Volume-Preserving Diffeomorphisms with Periodic Shadowing

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1 Int. Journal of Math. Analysis, Vol. 7, 2013, no. 48, HIKARI Ltd, htt://dx.doi.org/ /ijma Volume-Preserving Diffeomorhisms with Periodic Shadowing Manseob Lee Deartment of Mathematics, Mokwon University Daejeon, , Korea Coyright c 2013 Manseob Lee. This is an oen access article distributed under the Creative Commons Attribution License, which ermits unrestricted use, distribution, and reroduction in any medium, rovided the original work is roerly cited. Abstract We show that if a volume-reserving diffeomorhism belongs to the C 1 -interior of the set of all volume reserving diffeomorhisms having the eriodic shadowing roerty then it is Anosov. Mathematics Subject Classification: 37C10, 37C50, 37D20 Keywords: volume-reserving, star condition, eriodic shadowing, Anosov 1 Introduction The shadowing theory is closely related to the stability condition(see [8, 9]). We will study a kind of the shadowing roerty which is called eriodic shadowing roerty. The notion of the eriodic shadowing roerty was well studied by [7]. They showed that the following. Theorem 1.1 [7, Theorem] Let f Diff(M). The following are equivalent; (a) f has the Lischitz eriodic shadowing roerty. (b) f belongs to the C 1 -interior of the set of diffeomorhisms having the eriodic shadowing roerty. (c) f satisfies both Axiom A and the no-cycle condition.

2 2380 Manseob Lee Let M be a d-dimensional (d 2) Riemannian closed and connected manifold and let d(, ) denotes the distance on M inherited by the Riemannian structure. We endow M with a volume-form (cf.[5]) and let μ denote the Lebesgue measure related to it. Let Diffμ 1 (M) denote the set of volumereserving diffeomorhisms defined on M, i.e. those diffeomorhisms such that μ(b) = μ(f(b)) for any μ-measurable subset B. Consider this sace endowed with the C 1 Whitney toology. The Riemannian inner-roduct induces a norm on the tangent bundle T x M. We will use the usual uniform norm of a bounded linear ma A given by A = su v =1 Av. Let f Diffμ 1(M). Given δ>0, we say that a sequence of oints {x i } i Z M is a δ-seudo orbit of f if d(f(x i ),x i ) <δfor all i Z. We say that a δ-seudo orbit {x i } i Z is a δ-eriodic seudo orbit if x n+i = x i for some n Z. We say that f has the eriodic shadowing roerty if for any ɛ>0 there is δ>0such that for any eriodic δ-seudo orbit {x i } i Z with x n+i = x i, there is y P (f) such that d(f i (y),x i ) <ɛfor all i Z. Let Λ be a closed f-invariant set. We say that Λ is hyerbolic if the tangent bundle T Λ M has a Df-invariant slitting E s E u and there exist constants C>0and 0 <λ<1 such that D x f n E s x Cλ n and D x f n E u x Cλ n for all x Λ and n 0. If Λ = M then f is Anosov. We say that f has the C 1 -robustly eriodic shadowing roerty if there is a C 1 -neighborhood U(f) Diff μ (M) off such that for any g U(f), g has the eriodic shadowing roerty. In [2], Bessa roved that if f has the C 1 - robustly shadowing roerty then it is Anosov. Bessa, Lee and Wen shown in [3] that if f has the C 1 -robustly secification roerty then it is Anosov, and f is C 1 -robustly exansive then it is Anosov. From the results, we study the C 1 -robustly eriodic shadowing roerty. Then we have Theorem 1.2 Let f Diff μ (M). The following are equivalent: (a) f has the C 1 -robustly eriodic shadowing roerty, (b) f is Anosov. 2 Proof of Theorem 1.2 Let M be as before, and let f Diff μ (M). To rove, we will use the following version of the Franks lemma for the conservative case which is stated and roved in [4, Proosition 7.4]. Lemma 2.1 Let f Diff 1 μ (M), and U(f) be a C1 -neighborhood of f in Diff 1 μ (M). Then there exist a C 1 -neighborhood U 0 (f) U(f) of f and ɛ>0 such that

3 Volume-reserving diffeomorhisms with eriodic shadowing 2381 if g U 0 (f), any finite f-invariant set E = {x 1,...,x m }, any neighborhood U of E and any volume-reserving linear mas L j : T xj M T g(xj )M with L j D xj g ɛ for all j =1,...,m, there is a conservative diffeomorhism g 1 U(f) coinciding with f on E and out of U, and D xj g 1 = L j for all j =1,...,m. In the volume reserving case, the Axiom A condition is equivalent to the diffeomorhism be Anosov, since Ω(f) = M by Poincaré Recurrence Theorem. We define the set F μ (M) as the set of diffeomorhisms f Diff μ (M) which has a C 1 -neighborhood U(f) Diff μ (M) such that if for any g U(f), every eriodic oint of g is hyerbolic. Note that F μ (M) F(M) (see [1, Corollary 1.2]). Very recently, Arbieto and Catalan [1] roved that if a volume reserving diffeomorhism is contained in F μ (M) then it is Anosov. We can restate as follows. Theorem 2.2 [1, Theorem 1.1] If f F μ (M) then f is Anosov. To rove Theorem 1.2, it is enough to show that f F μ (M). Remark 2.3 From the Moser s Theorem (see [5]), there is a smooth conservative change of coordinates ϕ x : U(x) T x M such that ϕ x (x) = 0, where U(x) is a small neighborhood of x M. Lemma 2.4 Let f Diff μ (M). If f has the C 1 -robustly eriodic shadowing roerty, then f F μ (M). Proof. Suose that f has the C 1 -robustly eriodic shadowing roerty. Let U(f) Diff μ (M) beac 1 -neighborhood of f. Then for any g U(f), g has the eriodic shadowing roerty. To derive a contradiction, we may assume that f F μ (M). Then there is a nonhyerbolic eriodic oint P (g) for some g U(f). For simlicity, we assume that g() =. Then there is an eigenvalue λ of D g such that λ =1, and T M = E s Eu Ec, where Es is the eigensace corresonding to the eigenvalues of the smaller than 1, and E u is the eigensace corresonding to the eigenvalues of the greater than 1, and E c the eigensace corresonding to λ. Then we see that if λ R then dime c = 1, and if λ C then dime c =2. First, we consider dime c =1. For simlicity, we may assume that λ =1 (the other case is similar). By making use of the Lemma 2.1, we linearize g at with resect to Moser s Theorem; that is, by choosing α>0 sufficiently small we construct g 1 C 1 -nearby g such that { ϕ 1 D g 1 (x) = g ϕ (x) if x B α (), g(x) if x/ B 4α ().

4 2382 Manseob Lee Then g 1 () =g() =. Since the eigenvalue λ of D g 1 is one, D g 1 (v) =v for any v E c (α). Take v 0 E c (α) such that v 0 = α/4. We set and ϕ 1 I v0 = {t v 0 : 1 t 1} ϕ (B α ()), (I v0 )=J. Since g 1 (J )=J is the identity ma, ϕ 1 (I v0 )=J is g 1 -invariant. Take ɛ = α/8. Let 0 < δ = δ(ɛ) < ɛ be the number of the eriodic shadowing roerty of g 1. Take x 0 = x, x 1,...,x m J such that d(x i,x j ) < δ for all 0 i j m, and d(x, x m ) = 4ɛ. We have ξ 1 = {x(= x 0 ),x 1,...,x m,x m 1,...,x} is a finite δ-2m-eriodic seudo orbit of g 1. Then ξ = {...,ξ 1,ξ 1,...} = {x i } i Z is δ-eriodic seudo orbit of g 1 and it is clear ξ J. By the eriodic shadowing roerty, there is a eriodic shadowing oint y P (g 1 ) such that d(g1(y),x i i ) <ɛfor all i Z. If y P (g 1 ) \J then by Moser s Theorem, y = ϕ 1 (w) =ϕ 1 (ws,w u,w c ), where w =(w s,w u,w c ) E s Eu Ec. Since g 1 : J J is the identity ma, g i 1 (ϕ 1 (0, 0,wc )) = ϕ 1 (0, 0,wc ) J for all i Z. Thus one can find k Z such that d(g k 1(ϕ 1 (w s,w u, 0),g1(ϕ k 1 (0, 0,w c )) = d(g1(ϕ k 1 (w s,w u, 0),ϕ 1 (0, 0,w c )) ɛ. This is a contradiction by the eriodic shadowing roerty. Thus the eriodic shadowing oint have to be in J. But, g 1 : J J is the identity ma, for every oint y J is the fixed oint of g 1. Thus d(g1(y),x i i )=d(y, x i ) <ɛ for all i Z. Since d(x 0,x m )=2ɛthere is k>0such that d(g1 k(y),x k)= d(y, x k ) >ɛ. This is a contradiction by the eriodic shadowing roerty. Finally, if λ C, then dime c =2. To avoid the notational comlexity, we may assume that g() =. As in the first case, by Lemma 2.1, there are α>0 and g 1 V(f) such that g 1 () =g() = and { ϕ 1 D g 1 (x) = g ϕ (x) if x B α (), g(x) if x/ B 4α (). With a C 1 -small modification of the ma D g, we may suose that there is l>0 (the minimum number) such that D g l (v) =v for any v ϕ (B α ()) T M. Take v 0 ϕ (B α ()) such that v 0 = α/4, and set L = ϕ 1 ({t v 0 :1 t 1+α/4}). Then L is an arc such that (a) g1 i(l ) g j 1 (L )= for 0 i j l 1, (b) g1(l l )=L, and (c) g1 l L is the identity ma. Note that g 1 has the eriodic shadowing roerty if and only if g1 k has the eriodic shadowing roerty, for all k Z. As in the first case, we can show that g 1 does not have the eriodic shadowing roerty, which contradicts the fact that g 1 U(f). Thus,

5 Volume-reserving diffeomorhisms with eriodic shadowing 2383 if f belongs to the C 1 -interior of the set of a volume reserving diffeomorhism having the eriodic shadowing roerty, every eriodic oint of f is hyerbolic. Proof of Theorem 1.2. Let f Diff μ (M) has the C 1 -robustly eriodic shadowing roerty. Then by Lemma 2.4, f F μ (M). By Theorem 2.2, f is Anosov. Acknowledgement. This work is suorted by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology, Korea (No ). References [1] A. Arbieto and T. Catalan, Hyerbolicity in the volume reserving scenario, Ergod. Th. & Dynam. Sys. (to aear) DOI: htt://dx.doi.org/ /etds [2] M. Bessa, C 1 -stably shadowable conservative diffeomorhisms are Anosov, to aear in Bull. Korean Math. Soc. [3] M. Bessa, M. Lee and X. Wen, Shadowing, exansiveness and secification for C 1 -conservative systems, arxiv: v1. [4] C. Bonatti, L. J. Diáz and E. R. Pujals, A C 1 -generic dichotomy for diffeomorhism: weak forms of hyerbolicity or infinitely many sinks or sources, Ann. of Math., 116(2003), [5] J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc., 120(1965), [6] S. Newhouse, Quasi-elliitc eriodic oints in conservative dynamical systems, Amer. J. Math. 99(1977), [7] A. V. Osiov, S. Y. Pilyugin and S. B. Tikhomirove, Periodic shadowing and Ω-stability, Regular and Chaotic Dynam., 15(2010), [8] C. Robinson, Stability theorems and hyerbolicity in dynamical systems, Rocky Mountain J. Math., 7(1977), [9] K. Sakai, Pseudo-orbit tracing roerty and strong transversality of diffeomorhisms on closed manifolds, Osaka J. Math.,31(1994), Received: July 25, 2013