In this short note we prove that if a symplectomorphism f is C^1-stably shadowable, then f is Anosov. The same result is obtained for volume-preserving diffeomorphisms.info:eu-repo/semantics/publishedVersio

Bessa, Mário

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Bull. Korean Math. Soc. 50 (2013), No. 5, pp. 1495–1499http://dx.doi.org/10.4134/BKMS.2013.50.5.1495C1-STABLY SHADOWABLE CONSERVATIVEDIFFEOMORPHISMS ARE ANOSOVMário BessaAbstract. In this short note we prove that if a symplectomorphism fis C1-stably shadowable, then f is Anosov. The same result is obtainedfor volume-preserving diffeomorphisms.1. Introduction: basic definitions and statement of the results1.1. The symplectomorphisms frameworkDenote by M a 2d-dimensional manifold with Riemaniann structure andendowed with a closed and nondegenerate 2-form ω called symplectic form. Letµ stands for the volume measure associated to the volume form wedging ω d-times, i.e., ωd = ω∧· · ·∧ω. The Riemannian structure induces a norm ‖·‖ on thetangent bundle TxM . Denote the Riemannian distance by d(·, ·). We will usethe canonical norm of a bounded linear map A given by ‖A‖ = sup‖v‖=1 ‖A·v‖.By the theorem of Darboux (see e.g. [9, Theorem 1.18]) there exists an atlas{ϕj : Uj → R2d}, where Uj is an open subset of M , satisfying ϕ∗jω0 = ωwith ω0 =∑di=1 dyi ∧ dyd+i. A diffeomorphism f : (M,ω) → (M,ω) is called asymplectomorphism if it leaves invariant the symplectic structure, say f∗ω = ω.Observe that, since f∗ωd = ωd, a symplectomorphism f : M → M preservesthe volume measure µ. Symplectomorphisms arise naturally in the rationalmechanics formalism as the first return Poincaré maps of hamiltonian flows.For this reason, it has long been one of the most interesting research fields inmathematical physics. We suggest the reference [9] for more details on generalhamiltonian and symplectic theories. Let (Symp1ω(M), C1) denote the set ofall symplectomorphisms of class C1 defined on M , endowed with the usual C1Whitney topology.1.2. The shadowing propertyThe concept of shadowing in dynamical systems is inspired by the numericalcomputational idea of estimating differences between exact and approximateReceived January 23, 2012.2010 Mathematics Subject Classification. 37D20, 37C50.Key words and phrases. Anosov maps, shadowing, uniform hyperbolicity.c©2013 The Korean Mathematical Society14951496 M. BESSAsolutions along orbits and to understand the influence of the errors that wecommit and allow on each iterate. We may ask if it is possible to obtainshadowing of approximate trajectories in a given dynamical system by exactones. We refer Pilyugin’s book [11] for a completed description on the subject.There are, of course, considerable limitations to the amount of informationwe can extract from a given specific system that exhibits the shadowing prop-erty, since a C1-close system may be absent of that property. For this reason itis of great utility and natural to consider that a selected model can be slightlyperturbed in order to obtain the same property - the stably shadowable dy-namical systems.For δ > 0 and a, b ∈ [−∞,+∞] such that a < b, the sequence of points{xi}bi=a in M is called a δ-pseudo orbit for f if d(f(xi), xi+1) < δ for alla ≤ i ≤ b− 1.Let Λ ⊆ M be a closed and f -invariant set. The symplectomorphismf : M → M is said to have the shadowing property in Λ if for all ǫ > 0,there exists δ > 0, such that for any δ-pseudo orbit (xn)n∈Z in Λ, there is apoint x ∈ M which ǫ-shadows (xn)n∈Z, i.e., d(fi(x), xi) < ǫ.Let f ∈ Symp1ω(M) we say that f is C1-stably (or robustly) shadowable ifthere exists a neighborhood V of f in Symp1ω(M) such that any g ∈ V has theshadowing property.We point out that f has the shadowing property if and only if fn has theshadowing property for every n ∈ Z.1.3. Hyperbolicity and statement of the resultsWe say that any element f in the set Symp1ω(M) is Anosov if, there existsλ ∈ (0, 1) such that the tangent vector bundle over M splits into two Df -invariant subbundles TM = Eu ⊕Es, with ‖Dfn|Es‖ ≤ λn and ‖Df−n|Eu‖ ≤λn. We observe that there are plenty Anosov diffeomorphisms which are notsymplectic.Let us recall that a periodic point p of period π is said to be hyperbolic ifthe tangent map Dfπ(p) has no norm one eigenvalues.Let f ∈ Symp1ω(M) we say that f is in F1ω(M) if there exists a neighborhoodV of f in Symp1ω(M) such that any g ∈ V , has all the periodic orbits ofhyperbolic type. We define analogously the set F1µ(M) in the broader set ofvolume-preserving diffeomorphisms Symp1µ(M).The results in this note can be seen as a generalization of the result in [12]for symplectomorphisms and volume-preserving diffeomorphisms. Let us stateour first result.Theorem 1. If a symplectomorphism f defined in symplectic manifold M isC1-stably shadowable, then f is Anosov.It is well known that Anosov diffeomorphisms impose severe topological re-strictions to the manifold where they are supported. Thus, we present a simpleC1-STABLY SHADOWABLE CONSERVATIVE DIFFEOMORPHISMS 1497but startling consequence of Theorem 1 that shows how topological conditionson the phase space imposes numerical restrictions to a given dynamical system.Corollary 1.1. If M does not support an Anosov diffeomorphisms, then thereare no C1-stably shadowable symplectomorphisms.We end this introduction by recalling a result in the vein of ours provedin [4] - C1-robust topologically stable symplectomorphisms are Anosov. Wepoint out that, in the broader setting of dissipative diffeomorphisms, it wasproved, in [13, Theorem 4], that expansiveness and the (robust) shadowingproperty implies (robust) topologic stability. In [13, Theorem 11] it is alsoproved that topological stability, imply the shadowing property. Another resultwhich relates C1-robust properties with hyperbolicity is the Horita and Tahzibitheorem (see [6]) which states that C1-robustly transitive symplectomorphismsare partially hyperbolic.2. Proof of Theorem 1Theorem1 is a direct consequence of the following two propositions. Thefollowing result can be found in [10].Proposition 2.1 (Newhouse). If f ∈ F1ω, then f is Anosov.The following result is a symplectic version of [7, Proposition 1]. Actually,Moriyasu, while working in the dissipative context, considered the shadowingproperty in the non-wandering set, which, in the symplectic setting, and dueto Poincaré recurrence, is the whole manifold M .Proposition 2.2. If f is a C1-stably shadowable symplectomorphism, thenf ∈ F1ω.Proof. The proof is by contradiction; let us assume that there exists a C1-stably shadowable symplectomorphism f having a non-hyperbolic closed orbitp of period π.In order to go on with the argument we need to C1-approximate the sym-plectomorphism f by a new one, f1, which, in the local coordinates given byDarboux’s theorem, is linear in a neighborhood of the periodic orbit p. Toperform this task, in the sympletic setting, and taking into account [2, Lemma3.9], it is required higher smoothness of the symplectomorphism.Thus, if f is of class C∞, take g = f , otherwise we use [14] in order to obtaina C∞ C1-stably shadowable symplectomorphism h, arbitrarily C1-close to f ,and such that h has a periodic orbit q, close to p, with period π. We observethat q may not be the analytic continuation of p and this is precisely the casewhen 1 is an eigenvalue of the tangent map Dfπ(p).If q is not hyperbolic take g = h. If q is hyperbolic forDhπ(q), then, since h isC1-arbitrarily close to f , the distance between the spectrum of Dhπ(q) and theunitary circle can be taken arbitrarily close to zero. This means that we are in1498 M. BESSAthe presence of “poor” hyperbolicity, thus in a position to apply [6, Lemma 5.1]to obtain a new C1-stably shadowable symplectomorphism g ∈ Symp∞ω (M),C1-close to h and such that q is a non-hyperbolic periodic orbit.Now, we use the weak pasting lemma ([2, Lemma 3.9]) in order to obtaina C1-stably shadowable symplectomorphism f1 such that, in local canonicalcoordinates, f1 is linear and equal to Dg in a neighborhood of the periodic non-hyperbolic orbit, q. Moreover, the existence of an eigenvalue, λ, with modulusequal to one is associated to a symplectic invariant two-dimensional subspacecontained in the subspace Ecq ⊆ TqM associated to norm-one eigenvalues. Fur-thermore, up to a perturbation using again [6, Lemma 5.1], λ can be taken ra-tional. This fact assures the existence of periodic orbits arbitrarily close to thef1-orbit of q. Thus, there existsm ∈ N such that fmπ1 (q)|Ecq = (Dgmπ)q|Ecq= idholds, say in a ξ-neighborhood of q. Recall that, since f1 has the shadowingproperty fmπ1 also has. Therefore, fixing ǫ ∈ (0,ξ4), there exists δ ∈ (0, ǫ) suchthat every δ-pseudo fmπ1 -orbit (xn)n is ǫ-traced by some point in M . Take ysuch that d(y, q) = 3ξ4and a closed δ-pseudo fmπ1 -orbit (xn)n such that anyball centered in xi and with radius ǫ is still contained in the ξ-neighborhood ofq, moreover, take x0 = q and xs = y.By the shadowing property there exists z ∈ M such that d(fmπi1 (z), xi) < ǫfor any i ∈ Z. Moreover, we observe that d(fmπi1 (z), q) < ξ for every i ∈ Z.Therefore, z ∈ Ecq . Finally, we reach a contradiction by noting thatd(q, z) ≥ d(q, xs)− d(xs, z) = d(q, y)− d(xs, fmπs1 (z)) ≥3ξ4− ǫ >ξ2> ǫ.3. Volume-preserving diffeomorphismsTheorem 1 also holds on the broader context of volume-preserving diffeo-morphisms.Theorem 2. If a volume-preserving diffeomorphism f defined in a manifoldM is C1-stably shadowable, then f is Anosov.Proof. The proof follows the same steps as the one of Theorem 1. Next we givethe strategy for proving it by referring the fundamental pieces that replace theones used along the proof of Theorem 1.• The version of Proposition 2.1 for volume-preserving diffeomorphismsis proved in [1, Theorem 1.1];• Proposition 2.2 can be obtained by noting that:(1) the Darboux coordinates are switched by the Moser volume-preser-ving coordinates (cf. [8]);(2) the result in [14] are replaced by the smoothness result in [3];(3) the perturbation lemma in [6], are interchanged by its correspon-dent in the volume-preserving case proved in [5, Proposition 7.4];(4) and, finally, we should use [2, Theorem 3.6] instead of [2, Theorem3.9].C1-STABLY SHADOWABLE CONSERVATIVE DIFFEOMORPHISMS 1499We leave the details to the reader. Acknowledgements. The author was partially supported by National Fundsthrough FCT - “Fundação para a Ciência e a Tecnologia”, projects PEst-OE/MAT/UI0212/2011 and PTDC/MAT/099493/2008.References[1] A. Arbieto and T. Catalan, Hyperbolicity in the Volume Preserving Scenario, Ergod.Th. & Dynam. Sys. (at press) DOI: http://dx.doi.org/10.1017/etds.2012.111.[2] A. Arbieto and C. Matheus, A pasting lemma and some applications for conservativesystems, With an appendix by David Diica and Yakov Simpson-Weller., Ergodic TheoryDynam. Systems 27 (2007), no. 5, 1399–1417.[3] A. Avila, On the regularization of conservative maps, Acta Math. 205 (2010), no. 1,5–18.[4] M. Bessa and J. Rocha, A remark on the topological stability of symplectomorphisms,Appl. Math. Lett. 25 (2012), no. 2, 163–165.[5] C. Bonatti, L. Dı́az, and E. Pujals, A C1-generic dichotomy for diffeomorphisms: Weakforms of hyperbolicity or infinitely many sinks or sources, Ann. of Math. (2) 158 (2003),no. 2, 355–418.[6] V. Horita and A. Tahzibi, Partial hyperbolicity for symplectic diffeomorphisms, Ann.Inst. H. Poincaré Anal. Non Linéaire 23 (2006), no. 5, 641–661.[7] K. Moriyasu The topological stability of diffeomorphisms, Nagoya Math. J. 123 (1991),91–102.[8] J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965),286–294.[9] J. Moser and E. Zehnder, Notes on dynamical systems, Courant Lecture Notes in Math-ematics, 12. New York University, Courant Institute of Mathematical Sciences, NewYork; American Mathematical Society, Providence, RI, 2005.[10] S. Newhouse, Quasi-eliptic periodic points in conservative dynamical systems, Amer. J.Math. 99 (1977), no. 5, 1061–1087.[11] S. Pilyugin, Shadowing in Dynamical Systems, Lecture Notes in Math., 1706, Springer-Verlag, Berlin, 1999.[12] K. Sakai, C1-stably shadowable chain components, Ergodic Theory Dynam. Systems 28(2008), no. 3, 987–1029.[13] P. Walters, On the pseudo-orbit tracing property and its relationship to stability, Thestructure of attractors in dynamical systems (Proc. Conf., North Dakota State Univ.,Fargo, N.D., 1977), 231–244, Lecture Notes in Math., 668, Springer, Berlin, 1978.[14] E. Zehnder, Note on smoothing symplectic and volume-preserving diffeomorphisms, Ge-ometry and topology (Proc. III Latin Amer. School of Math., Inst. Mat. Pura AplicadaCNPq, Rio de Janeiro, 1976), pp. 828–854. Lecture Notes in Math., Vol. 597, Springer,Berlin, 1977.Universidade da Beira InteriorRua Marquês d’Ávila e Bolama, 6201-001 Covilhã PortugalE-mail address: bessa@ubi.pt

C1-stably shadowable conservative diffeomorphisms are Anosov

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C1-stably shadowable conservative diffeomorphisms are Anosov

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