R E S E A R C H
Open Access
Hamiltonian systems with orbital, orbital
inverse shadowing
Manseob Lee
**Correspondence:
lmsds@mokwon.ac.kr Department of Mathematics, Mokwon University, Daejeon, 302-729, Korea
Abstract
We show that if a Hamiltonian system has the robustly orbitally shadowing property, then it is Anosov. Moreover, if a Hamiltonian system has the robustly orbitally inverse shadowing property with respect to the class of continuous methods, then it is Anosov.
MSC: 37C10; 37C50; 37D20
Keywords: Hamiltonian systems; Hamiltonian star systems; shadowing; orbital shadowing; inverse shadowing; orbital inverse shadowing; Anosov
1 Introduction
The various shadowing theory (shadowing, orbital shadowing, inverse shadowing, orbital inverse shadowing property) is close to the stability theory. In fact, Sakai has shown in [] that if a diffeomorphism belongs to theC-interior of the set of all diffeomorphisms
having the shadowing property, then it is a structurally stable diffeomorphism. Pilyuginet al.[] proved that if a diffeomorphism belongs to the set of all diffeomorphisms having the orbital shadowing property, then it is a structurally stable diffeomorphism. It extends the result of the shadowing property. For the orbital shadowing property, we can find many results (see [–]).
The notion of the inverse shadowing property is a dual notion of the shadowing prop-erty which was introduced by Corless and Pilyugin in []. Pilyugin [] and Lee [] proved that if a diffeomorphism belongs to the set of all diffeomorphisms having the inverse shad-owing property with respect to the class the continuous method then it is a structurally stable diffeomorphism. The qualitative theory of dynamical systems with the property was developed by various researchers (see [–]). The inverse shadowing property is related to topological stability. In fact, if a dynamical system is topologically stable then it has the inverse shadowing property with respect to the continuous method, but the converse does not hold in general. The following can be found in [].
Example . Letf be a diffeomorphism on a manifoldM. Then we consider the equiv-alence relation ‘∼’ defined by (x, )∼(f(x), ) onM×[, ], forx∈M. SetM=M× [, ]/∼. We define a flowKf onMby setting
Kf
(x,s),t=f[t+s](x),t+s– [t+s]
for (x,s)∈Mandt∈R, where [t] denotes the greatest integer less than or equal tot. Then the flowKf onMis called thesuspension flowoff.
Let f :S→S be a diffeomorphism such that f(x) =x+xsin(π/x) forx= , and
f(x) = forx= , where> is sufficiently small. Then we know thatf has the shad-owing property. By Lee and Park [], it has the inverse shadshad-owing property with respect to the continuous method. But the diffeomorphism is not topologically stable (see []). By Thomas [], the suspension flowKf off is topologically stable if and only iff is
topo-logically stable. Thus the inverse shadowing is a general notion of topological stability. The notion of the orbital inverse shadowing property was introduced by []. It was proved in [] that if a diffeomorphism belongs to theC-interior of the set of all diffeomorphisms
having the orbital inverse shadowing property with respect to the continuous methods, then it is a structurally stable diffeomorphism. For vector fields, Leeet al.[] proved that if a vector field belongs to theC-interior of the set of all vector fields having the orbital in-verse shadowing property with respect to the continuous methods, then it is a structurally stable vector field.
In this paper, we study orbital shadowing and the orbital inverse shadowing property for a Hamiltonian system.
2 Hamiltonian systems
Let (M,ω) be a symplectic manifold, whereMis a n(≥)-dimensional, compact, bound-aryless, connected, and smooth Riemannian manifold, endowed with the symplectic formω. AHamiltonian H:M→Ris a real valuedCr (r≥) function onM. Denote
byCr(M,R) the set ofCr-Hamiltonian onM. In this paper, we consider theC-topology, thus we setr= . Given a HamiltonianH, we define theHamiltonian vector field XHas
follows: for allv∈TpM
ωXH(p),v
=dpH(v),
which generates the Hamiltonian flowXHt . Note that a Hamiltonian vector fieldXHisC
if and only if the Hamiltonian functionHisC. There exists a formulation in terms of the
Hamiltonian equations, that is, the usual Hamiltonian equations are
˙ qi=
∂H
∂pi
, p˙i= – ∂H
∂qi
,
where (qi,pi)∈Mandi= , . . . ,n. Denote bySing(XH) the set of all singularities ofXH.
SinceHis smooth andMis compact,Sing(XH)= ∅. A scalare∈H(M)⊂Ris called the
energyof H. Anenergy hypersurfaceEH,e is a connected component of H–({e}) called
anenergy level set. The energy level setH–({e}) is said to beregular if any energy hy-persurface ofH–({e}) is regular, which means it does not contain singularities. Clearly a
regular energy hypersurface is aXt
H-invariant, compact, and (n– )-dimensional
mani-fold. We say a Hamiltonian level (H,e) isregularif the energy level setH–({e}) is regular.
AHamiltonian systemis (H,e,EH,e), whereHis the Hamiltonian,eis the energy, andEH,e
is a regular connected component ofH–({e}). ThenH–({e}) corresponds to the union of
a finite number of closed connected components, that is,H–({e}) =ni=EH,e,i, forn∈N.
theLiouville theorem. Thus the n-formωn=ω∧ · · · ∧ω(n-times) is a volume form and
induces a measureμonM, which is called the Lebesgue measure associated toωn. Then the measureμonMis invariant by the Hamiltonian flow. For a regular Hamiltonian level (H,e), we define a volume formωEH,eon each energy hypersurfaceEH,e⊂H–({e}), where
for allx∈EH,e,
ωEH,e(u,v,w) =ωn(∇Hx,u,v,w),
whereωEH,e:TxEH,e×TxEH,e×TxEH,e→R. The volume formωH,eisXHt-invariant. Since
any energy hypersurface is compact, it induces an invariant volume measureμEH,eonEH,e
which is a finite measure. Now we consider the transversal linear Poincaré flow. Given a Hamiltonian vector fieldXHand a regular pointx∈M, lete∈H(x). DefineNx=Nx∩
TxH–({e}), whereTxH–({e}) =KerdH(x) is the tangent space to the energy level set. Then Nxis a (dimM– )-dimensional bundle. Thetransversal linear Poincaré flowassociated toH,PtH(x) :Nx→NXt
for the Hamiltonian flowXHt associated toH.
3 Orbital shadowing
LetAandBbe closed sets ofM. Then we can define theHausdorff distanceas follows:
We introduce the notion of the orbital shadowing property. For x∈M, we denote by OrbXH(x) the orbit ofH throughx; that is,OrbXH(x) ={X
t
H(x) :t∈R}. We say that
(H,e,EH,e) hasthe orbital shadowing propertyif for any> there isδ> such that for
any (δ, )-pseudo-orbitξ={(xi,ti) :ti≥,i∈Z}there is a pointy∈EH,esuch that
dH
OrbXH(x),ξ
<.
This means that
ξ⊂B
OrbXH(y), and OrbXH(y)⊂B(ξ),
whereB(A) is a neighborhood ofA.
Note that the orbital shadowing property is a weak version of the shadowing property: the difference is that we do not require a pointxiof a pseudo-orbitξ and the pointXHti(y)
of an exact orbitOrbXH(y) to be close ‘at any time moment’; instead, the sets of the points ofξ andOrbXH(y) are required to be close. We say that (H,e,EH,e) has therobustly or-bitally shadowing property if there is a neighborhoodU of (H,e,EH,e) such that every
(H,e,EH,e)∈Uhas the orbital shadowing property.
In [], Bessaet al.proved that if a Hamiltonian system has the robustly shadowing property then it is Anosov. From this fact, we have the following result, which is more general than the result of [].
Theorem . Let(H,e,EH,e)be a Hamiltonian system.If(H,e,EH,e)has the robustly
or-bitally shadowing property,then it is Anosov.
4 Orbital inverse shadowing LetH∈C(M,R), and let (H,e,EH
,e) be a Hamiltonian system. A mapping:R×M→M
is called a (δ,T)-methodforHif for anyx∈M, the mapx:R→Mdefined by
x(t) =(t,x), t∈R,
is a (δ,T)-pseudo-orbit ofH.is said to becompleteif(,x) =xfor anyx∈M. Then a (δ,T)-method forHcan be considered as a family of (δ,T)-pseudo-orbit ofH. A method
forHis said to becontinuousif the map:M→MR given by(x)(t) =(t,x) for x∈M andt∈R, is continuous under the compact open topology onMR, whereMR denotes the set of all functions fromRtoM. The set of all complete (δ, )-methods for H∈C(M,R) will be denoted byTa(δ,H). LetTd(δ,H) be the set of all complete continuous
(δ, )-methods forHwhich are induced by Hamiltonian systemsH withdC(H,H) <δ,
wheredCis theCmetric onC(M,R). We say that a Hamiltonian system (H,e,EH,e) has
theinverse shadowing propertywith respect to the classTdif for any> there isδ> such that for any (δ,T)-method∈Td(δ,H) and any pointx∈M, there arey∈Mand an increasing homeomorphismh:R→Rwithh() = such that
dXh(t)(x),(t,y)<, t∈R.
We say that (H,e,EH,e) has theorbital inverse shadowing propertywith respect to the
classTdif for any> there isδ> such that for any (δ, )-method∈Td(δ,H) and any pointx∈Mthere isy∈Msuch that
dH
OrbXH(x),Orb(y,)
<,
whereOrb(y,) ={(t,y) :t∈R}. Note that ifHhas the inverse shadowing property with respect to the classTd, thenH has the orbital shadowing property with respect to the classTd. But the converse is not true. Indeed, an irrational rotation map does not have the inverse shadowing property, and the map has the orbital shadowing property. We say that (H,e,EH,e) has therobustly orbitally inverse shadowing propertywith respect to the class Tdif there is a neighborhoodUof (H,e,EH,e) such that for any (H,e,EH,e)∈U has the
orbital inverse shadowing property with respect to the classTd.
In [], Bessaet al.proved that if a Hamiltonian system is robustly topologically stable then it is Anosov. Note that by the definition of topological stability, the inverse shadowing property with respect to the class of the continuous method is a general notion of topo-logical stability. It means that if a system is topotopo-logically stable, then it has the inverse shadowing property with respect to the class of the continuous method, but the converse is not true. From these facts, we have the generalization results as follows.
Theorem . Let(H,e,EH,e)be a Hamiltonian system.If H has the robustly orbitally
in-verse shadowing property with respect toTd,then it is Anosov.
5 Proof of Theorem 3.1 and Theorem 4.1
A Hamiltonian system (H,e,EH,e) is aHamiltonian star systemif there is a neighborhood U of (H,e,EH,e) such that for any (H,e,EH,e)∈U, the corresponding regular energy
hy-persurfaceEH,ehas all hyperbolic closed orbits.
In [], Bessaet al.showed that if we have a Hamiltonian star system on a four dimen-sional manifold, then it is Anosov. Afterwards, Bessaet al.proved the following.
Lemma .[, Theorem ] If(H,e,EH,e)is a Hamiltonian star system,then it is Anosov.
To prove Theorem . and Theorem ., it is enough to show that a Hamiltonian system (H,e,EH,e) is a Hamiltonian star system. For this, we need the following proposition.
Proposition . Let(H,e,EH,e)be a Hamiltonian system.If the following hold: (a) (H,e,EH,e)has the robustly orbitally shadowing property,
(b) (H,e,EH,e)has the robustly orbitally inverse shadowing property with respect to theTd,
then(H,e,EH,e)is a Hamiltonian star system.
The next lemma is called a pasting lemma; it was established in [].
Lemma .[, Theorem .] Let H∈Cr(M,R), ≤r≤ ∞,and let K be a compact subset
of M,and U a small neighborhood of K.Given > there existsδ> such that if H∈
Cl(M,R),for≤l≤ ∞isδ–Cmin{r,l}-close to H on U,then there exist H∈Cl(M,R)and
(a) K⊂V⊂U, (b) H=HonV,
(c) H=HonUc,
(d) His-Cmin{r,l}-close toH.
Letx∈M\Sing(XH). We defineX[Ht,t](x) ={XHt(x) :t∈[t,t]}. LetNxbe a transversal
section to the flow atx, a flow box associated toNxis defined byF(x) =XH[–τ,τ](Nx), where τ,τare taken small such thatF(x) is a neighborhood ofxfoliated by regular orbits. The
following lemma is a version of Frank’s lemma for Hamiltonians.
Lemma .[, Theorem ] Let H∈Cr(M,R), ≤r≤ ∞,> ,τ> and x∈M.Then
there isδ> such that for any flowboxF(x)of an injective arc of orbit XH[,t](x) (t≥τ) and a transversal symplecticδ-perturbation of Pt
H(x),there is H∈Cl(M,R)with l=
max{,k– }such that
(a) His-C-close toH, (b) PHt (x) =,
(c) H=HonXH[,t](x)∪(M\F(x)).
Let (H,e,EH,e) be a Hamiltonian system and letpbe a periodic point inEH,e with
pe-riodπ(p). For a pointp∈EH,e,Np⊂Mis transverse to the flow, that is, a local (n–
)-submanifold for whichXHis nowhere tangency. Define the n– symplectic submanifold Np=Np∩EH,e. Forx∈Np,TxEH,e=TxNp⊕ XH(x). LetU⊂Mbe an open
neighbor-hood ofpandV=U∩M. Letf :V→Npbe the Poincaré map ofXHt toNp such that f(x) =XτH(x)(x) for allx∈V, whereτ(x) is the return time toNp defined by the relation XHτ(x)(x)∈Npandτ(p) =π(p). Thenf is aC-symplectic diffeomorphism. The following
lemma was established in [, Theorem .].
Lemma . Let H∈C∞(M,R)be the Poincaré map f at a periodic point p.Then for any
> there isδ> such that for any symplectic diffeomorphism gδ-C-close to f there is a
Hamiltonian H-C-close with a Poincaré map g.
A pointx∈EH,eis anon-wandering pointofHif for any neighborhoodUofxinEH,e
there isT> such thatXT
H(U)∩U=∅. Denote by(H|EH,e) the set of non-wandering points ofH on the energy hypersurfaceEH,e. We say thatx∈EH,e is aperiodic point of
Hif there isT> such thatXT
H(x) =x. Denote byP(H|EH,e) the set of all periodic points ofH. Given ≤k≤n– , we recall that ak-elliptic closed orbit has ksimple non-real eigenvalues of the transversal linear Poincaré flow at the period of norm , and the norm of its remaining eigenvalues different from . Ifk=n– theelliptic closed orbitshave all eigenvalues at the period of norm , simple and non-real.
By Abraham’s and Marsden’s [] result-the symplectic eigenvalue theorem-ifλis an eigenvalue ofDpfπ of multiplicityk, then /λis an eigenvalue ofDpfπ of multiplicityk.
Moreover, if the multiplicity of the eigenvalues and –, then it is even.
Recall that if we letWbe a small neighborhood of a regular energy hypersurfaceEH,e,
Remark . By Robinson’s version of the Kupka-Smale theorem [], aC-generic
Hamil-tonian has all closed orbits of hyperbolic or elliptic type. Thus, we can see that there is a Hclose toHsuch thatHhas ak-elliptic periodic orbitpof periodπ(p). Therefore, there
is a splitting of the normal subbundle cp along the orbitpinto k-subspaces such that
c
p=c⊕ · · · ⊕ck, wheredimci= fori= , . . . ,k. Letλi=exp(θij) is an eigenvalue of
PπH(p)|ci. Ifθiis an irrational, then, by Frank’s lemma, there isHC-close toHsuch that
each appropriate restriction of the Poincaré mapfH|Nc
i is conjugate to a rational rotation, fori= , . . . ,k.
The following is the proof of Proposition .(a).
Lemma . Let(H,e,EH,e)be a Hamiltonian system.If(H,e,EH,e)has the robustly orbitally
shadowing property then(H,e,EH,e)is a Hamiltonian star system.
Proof Letp∈γ ∈P(H|EH,e) withXHπ(p)(p) =p(π(p) > ) associated toEH,e. Suppose thatp
is not hyperbolic forH. Then there is an eigenvalueλofPπH(p)(p) such that|λ|= . Then by Lemma . and Lemma ., we can findH C-close toH such thatH has a
non-hyperbolic periodic pointpclose topwith periodπ(p) close toπ(p). It is clear thatpis
not the analytic continuation top. Then as in the proof of the [, Theorem .], we can make the Poincaré mapf atpassociated to the Hamiltonian flowXHt aC∞local
sym-plectic diffeomorphism, that is,f :Np,r→Npfor somer> . For simplicity, we assume
thatpis -elliptic. ThenTpNp,r=cp⊕ps ⊕up, wherecpis associated toλ,cpis
associated to an eigenvalues less than andupassociated to an eigenvalue greater than . By Lemma ., Lemma . and the Darboux theorem, there are> and a linear map
A:Np→Npsuch thatg(x) =ϕp◦A◦ϕp–(x) forx∈B(p)∩Np,randg(x) =f(x) for
x∈/B(p)∩Np,r, whereϕp:B(p)∩Np,r→TpNpwithϕp(p) =
– →
. It is clear that g:Np,r→Np is the Poincaré mapH. Using Lemma .,λcan be taken as a rational
rotation, that is, there isl> such that for anyx∈ϕp–(cp)∩B(p)⊂Np,r, we have
gl(x) =x. PutNc
p=ϕp–(cp)∩B(p). For simplicity, we may assume thatgl=g. Then
xis a fixed point forg, and the orbit ofxis periodic, that is,XπH(x)(x) =x, whereπ(x) is the period ofx. Letx=p. Take=min{/,r/}, and let <δ<be the number of
the orbital shadowing property. Takey∈Npc such thatd(p,y) = . Now, we construct a δ-pseudo-orbit{(xi,ti) :ti> ,i∈Z} ⊂Npc as follows: (i)ti=π(p) for alli∈Z, andxn=y
for somen∈N, (ii)xi=pfori≤, (iii)d(g(xi),xi+) <δfor ≤i≤n– , and (iv)xi=yfor
i≥n. By the orbital shadowing property, there isz∈EH,esuch that
dH
OrbXH(z),
(xi,ti) :ti=π(p),i∈Z
<.
Ifz∈EH,e\Npc, then, by the hyperbolicity, there isk∈Zsuch that
dXHkπ(p)(z),(xi,ti) :ti=π(p),i∈Z
>.
This is a contradiction. Thus the orbital shadowing pointz∈Nc
p. Sinceg(z) =z, we have
Z}such that
dxl,X jπ(p) H (z)
=d(xl,z) >,
for somej∈Z. Thus{XHt (z) :t∈R} ⊂B({(xi,ti) :ti=π(p),i∈Z}), which is a
contradic-tion.
Letp∈Hbe a non-hyperbolic periodic point. Then there is a HamiltonianHC-close
to H such thatH has a continuation of periodic points close top. For a Hamiltonian
system (H,e,EH,e) and a periodic pointp∈EH,ewith periodπ(p), letNpcbe a submanifold Npassociated top. Then we have the following.
Lemma .[, Lemma .] Let(H,e,EH,e)be a Hamiltonian system,and let p∈EH,ebe a
non-hyperbolic periodic point.Then there is a Hamiltonian system(H,e,EH,e)C-close
to(H,e,EH,e)such that Hhas a non-hyperbolic periodic point q∈EH,eclose to p and every
point in a small neighborhood of q∈Nc
p is a periodic point of H.
The following is the proof of Proposition .(b).
Lemma . Let(H,e,EH,e)be a Hamiltonian system.If(H,e,EH,e)has the robustly orbitally
inverse shadowing property with respect to the classTd,then(H,e,EH,e)is a Hamiltonian
star system.
Proof LetU ⊂C(H,R) be aC-neighborhood ofH. Suppose thatp∈EH
,eis not a
hy-perbolic periodic point. By Lemma ., there areH∈U andη> such that (i)Hhas a
non-hyperbolic periodic pointq∈EH,e, and (ii) everyγ ∈Bη(q)∩Nqcis a periodic point
of H. Take < <η/, and let <δ< be as in the definition of the orbital inverse
shadowing property ofH. LetfHbe the Poincaré map atqassociated toPtH. Since every γ ∈Bη(q)∩Nqcis a periodic point ofH, we know thatfH(γ) =γ.
Take <τ < such that it is sufficiently small. Then we define the Poincaré mapgH
nearqbygH(x) = (τx,x) forx∈Bη(q)∩Nqcsuch thatdC(fH,gH) <δ, wherexis the
-component ofxandxis the other component. By Lemma ., there is a Hamiltonian Hwhich is associated to the Poincaré mapgH, anddC(H,H) <δ. Takey∈Bη(q)∩Nqc
such thatd(y,q) = . Then for anyz∈EH,e,
dH
OrbXH
(y),OrbXH(z)
<.
Ify=zthenfH(y) =yandgHi (y) = (τiy,y) for alli∈Z. Then there isk∈Zsuch that for
gH|Nc q,
gHk (y) =τky,y
/ ∈B
fHi (y) :i∈Z.
Then we know thatdH(OrbXH(y),OrbXH(z)) >. This is a contradiction, sinceHhas the
orbital inverse shadowing property. Finally we consider y=z. Set z=Xt
H(z)∈Bη(q)∩Nqc, where |t|=min{|t|:XHt (z)∈
Bη(q)∩Nqc}. Since
dH
OrbXH(y),OrbXH(z)
we haved(XHt (y),gHi (z)) <for somet∈R. SincedC(H,H) <δ, we can choose a
se-quence{gi
H(z)} ⊂B(q)∩Nqcsuch thatd(fHi (y),gHi (z)) <for alli∈Z. By hyperbolicity,
there isk> such that
gHk (z) /∈B
fHi (y) :i∈Z.
ThusB({gHi (z) :i∈Z})⊂B(q)∩Nqc. This is a contradiction by the orbital inverse
shad-owing property.
Competing interests
The author declares that they have no competing interests.
Acknowledgements
This work is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (No-2014R1A1A1A05002124).
Received: 26 February 2014 Accepted: 16 June 2014 Published:23 Jul 2014
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