Chain components with stably limit shadowing property are hyperbolic

(1)

R E S E A R C H

Open Access

Chain components with stably limit

shadowing property are hyperbolic

Manseob Lee

1

_{and Junmi Park}

2*

*_{Correspondence: [email protected]} 2_{Department of Mathematics,}

Chungnam National University, Daejeon, 305-764, Korea Full list of author information is available at the end of the article

Abstract

Letfbe a diﬀeomorphism on a closed smooth manifoldM. In this paper, we show thatfhas theC1_{-stably limit shadowing property on the chain component}_C

f(p) off

containing a hyperbolic periodic pointp, if and only ifCf(p) is a hyperbolic basic set.

MSC: Primary 37C50; secondary 34D10

Keywords: hyperbolic; limit shadowing; shadowing; homoclinic class; chain component

1 Introduction

Various closed invariant sets (transitive set, chain transitive set, homoclinic class, chain component, etc.) in dynamical systems are natural candidates to replace Smale’s hyper-bolic basic sets in non-hyperhyper-bolic theory of diﬀerentiable dynamical systems see [–]). To investigate the above, we deal with the shadowing property. It usually plays an impor-tant role in the stability theory and ergodic theory (see []).

LetMbe a closedC∞manifold, and letDiff(M) be the space of diﬀeomorphisms ofM

endowed with theC_{-topology. Denote by}_d_{the distance on}_M_{induced from a}

Rieman-nian metric · on the tangent bundleTM. Letf∈Diff(M). Letbe a closedf-invariant set. Forδ> , a sequence of points{xi}b_i₌_a(–∞ ≤a<b≤ ∞) inMis called aδ-pseudo orbit

off ifd(f(xi),xi+) <δfor alla≤i≤b– . For givenx,y∈M, we writexyif for anyδ> , there is aδ-pseudo orbit{xi}bi=a(a<b) off such thatxa=xandxb=y. We writexyif

xyandyx. The set of points{x∈M:xx}is called thechain recurrent setoff

and is denoted byR(f). DenoteCf(p) ={x∈M:xpandpx}thechain componentof

f containingp. For a closedf-invariant set⊂M, we say thatischain transitiveif for any pointx,y∈andδ> , there exists aδ-pseudo orbit{xi}bδ

i=aδ⊂(aδ<bδ) off such

thatxa_δ=xandxb_δ=y.

Let⊂Mbe a closedf-invariant set. We say thatf has theshadowing propertyonif

for every> , there isδ>  such that for anyδ-pseudo orbit{xi}_ib₌_a⊂off (–∞ ≤a<

b≤ ∞), there is a pointy∈Msuch thatd(fi₍_y_),_xi_{) <}_{for all}_a_≤_i_≤_b_.

Now, we introduce the limit shadowing property which was introduced and studied by Lee []. We say thatf has thelimit shadowing property onif there existsδ>  with the following property: if a sequence{xi}i∈Z⊂is aδ-pseudo orbit off for which relations

d(f(xi),xi+)→ asi→+∞, andd(f–(xi+),xi)→ asi→–∞hold, then there is a point y∈Msuch thatd(fi(y),xi)→ asi→ ±∞. Here, the sequence{xi}i∈Zis called aδ-limit pseudo orbit of f. It is easy to see thatf has the limit shadowing property onif and only

(2)

iffn_{has the limit shadowing property on}_for_n_∈_Z_{\ {}__}_{, and the identity map does not}

have the limit shadowing property.

Note that the above deﬁnition is not the shadowing property, also it is not the notion of the original limit shadowing property in (see [, Examples , ] and [, ]).

We say thatislocally maximalif there is a compact neighborhoodUofsuch that

n∈Zfn(U) =. We say thatf has theC-stably limit shadowing propertyonif there

are aC_{-neighborhood}_U₍_f_{) of}_f _{and a compact neighborhood}_U_of_{such that}

() =f(U) =n∈Zfn(U)(locally maximal),

() for anyg∈U(f),ghas the limit shadowing property ong(U), where

g(U) =

n∈Zgn(U)is the continuation of=f(U).

It is well known that ifpis a hyperbolic periodic point off with periodkthen the sets

Ws(p) =x∈M:fkn(x)→pasn→ ∞ and

Wu(p) =x∈M:f–kn(x)→pasn→ ∞

areC_{-injectively immersed submanifolds of}_M_{. A point}_x_∈_Ws₍_p₎_∩_Wu₍_p_{) is called a} homoclinic pointoffassociated top, and it is said to be atransversal homoclinic pointoff

if the above intersection is transverse. The closure of the homoclinic points off associated to pis called thenon-transversal homoclinic classoff associated top, say, generalized homoclinic class, and it is denoted byHf(p), and the closure of the transversal homoclinic points off associated topis called thetransversal homoclinic classoff associated top, and it is denoted byHf(p). Letp,qbe hyperbolic periodic points off. We say thatpandq

arehomoclinically related, and writep∼qif

Ws(p)Wu(q)=∅ and Wu(p)Ws(q)=∅.

It is clear that ifp∼qthenindex(p) =index(q);i.e.,dimWs(p) =dimWs(q). By Smale’s transverse homoclinic point theorem,Hf(p) coincides with the closure of the set of

hyper-bolic periodic pointsqoff such thatp∼q. In this paper, we consider all periodic points of the saddle type, because, ifp∈P(f) is a sink or a source, thenCf(p) is the periodic orbit ofpitself.

Note that ifpis a hyperbolic periodic point off then there is a neighborhoodUofpand aC_{-neighborhood}_U₍_f_{) of}_f _{such that for any}_g_∈_U₍_f_{), there exists a unique hyperbolic} periodic pointpgofginUwith the same period aspandindex(pg) =index(p). Such a point

pgis called thecontinuationofp=pf.

Letbe a closedf-invariant set. We say thatishyperbolicif the tangent bundleTM

has aDf-invariant splittingEs_⊕_Eu_{and there exist constants}_C_{>  and  <}_λ_{<  such that}

Dxfn|Exs≤Cλ

n _and _D

xf–n|Eux≤Cλ

n

for allx∈andn≥. Moreover, we say thatadmits adominated splittingif the tangent

bundleTMhas a continuousDf-invariant splittingE⊕Fand there exist constantsC>  and  <λ<  such that

Dxfn|E(x)·Dxf–n|F(fn₍_x₎₎≤Cλn

(3)

The following is the main theorem in this paper.

Theorem . Let p be a hyperbolic periodic point of f,and let Cf(p)be the chain component of f associated to p.Then f has the C_{-stably limit shadowing property on Cf}₍_p₎_{if and only}

if Cf(p)is hyperbolic.

Letbe a locally maximal subset ofM. In [], Lee showed that ifis hyperbolic then it

is limit shadowable. Note that a hyperbolic sethas the local product structure if and only if it is locally maximal. Since the chain componentCf(p) has the local product structure, if

Cf(p) is hyperbolic,Cf(p) is locally maximal. Thus by the hyperbolicity of the chain

com-ponentCf(p),f has theC-stably limit shadowing property. Thus, in this paper, we show that iff has theC_{-stably limit shadowing property on}_Cf₍_p_{), then}_Cf₍_p_{) is hyperbolic.}

2 Proof of Theorem 1.1

LetMbe as before, and letf ∈Diff(M).

Lemma . Letbe a locally maximal subset of M.If f has the limit shadowing property

onthen the shadowing points are taken from.

Proof Letδ>  be the number of the limit shadowing property off, and letUbe a locally

maximal neighborhood of. Suppose thatf has the limit shadowing property on. Let

{xi}i∈Z⊂be aδ-limit pseudo orbit off. To derive a contradiction, we may assume that

there isy∈M\such that

dfi(y),xi→ asi→ ±∞.

Sinceis compact, there isη>  such thatBη()⊂U, whereBη() is aη-neighborhood

of. Since{xi}i∈Z⊂and by the limit shadowing property, we can ﬁndl∈Zsuch that

fl₍_y₎_∈_B

η(). Sinceis locally maximal inUandf-invariant,

=

n∈Z

fn₍₎_⊂ n∈Z

fn_B

η()

⊂

n∈Z

fn₍_U_{) =}_.

Then for alln∈Z,fn₍_fl₍_y_{)) =}_fl+n₍_y₎_∈_{. Since}_is_f_-invariant,_y_∈_f–n–l_{() =}_{, this is a}

contradiction. Thus the limit shadowing points are in.

Let us recall some notions for the proof of the following lemma. A compact invariant setisattractingif=_n_≥_fn₍_U_{) for some neighborhood}_U_of_satisfying_fn₍_U₎_⊂_U

for alln> . Anattractoroff is a transitive attracting set offand a repeller is an attractor forf–n_{. We say that}_{is a}_{proper attractor}_{or repeller if}_∅₌₌_M_{. A sink (source) of}_f _is

an attracting (repelling) critical orbit off.

Lemma .([, Proposition ]) Letbe a locally maximal set.f|is chain transitive if

and only ifhas no proper attractor for f.

Lemma . Letbe a locally maximal set.If f has the limit shadowing property on

(4)

Proof Supposehas a proper attractorPin. ThenP=∅and\P=∅. SincePis an attractor, there existsδ>  such thatPattracts the openδ/-neighborhoodBδ/(P) ofP in. Chooseq∈\Bδ/(P) andp∈Psuch thatd(q,p) <δ. Consider a sequence

xi=fi(p), i≤, xi=fi(q), i> 

with i∈Z. Clearly, the sequence {xi}i∈Z is a δ-limit pseudo orbit of f in. Then by

Lemma ., there isy∈such that

dfi(y),xi

→ asi→ ±∞.

Then there existsN>  large enough such thatf–N₍_y₎_∈_B

δ/(P). Therefore,fn(f–N(y))∈

Bδ/(P) forn> , sincePis an attractor. Taking –N= –ik, we have thaty=fik(f–ik(y))∈ Bδ/(P). Thus, by deﬁnition ofBδ/(P), we have that

dfi(y),fi(q) asi→ ∞.

This contradicts the deﬁnition of the limit shadowing property and completes the proof.

Lemma . Letbe a locally maximal set.Suppose f has the limit shadowing property on.Then for any hyperbolic periodic points p,q in,

Ws(p)∩Wu(q)=∅ and Wu(p)∩Ws(q)=∅.

Proof Supposef has the limit shadowing property on locally maximal, and letp,q∈ be hyperbolic periodic points forf. We will show thatWs₍_p₎_∩_Wu₍_q₎₌_∅_{. Other case is}

similar. Sincefhas the limit shadowing property on locally maximal, by Lemma ., we

can take aδ-chain{xi}n_i_=fromptoqsuch thatx=p,xn=q. Then we can construct a δ-limit pseudo orbitξas follows: (i)xi=fi₍_p_),_i_{< , (ii)}_d₍_f₍_xi_),_xi

+) <δ,i= , . . . ,n–  and (iii)xn+i=fi(q),i≥. Then

ξ=. . . ,f–(p),x=p,x, . . . ,xn–,xn=q,f(q), . . .

.

Clearly,ξis aδ-limit pseudo orbit off in. Then, by Lemma ., there exists a pointy∈ such that

dfi(y),xi→ asi→ ±∞.

This implies thaty∈Wu(p) andfn(y)∈Ws(q) (y∈Ws(q)). ThusWu(p)∩Ws(q)=∅. The following so-called Franks lemma will play essential roles in our proof.

Lemma . LetU(f)be any given C_{-neighborhood of f}_._{Then there exist}_{> }_{and a C}

-neighborhood U(f)⊂U(f)of f such that for given g∈U(f),a ﬁnite set {x,x, . . . ,xN},

(5)

Dxig ≤for all≤i≤N,there exists g∈U(f)such that g(x) =g(x)if x∈ {x,x, . . . ,xN}∪ (M\U)and Dx_ig=Lifor all≤i≤N.

Proof See the proof of Lemma . [].

Lemma .([, Lemma .]) Letbe locally maximal in U,and letU(f)be given.If p∈g(U)∩P(g) (g∈U(f))is not hyperbolic,then there is g∈U(f)possessing hyperbolic

periodic points qand qing(U)with diﬀerent indices.

In this section, we will prove Theorem . by making use of the technique developed by Mañé in []. That is, we use the notion of uniform hyperbolicity for a family of periodic

sequences of linear isomorphisms ofRdimM_{. For this, we need several lemmas.}

We say that a diﬀeomorphismf is Kupka-Smale if for any periodic point off is

hyper-bolic and their invariant manifolds intersect transversely and denote the set of Kupka-Smale diﬀeomorphisms byKS(M). It is well known thatKS(M) is residual inDiff(M).

Lemma . Let f ∈Diff(M),and letbe a closed f -invariant set.Suppose that f has the C-stably limit shadowing property on .Then there exist a C-neighborhoodU(f)of f and a compact neighborhood U ofsuch that for any g∈U(f),every p∈g(U)∩P(g)is hyperbolic for g,whereg(U) =n∈Zgn(U).

Proof Since f has the C_{-stably limit shadowing property on} _{, there exist a} _C

-neighborhoodU(f) off and a compact neighborhoodUofsuch that for anyg∈U(f),

g has the limit shadowing property ong(U) =

n∈Zgn(U). Let>  andU(f)⊂U(f)

be the corresponding number andC-neighborhood off given by Lemma . with

re-spect toU(f). Suppose there is a pointq∈g(U)∩P(g) which is not hyperbolic. Then

by Lemma ., we can chooseg∈U(f) such thatindexpg=indexqg, wherepg,qg∈ g(U)∩P(g). ThendimWs(pg) +dimWu(qg) <dimMordimWu(pg) +dimWs(qg) < dimM. We may assume thatdimWs₍_pg_{) +}_dim_Wu₍_qg_{) <}_dim_M_{. By Lemma ., we can}

takeh∈U(g)∩KS(M) such thatindex(pg) =index(ph) andindex(qg) =index(qh) where

ph,qh are the continuation ofpg,qg forh, respectively. Then, sincehis Kupka-Smale, Ws(ph)∩Wu(qh) =∅. On the other hand, sinceh∈U(f),h|h(U) satisﬁes the limit shad-owing property so thatWs₍_ph₎_∩_Wu₍_qh₎₌_∅_{by Lemma .. This is a contradiction and}

completes the proof.

It is a well-known result that the transversal homoclinic classHf(p) is a subset of the gen-eralized homoclinic classHf(p), and it is a subset of the chain componentCf(p). However, under the notion of the limit shadowing property with locally maximal,Hf(p) =Cf(p). It

is obtained by the following lemma.

Lemma . Let U be a locally maximal neighborhood of Cf(p).If f has the limit shadowing property on Cf(p)then Cf(p) =Hf(p).

Proof Letpbe a hyperbolic saddle. For simplify we may assume thatf(p) =p. LetUbe a

locally maximal neighborhood ofCf(p). Suppose thatf has the limit shadowing property

on a locally maximalCf(p). For anyx∈Cf(p), we show thatx∈Hf(p). Letδ>  be the

(6)

orbit{xi}k_i_=–_loff such thatx–l=p,x=xandxk=pfor somel=l(δ),k=k(δ) > . Then the periodicδ-pseudo orbit{xi}k

–l⊂Cf(p) (see [, Proposition .]). Now we construct

aδ-limit pseudo orbit as follows: (i)x–l–i=f–i(p) for alli≥, and (ii)xk+i=fi(p) for all

i≥. Then we know theδ-limit pseudo orbit

neighborhood ofx. Thus we conclude that

y∈Ws(p)∩Wu(p)∩Bη(x).

This meansCf(p)⊂Hf(p), and thereforeCf(p) =Hf(p).

It is well known that a dominated splitting is always extended to a neighborhood. More

precisely, letbe a closedf-invariant set. Then ifadmits a dominated splittingTM=

E⊕Fsuch thatdimEx(x∈) is constant, then there are aC-neighborhoodU(f) off

and a compact neighborhoodU ofsuch that for anyg∈U(f),_n_∈_Zgn₍_U_{) admits a}

dominated splitting

T_n_∈_Zgn₍_U₎M=E(g)⊕F(g)

withdimE(g) =dimE.

From Lemma ., the family of periodic sequences of linear isomorphisms ofRdimM

gen-erated byDg(g∈U(f)) along the hyperbolic periodic pointsp∈g(U)∩P(g) is uniformly

hyperbolic. That is, there exists>  such that for anyg∈U(f),p∈g(U)∩P(g), and any

sequence of linear mapsLi:Tgi₍_p₎M→T_gi+₍_p₎MwithLi–D_gi₍_p₎g<for ≤i≤π(p) – ,

π(p)–

i= Liis hyperbolic. HereU(f) is theC-neighborhood off given by Lemma .. Thus by Proposition II. in [] and Lemma . above, we get the following proposition.

Proposition . Suppose that f has the C_{-stably limit shadowing property on the}

(7)

Remark From Proposition .(b) and Lemma .,Cf(p) =Hf(p) =Hf(p).

In general, a non-hyperbolic homoclinic classHf(p) contains saddle periodic points with diﬀerent indices. Letpbe a hyperbolic periodic point off.

Proposition . Suppose that f has the C-stably limit shadowing property on Cf(p).

Then for any q∈Cf(p)∩P(f),

index(p) =index(q),

whereindex(p) =dimWs₍_p_).

Proof Suppose thatf has theC-stably limit shadowing property onCf(p). LetU be a compact neighborhood of Cf(p), and letU(f) be aC_{-neighborhood of}_f_{. Then for any}

g∈U(f),ghas the limit shadowing property ong(U) =

n∈Zgn(U). By Lemma ., for

anyq∈Cf(p)∩P(f),qis hyperbolic. By contradiction, suppose that there isq∈Cf(p)∩P(f)

such thatindex(p)=index(q). This implies that

dimWs(q) +dimWu(p) <dimM or dimWu(q) +dimWs(p) <dimM.

Then we can choose g∈U(f)∩KS(M) such thatindex(pg) =index(p) andindex(qg) =

index(q) for the continuationspg,qg∈g(U)∩P(g) ofp,q, respectively. Then we may

assume thatdimWs(qg) +dimWu(pg) <dimM. Other case is similar. Since g is Kupka-Smale,dimWs₍_qg_{) +}_dim_Wu₍_pg_{) <}_dim_M_{implies that}_Ws₍_qg₎_∩_Wu₍_pg_{) =}_∅_{. On the other}

hand, by the deﬁnition of theC-stably limit shadowing property, forpg,qg∈g(U)∩P(g),

Ws(qg)∩Wu(pg)=∅.

This is a contradiction and completes the proof.

Note that for any hyperbolic periodic pointqinCf(p) for a hyperbolic periodic pointp, there exist aC-neighborhoodU(f) off and a neighborhoodUofCf(p) such that for any

g∈U(f), there is uniquepg∈Cg(pg)∩P(g) which contained ing(U)∩P(g), wherepgis

the continuation ofpforg.

We denote theindex(p) byj( <j<dimM) and letPj(f|Hf(p)) be the set of periodic points

q∈Hf(p)∩P(f) such thatindex(q) =jfor all  <j<dimM. Set j(f) =Pj(f|Hf(p)), then

Hf(p) =j(f) =Cf(p).

Lemma . LetU(f) be the C_{-neighborhood of f given by Lemma} _._and

Proposi-tion.and letV(f)⊂U(f)be a small connected C-neighborhood of f.If g∈V(f) satis-fying g=f on M\Uj,then

index(q) =index(p)

(8)

Proof Suppose the property is not true then there areg∈V(f) andq∈g∩P(g) such

thatg=f onM\Ujandindex(q)=index(p). Suppose that (g)n(q) =q,i=index(q), and deﬁneϕ:V(f)→Zby

ϕ(g) =y∈g(U)∩P(g) :gn(y) =yandindex(y) =i

,

whereAis the number of elements ofA. By Lemma ., the functionϕ is continuous,

and sinceV(f) is connected, it is constant. But the property ofgimpliesϕ(g) >ϕ(f). This

is a contradiction, so that the lemma is proved.

For any> , denote byB(x,f) a-tubular neighborhood off-orbit ofx, that is,

B(x,f) =

y∈M:dfn(x),y<, for somen∈Z.

We say that a point x∈M is well closablefor f ∈Diff(M) if for any >  there are

g ∈Diff(M) withd(f,g) < andp∈M such thatp∈P(g), g =f on M\B(x,f) and

d(fn₍_x_),_gn₍_p₎₎_≤ _{for any }_≤_n_≤_π(_p_{), where}_π(_p_{) is the period of}_p_{, and}_d

is theC

-metric. Let f denote the set of well closable points off. Then we know the following

fact.

Lemma . ([, Theorem A]) For any f -invariant probability measure μ, we have

μ(f) = .

Proof of Theorem. Suppose thatf has theC_{-stably limit shadowing property on}_Cf₍_p_).

Then there are aC-neighborhoodU(f) off and a compact neighborhoodU ofCf(p)

as in the deﬁnition. Let U(f)⊂U(f) off given by Lemma . and Proposition ..

Deﬁne j as the set such that every periodic orbit in it has indexj. To get the

conclu-sion, it is suﬃcient to show thatj(f) is hyperbolic sinceHf(p) =Cf(p) =j(f), where

 <j=index(p) <dimM. NowCf(p) admits a dominated splittingTCf(p)M=E⊕F such thatdimE=index(p) by Proposition .(b). Thus, as in the proof of [, Theorem B], we can show that

lim inf n→∞Dxf

n_|

E(x)=  and lim inf

n→∞Dxf

–n_|

F(x)= ,

for allx∈Cf(p) and therefore the splitting is hyperbolic.

More precisely, we will prove the case oflim infn→∞Dxfn|E(x)=  (other case is simi-lar). It is enough to show that for anyx∈Cf(p), there existsn=n(x) >  such that

n–

j=

Dfm|E_{f mj}₍_x₎< .

If it is not true, then there isx∈Cf(p) such that

n–

j=

(9)

(10)

Thus,Cf(p) =Cf(p)∩(f) almost everywhere. Therefore,

(see [, pp.-]). On the other hand, by Proposition ., we see that

k–

The authors declare that they have no competing interests.

Authors’ contributions

The authors carried out the proof of the theorem and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, Mokwon University, Daejeon, 302-729, Korea.}2_{Department of Mathematics, Chungnam}

(11)

Acknowledgements

ML 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). JP was supported by BK21 math vision 2020 project.

Received: 27 December 2013 Accepted: 24 March 2014 Published:04 Apr 2014

References

1. Lee, K, Lee, M: Hyperbolicity ofC1_{-stably expansive homoclinic classes. Discrete Contin. Dyn. Syst.}₂₇_{, 1133-1145} (2010)

2. Lee, K, Moriyasu, K, Sakai, K:C1_{-Stable shadowing diﬀeomorphisms. Discrete Contin. Dyn. Syst.}₂₂_{, 683-697 (2008)} 3. Sakai, K:C1_{-Stably shadowable chain components. Ergod. Theory Dyn. Syst.}₂₈_{, 987-1029 (2008)}

4. Sakai, K: Pseudo-orbit tracing property and strong transversality of diﬀeomorphisms on closed manifolds. Osaka J. Math.31, 373-386 (1994)

5. Sambarino, M, Vieitez, J: OnC1_{-persistently expansive homoclinic classes. Discrete Contin. Dyn. Syst.}₁₄_{, 465-481} (2006)

6. Wen, X, Gan, S, Wen, L:C1_{-Stably shadowable chain components are hyperbolic. J. Diﬀer. Equ.}₂₄₆_{, 340-357 (2009)} 7. Pilyugin, SY: Shadowing in Dynamical Systems. Lecture Notes in Math., vol. 1706. Springer, Berlin (1999) 8. Lee, K: Hyperbolic sets with the strong limit shadowing property. Math. Inequal. Appl.6, 507-517 (2001)

9. Pilyugin, SY: Sets of dynamical systems with various limit shadowing properties. J. Dyn. Diﬀer. Equ.19, 747-775 (2007) 10. Ribeiro, R: Hyperbolicity and types of shadowing forC1_{generic vector ﬁelds. Discrete Contin. Dyn. Syst.}₃₄_,

2963-2982 (2014)

11. Franks, J: Necessary conditions for stability of diffeomorphisms. Trans. Am. Math. Soc.158, 301-308 (1971) 12. Sakai, K, Sumi, N, Yamamoto, K: Diffeomorphisms satisfying the specification property. Proc. Am. Math. Soc.138,

315-321 (2010)

13. Mañé, R: An ergodic closing lemma. Ann. Math.116, 503-540 (1982)

14. Shimomura, T: On a structure of discrete dynamical systems from the view point of chain components and some applications. Jpn. J. Math.15, 99-126 (1989)

15. Kulczycki, M, Kwietniak, D, Oprocha, P: On almost speciﬁcation and average shadowing properties. arXiv:1307.0120v1

10.1186/1687-1847-2014-104