
Liao
,
A basic property of a certain class of differential systems
,
Acta Math. Sinica
22
(
1979
),
316
343
.
8.
S. T.
Liao
,
An existence theorem for periodic orbits
,
Acta Sci. Natur. Univ. Pekinensis
1
(
1979
),
1
20
.
9.
S. T.
Liao
,
Obstruction sets II
,
Acta Sci. Natur. Univ. Pekinensis
2
(
1981
),
1
36
.
10.
R.
Mañé
,
Contributions to the stability conjecture
,
Topology
17
(
1978
),
397
405
.
11.
R.
Mañé
,
An ergodic closing lemma
,
Ann. Math.
116
(
1982
),
503
540
.
12.
R.
Mañé
,
A proof of the C1 Stability Conjecture
,
Publ. Math. IHES
66
(
1988
),
161
210
.
13.
R.
Mañé
,
Expansive homeomorphisms and topological dimension
,
Trans. Ame. Math. Soc.
252
(
1979
),
313
319
.
14.
V. A.
Pliss
,
On the conjecture due to Smale
,
Diff. Uravnenija
8
(
1972
),
268
282
.
15.
V. A.
Pliss
,
Analysis of the necessity of the conditions of Smale and Robbin for structural stability of periodic systems of differential equations
,
Diff. Uravnenija
8
(
1972
),
553
555
.
16.
S.
Smale
,
Differentiable dynamical systems
,
Bull. Amer. Math. Soc.
73
(
1967
),
747
817
.
17.
X.T.
Tian
,
Minimal diffeomorphisms cannot satisfy generalized dominated splitting
, (
2011
) arXiv:1106.3737v1.
18.
L.
Wen
,
Homoclinic tangengcies and dominated splittings
,
Nonlinearity
15
(
2002
),
1445
1469
.
19.
P. F.
Zhang
,
Diffeomorphisms with global dominated splittings cannot be minimal
,
Proceedings of A.M.S.
140
, No.
2
(
2012
),
589
593
.
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