Int. Journal of Math. Analysis, Vol. 7, 2013, no. 23, 1149-1154 HIKARI Ltd, www.m-hikari.com Continuum-Wise Expansive and Dominated Splitting Manseob Lee Department of Mathematics Mokwon University Daejeon, 302-729, Korea lmsds@mokwon.ac.kr Copyright c 2013 Manseob Lee. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, we show that if a nontrivial transitive set is C 1 -stably continuum-wise expansive, then it admits a dominated splitting. Mathematics Subject Classification: 37D30 Keywords: transitive expansive, continuum-wise expansive, dominated splitting, 1 Introduction Let Diff(M) be the space of diffeomorphisms of M endowed with the C 1 - topology, and let d denote the distance on M induced from a Riemannian metric on the tangent bundle T M. For any closed f-invariant set Λ, We say that Λ is expansive for f if there is α>0 such that for any pair of distinct points x, y Λ, d(f n (x),f n (y)) >αfor some n Z. The number α>0 is called an expansive constant for f Λ. We introduce the notion of continuum-wise expansivity was sstudied by Kato in [6]. By a subcontinuum in M, we mean a compact connected nondegenerate subset A of M. We say that Λ is continuumwise expansive if there is a constant e>0 such that for any nondegenerate subcontinuum A of Λ, there is an integer n = n(a) such that diamf n (A) e, where diama = sup{d(x, y) :x, y A} for any subset S of M. Such a constant e is called a continuum-wise expansive constant for f Λ. Clearly every
1150 Manseob Lee expansive homeomorphism is continuum-wise expansive, but its converse is not true. Kato gave an example to show that the continuum-wise expansivity does not imply the expansivity from homeomorphisms viewpoint (see [7, Example 3.5]). From diffeomorphisms viewpoint, the class of continuum-wise expansive diffeomorphisms is strictly larger than that of expansive diffeomorphisms. In fact, it is well-known that S 2 does not admit an expansive diffeomorphism, but it admits a continuum-wise expansive diffeomorphisms as we can see in the following example which is introduced by [4]. Very recently, in [4], the authors showed that if the homoclinic class is C 1 -stably continuum-wise expansive, then it is hyperbolic. Note that the homoclinic class containing hyperbolic periodic point is a transitive set. But still we don t know that a nontrivial transitive set is C 1 -stably continuum-wise expansive then is it hyperbolic? In this paper, we study continuum-wise expansive and dominated splitting. In differentiable dynamical system, dominated splitting is a nature generalization of hyperbolicity (see [1, 2, 3, 5, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21]). Let Λ be a closed f-invariant set. We say that Λ is locally maximal if there is a neighborhood U of Λ such that Λ = n Z f n (U). Definition 1.1 Let Λ be a closed invariant set of f Diff(M). We say that Λ is C 1 -stably continuum-wise expansive if there are a compact neighborhood U of f and a C 1 -neighborhood U(f) of f such that Λ=Λ f (U) = n Z f n (U)(locally maximal), and for any g U(f), Λ g (U) is continuum-wise expansive, where Λ g (U) = n Z gn (U) is the continuation of Λ. We say that Λ is transitive set if there is a point x Λ such that ω f (x) =Λ, where ω f (x) is the ω-limit set of x. Let Λ M be an f-invariant closed set. We say that Λ admits a dominated splitting if the tangent bundle T Λ M has a continuous Df-invariant splitting E F and there exist constants C > 0 and 0 <λ<1 such that D x f n E(x) D x f n F (f n (x)) Cλ n for all x Λ and n 0. We say that Λ is nontrivial if Λ is not one orbit. In this paper, we have Theorem 1.2 Let Λ be a nontrivial transitive set of f Diff(M). If Λ is C 1 -stably continuum-wise expansive, then it admits a dominated splitting. 2 Proof of Theorem 1.2 Now, we introduce the notation of pre-sink (resp. pre-source). A periodic point p of f is called a pre-sink (resp. pre-source) if Df π(p) (p) has a multiplicity one eigenvalue with modulus 1 and the other eigenvalues has norm strictly less than
Continuum wise expansive and dominated splitting 1151 1 (resp. bigger than 1). Theorem 1.2 is all base on the following proposition. Thus it is enough to show the following proposition. Proposition 2.1 Let Λ be a closed invariant set of f, ifλ is C 1 -stably continuumwise expansive for f, and there exist a sequence g n goes to f and periodic orbits P n of g n which converges to Λ in Hausdorff limits, then Λ admits a dominated splitting. To prove Proposition 2.1, it is enough to show that the following lemma. Lemma 2.2 Let Λ be a closed set of f. Suppose that Λ is C 1 -stably continuumwise expansive for f. LetU and U(f) be given in the Definition 1.1, then for any g U(f), g has neither pre-sink nor pre-sources with the orbit staying in U. Proof. We prove the lemma by contradiction. Assume that there is g U(f) such that g has a pre-sink p with Orb(p) U. By the Franks Lemma, we can linearize g at p with respect to the exponential coordinates exp p, i.e, after an arbitrarily small perturbation, we can get a diffeomorphism g 1 U(f) such that there is ɛ 1 > 0 small enough with B ɛ1 (Orb(p)) U such that g 1 Bɛ1 (g i (p)) = exp g i+1 (p) D g i (p)g exp 1 g i (p) B ɛ1 (g i (p)), for any 0 i π(p) 1. Since p is pre-sink of g, D p g π(p) has a multiplicity one eigenvalue such that λ = 1 and other eigenvalues of D p g π(p) have moduli less than 1. Denote by Ep c the eigenspace corresponding to λ, and Es p the eigenspace corresponding to the eigenvalues with modulus less than 1. Thus T p M = Ep c Es p. If λ R then dimep c = 1, and if λ C then dimep c =2. At first, we consider the case dimep c = 1. For simplicity, we suppose that λ =1, and g π(p) 1 (p) =p. The case of λ = 1 can be proved similarly. Since the eigenvalue λ =1, there is a small arc I p B ɛ1 (p) exp p (E c p (ɛ 1)) centered at p such that g π(p) 1 Ip is the identity map. Here E c p(ɛ 1 ) is the ɛ 1 -ball in E c p center at the origin O p. Since g π(p) 1 Ip is the identity map, we know that g π(p) 1 Ip is not continuum-wise expansive. This is a contradiction. Finally, we consider the case dimep c = 2. In this proof, to avoid the notational complexity, we may assume that g(p) =p. As in the first case, by Franks Lemma, there are ɛ 1 > 0 and g 1 U(f) such that g 1 (p) =g(p) =p and g 1 (x) = exp p D p g exp 1 p (x) if x B ɛ1 (p). With a C 1 -small modification of the map D p g, we suppose that there is l>0such that D p g l (v) =v for any v Ep(ɛ c 1 ) exp 1 p (B ɛ (p)). By our
1152 Manseob Lee assumption, g 1 expp (E s p (ɛ 1)) B ɛ1 (p)) of the map is contraction. Take v E c p (ɛ 1) such that v = ɛ 1 /4, and set D p = exp p ({t v :1 t 1+ɛ 1 /4}) B ɛ1 (p). Then g1(d l p )=D p and g1 l Dp is the identity map. Then by similar arguments as above, we get the contradiction. Since the proof is essentially the same as that of [12], we omit the proof here. From the above lemmas and main conclusion of [3], one can get the following lemma. Lemma 2.3 [12, Lemma 3.3] Let Λ,g n and P n be given as in the assumption of Proposition 2.1. Then for any ɛ>0 there are n(ɛ),l(ɛ) > 0 such that for any n>n(ɛ) if P n does not admit an l(ɛ) dominated splitting, then one can find g n C 1 ɛ-close g n and preserving the orbit of P n such that P n is pre-sink or pre-source respecting g n. Let GL(n) be the group of linear isomorphisms of R n. A sequence ξ : Z GL(n) is called periodic if there is k>0 such that ξ j+k = ξ j for k Z. We call a finite subset A = {ξ i :0 i k 1} GL(n) isaperiodic family with period k. For a periodic family A = {ξ i :0 i n 1}, we denote C A = ξ n 1 ξ n 2 ξ 0. We consider about uniformly contracting family. Let A = {ξ i :0 i k 1} GL(n) be a periodic family. We say the sequence A is uniformly contracting family if there is a constant δ>0 such that for any δ-perturbation of A are sink, i.e., for any B = {ζ i :0 i k 1} with ζ i ξ i <δ,all eigenvalue of C B have moduli less than 1. Similarly, we can define the uniformly expanding periodic family. Let P n be a periodic orbit sequence of f. Choose p n P n, then we get a linear map sequence A n = {D pn f,d f(pn)f,...,d f π(pn) 1 (p n)f}. Lemma 2.4 [12, Lemma 3.2] If Λ is not a periodic orbit and A n is given in above. Then for any ɛ>0there exists an n 0 (ɛ) > 0 such that for any n>n 0 (ɛ), A n is neither ɛ-uniformly contracting nor ɛ-uniformly expanding. Lemma 2.5 [21, Corrollary 2.7.1] Let Λ be a transitive set. Then there are a sequence {g n } of diffeomorphism and a sequence {P n } of periodic orbits of g n with period π(p n ) such that g n f in the C 1 -topology and P n H Λ as n, where H is the Hausdorff limit, and π(p n ) is the period of P n. From the above lemmas and the next property of dominated splitting, we can get Proposition 2.1.
Continuum wise expansive and dominated splitting 1153 Lemma 2.6 [2, Lemma 1.4] Let g n converges to f and if Λ n be a closed g n - invariant set such that the Hausdorff limit of Λ n equal to Λ. If Λ gn (U) admits a l-dominated splitting respecting g n, then Λ admits an l-dominated splitting respecting f. Acknowledgements The author wishes to express his deepest appreciation to the referee for his careful reading of the manuscript. This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2011-0007649). References [1] Abdenur. F, Bonatti. C and Corvisier. S, Global dominated splitting and the C 1 -newhouse phenomenon, Proc. Amer. Math. Soc. 134 (2006), 2229-2237. [2] Bonatti. C, Díaz. L. J and Pujals. E, A C 1 -generic dichotomy for diffeomorphisms: weak forms of hyperbolicity or infinitely many sinks or sources, Ann. of Math. 158 (2003), 187-222. [3] Bonatti. C, Gourmelon. N and Vivier. T, Perturbations of the derivative along periodic orbits, Ergodi. Th. & Dynm. Syst 26 (2006), 1307-1337. [4] T. Das, K. Lee and M. Lee, Hyperbolicity of C 1 -stably continuum-wise expansive homoclinic classes, preprint. [5] S. Gan, K. Sakai and L. Wen, C 1 -stably weakly shadowing homoclinic classes, Disc. Contin. Dyan. Syt. 27(2010), 205-216. [6] H. Kato, Continuum-wise expansive homeomorphisms, Canad. J. Math. 45 (1993), 576-598. [7] H. Kato, Concerning continuum-wise fully expansive homeomorphisms of continua, Topology Appl. 53 (1993), 239-258. [8] J. Lewowicz, Expansive homeomorphisms on surfaces, Bol. Soc. Bras. Mat. Nova Ser. 20 (1989), 113-133. [9] K. Lee and M. Lee, Hyperbolicity of C 1 -stably expansive homoclinic classes, Discrete and Continuous Dynam. Sys. 27 (2010), 1133-1145.
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