Chaotic dynamics of continuous-time topological semi-flows on Polish spaces

0.11 MB·English

by  Xiongping Dai

Chaotic dynamics of continuous-time topological semi-flows on
Polish spaces
XiongpingDai
DepartmentofMathematics,NanjingUniversity,Nanjing210093,People’sRepublicofChina
5
1
0
2
Abstract

n
a DifferentlyfromLyapunovexponents,Li-Yorke,Devaneyandothersthatappearedintheliter-
J ature,weintroducetheconcept,chaos,foracontinuoussemi-flow f: R ×X → X onaPolish
+

9 spaceXwithametricd,whichisusefulinthetheoryofODEandisinvariantundertopological
equivalenceofsemi-flows. OurdefinitionisweakerthanDevaney’sonesincehere f mayhave

]
S neitherfixednorperiodicelements;butitimpliesrepeatedlyobservablesensitivedependenceon
D initialdata: thereisanǫ > 0suchthatforany x ∈ X,therecorrespondsadenseGδ-setSuǫ(x)in
X satisfying
.
h limsupd(ft(x), ft(y))≥ǫ ∀y∈Su(x).
at t→+∞ ǫ
m ThissensitivityisobviouslystrongerthanGuckenheimer’sonethatrequiresonlyd(ft(x), ft(y))≥
[ ǫ forsomemomentt>0andsomeyarbitrarilyclosetox.

1 Keywords: Semi-flow,sensitivity,chaos,quasi-transitivity,maximalchaoticsubsystem.
v
1 2010MSC: 34C28,37D45,37B05,37B20,54H20
2
1
2
0
1. Introduction
.
1
0 Lyapunovstabilityandchaoticbehaviorofmotions f(t,x)ofadynamicalsystemonametric
5
space(X,d)withcontinuous-time0≤t<∞havebeenthemostfascinatingaspectsinnonlinear
1
science.
:
v TheLyapunovstabilityofamotion f(t,x)isdefinedfromA.A.Markovasfollows:givenany
i
X ǫ >0thereexistsanumberδ> 0sothatforanyywithinadistanceδof x,d(f(t,x), f(t,y))<ǫ
forallt > 0; see [9, DefinitionV.8.03]. Althoughtherehavebeenvariousdefinitionsofchaos
r
a fordiscrete-timedynamicalsystemssinceLiandYorke1975,yetasfarasweknowthereisno
rigorous mathematical definition of chaos for continuous-timeintegrated flow or semi-flow of
a differential system on Riemannian manifolds like Rn. In fact, there are essential differences
between the chaotic dynamics of discrete-time and continuous-time dynamical systems. For
example, if X has no isolated points and if a topologically transitive continuous map T of X
Emailaddress:[email protected](XiongpingDai)
PreprintsubmittedtoJDE(ReceivedMay312013;revised23November2014) January12,2015
1 INTRODUCTION 2
has a fixed or periodicpoints, then T is Li-Yorkechaotic; thatis, one can find an uncountable
scrambledsetS ⊂ XforT suchthatforallx,y∈S with x,y,
liminfd(Tn(x),Tn(y))=0 and limsupd(Tn(x),Tn(y))>0
n→+∞ n→+∞
bytheworkofW.HuangandX.Ye2002[6]. However,thisisnotthecaseforcontinuous-time
flows as shown by a counterexample as follows: if f: R × T1 → T1 is the continuous-time
rotation of the unit circle T1, then it is topologically transitive and the underlying space T1
exactlyconsistsofaperiodicorbit,butdoesnothaveanyYi-Yorke’sscrambledpair(x,y).
In this paper, we will introduce a kind of chaos which may characterize the opposite side
of stability—instability: if the motion f(t,x) is chaotic in a region Σ, then there exists an
ǫ > 0 such that for any δ > 0 one can find some y ∈ Σ within a distance δ of x such that
limsup d(f(t,x), f(t,y)) ≥ ǫ. Thatisto say, f(t,x) hasthesensitive dependenceoninitial
t→+∞
condition. Importantly, this sensitivity is repeatedly observable; since we can choose a Borel
subsetO of[0,∞)withLebesquemeasure∞suchthatd(f(t,x), f(t,y))≥ǫ forallt∈O.
Moreprecisely,let f: R×Rn → Rn betheintegratedflowofacompleteC1-vectorfield X
onthen-dimensionaleuclideanspaceRn;i.e.,
d
f(t,x)= X(f(t,x)), ∀t∈Randx∈Rn.
dt
To study the unpredictability of the dynamical behavior of a given Lagrangian stable motion
f(t,x)ast →+∞,theclassicalwayinstatisticalmechanicsistoseeiftheLyapunovexponent,
definedas
1 ∂f(t,x)
χ(f,x)=limsup log ,
t ∂x
t→+∞ (cid:13) (cid:13)
(cid:13) (cid:13)
ispositiveornot.Differentlyfromthisanalyticmetho(cid:13)d,wepres(cid:13)entatopologicalformulationof
(cid:13) (cid:13)
theunpredictabilityofthemotion f(t,x).
LetΣ betheclosureoftheforwardorbit f([0,∞),x)ofthemotion f(t,x),forany x ∈ Rn.
x
Wesay f(t,x)ischaoticinΣ ifthefollowingtwotopologicalconditionsaresatisfied:
x
• (Minimal-setsdensity)the(Birkhoff)recurrentmotionsaredenseinΣ ;
x
• (Non-minimality)Σ isnotminimal.
x
Thenwecanobtainthefollowingsensitivitytheorem:
TheoremA. IftheLagrangianstablemotion f(t,x)ischaoticinΣ ,thenitisobservablysensi-
x
tiveinthesensethatthereexistsanǫ >0havingtheproperty: foranyδ>0,onecanfindsome
y ∈ Σ withkx−yk < δandaBorelsubsetO = O(y)of[0,∞)withLebesquemeasure∞,such
x
thatkf(t,x)− f(t,y)k>ǫ foreveryt∈O.
WenoteherethatsincetheC1-vectorfieldXisboundedrestrictedtoΣ ,toproveTheoremA
x
itissufficienttoprovethefollowingweakresult:
PropositionB. IftheLagrangianstablemotion f(t,x)ischaoticinΣ ,thenthereexistsanǫ >0
x
suchthatforanyδ>0,onecanfindsomey∈Σ withkx−yk<δsothatlimsup kf(t,x)−
x t→+∞
f(t,y)k>ǫ.
1 INTRODUCTION 3
Consequentlyifthemotion f(t,x)ischaotic,thenitisnotLyapunovasymptoticallystable.
Infact,wewillprovethispropositioninamoregeneralframework;seeTheorem4inSection2.
As shownby the motionof the irrationalrotationon the 2-dimensionaltorusT2, the weak
non-minimalityconditionis notignorablefor our chaos. In addition, the minimal-setsdensity
conditionisnotignorabletoo.
1.1. Topologicalsemi-flows
Wenowformalizeourstudyframeworkinthesequelofthispaper. LetX beaPolishspace
(i.e.,acompleteseparablemetricspace)withametricd,andletR =[0,∞)bethenonnegative
+
timeaxisofthedynamicalsystemsweconsiderhere. Fromnowonwelet
f: R ×X →X; (t,x)7→ f(t,x)= ft(x)
+
beacontinuoussemi-flowonX;thatis, f(t,x)satisfiesthefollowingthreeconditions.
(I) Theinitialcondition: f(0,x)= xforallx∈X.
(II) The condition of continuity with respect to the variables t and x: if there be given two
convergent sequences t → t in R and x → x in X, then f(t ,x ) → f(t ,x ) as
n 0 + n 0 n n 0 0
n→∞.
(III) Thesemigroupcondition: f(t , f(t ,x))= f(t +t ,x)foranyx∈ X andanytimest ,t ∈
2 1 1 2 1 2
R .
+
Thenthe pair (X, f) is namedas a continuous-timetopologicaldynamicalsystem. Thisframe-
workisapplicableforthetheoryofordinarydifferentialequations.
Fromcondition(II)thereisobtainedasacorollarythefollowingproperty:
(II)′ Foranyq∈X,anynumberT >0(arbitrarilylarge)andanyε>0(arbitrarilysmall),there
can be founda number δ = δ(q,T,ε) > 0 such that if d(q,x) < δ and 0 < t ≤ T, then
d(f(t,q), f(t,x))<ε.
Proof. Ifthispropertywerefalse, thentherecouldbefoundasequenceofpointsq → qin X
n
andacorrespondingsequenceofnumberst →t in[0,T],suchthatd(f(t ,q), f(t ,q ))≥αfor
n 0 n n n
someconstantsα>0. Thenbythetriangleinequality,
d(f(t ,q), f(t ,q ))≤d(f(t ,q), f(t ,q))+d(f(t ,q), f(t ,q )).
n n n n 0 0 n n
Hencebycondition(II),itfollowsthatd(f(t ,q), f(t ,q)) ≥ α,acontradiction. Thisprovesthe
0 0
assertion.
Asusual,apointp∈ Xiscalledaperiodicpointofperiodπof f if fπ(p)= pand ft(p), p
forall0 < t < π,forsomenumberπ > 0;byPer(f)wemeanthesetofallperiodicpointsof f.
If ft(p) = pforallt > 0then piscalledafixedpointof f andwewriteFix(f)forthesetofall
fixedpointsof f.
Werecallthat f istopologicallytransitiveifthereisan(strictlypositive)orbitof f
O (f,x):= ft(x)|ε≤t<∞ , forsomeε>0,
ε+
that is dense in X; i.e., O (f,x) =(cid:8)X. Accordingto(cid:9)a well-knowntheorem of Birkhoff, there
ε+
followsthat
1 INTRODUCTION 4
• (X, f)istopologicallytransitiveifandonlyifforeverypairofnonemptyopensubsetsU
andV ofX,therefollowsV∩ ft(U),∅forsomet≤−1.See,e.g.,[11,Theorem5.9]for
thediscrete-timeZ-actioncase.
Proof. Let (X, f) be topologically transitive with O (f,x) = X and U,V two non-void open
ε+
subsetsofX. Thenonecanfind0<t1 <t1+2<t2suchthat ft1(x)∈V and ft2(x)∈U,andthus
ft1−t2(U)∩V ,∅.
Conversely,assumethatwheneverU,V arenonemptyopensetsthenthereexistst≤−1with
V ∩ ft(U) , ∅. Let U ,U ,... be a countable base for X. Then x∈X|O (f,x)= X =
1 2 1+
∞ ft(U )and ft(U )isclearlydensebythehypothesis. Hencetheresult
n=1 −∞<t≤−1 n −∞<t≤−1 n (cid:8) (cid:9)
follows.
T S S
Wenowintroduceakindofweakertransitivity.
Definition 1. The topological semi-flow f: R × X → X is said to be topologically quasi-
+
transitiveifthereisan(positive)orbitof f,O (f,x):= ft(x)|0≤t<∞ ,whichisdenseinX,
+
i.e.,O (f,x)= X.
+ (cid:8) (cid:9)
Itshouldbenotedthatsincehereeverysamples f(t,·): X →Xarenotnecessarilysurjective,
the abovetwo transitivity propertiesare different. For example, the closure of a positive orbit
itself is topologically quasi-transitive, but not topologically transitive except it is Poissonwise
recurrent. Theessentialdifferenceisthatif f istopologicallytransitivethereisaresidualsetof
transitivepoints;butthisisnotthecasefortopologicalquasi-transitivity.
Asusual,apoint p ∈ Fix(f)∪Per(f)iscalledacriticalelementof f anditscorresponding
closedorbitO (f,p)acriticalorbitof f.
+
Recall from [9, V.7.05] that a motion f(t,x) is called recurrent in the sense of Birkhoff if
foranyε > 0 there existsa T = T(ε) > 0 such thatforanytwo momentsτ,υ ∈ R therecan
+
be found a number t with τ < t < τ+T such that d(fυ(x), ft(x)) < ε. Clearly the fixed and
periodicmotions f(t,p)bothareBirkhoffrecurrent.WenoteherethataBirkhoffrecurrentpoint
issometimescalledanalmostperiodicpointorauniformlyrecurrentpointforthediscrete-time
dynamicalsystemsinmostoftheliterature;see,e.g.,[4]and[11].
Since the underlying space X of our topological dynamical system (X, f) is complete, the
connectionbetweenrecurrentmotionsandminimalsetsisestablishedbythefollowingtheorem
ofBirkhoff:
• Amotion f(t,x)isBirkhoffrecurrentifandonlyifO (f,x)isacompactminimalinvariant
+
setof f;see,e.g.,[9,TheoremsV.7.06andV.7.07].
ItiseasytoshowthataBirkhoffrecurrentmotionisPoissonstableandLagrangianstable.1 In
addition,wesaytwomotions f(t,x)and f(t,y)arefarawayeachotherifO (f,x) O (f,y) =
+ +
∅. Soanytwodistinctcriticalorbitsarefarawayeachother.
T
1.2. Definitionofchaosoftopologicalsemi-flow
To describethe unpredictabilityand disorderof the dynamicalbehaviorof the continuous-
timedynamicalsystem(X, f),differentlyfromLi-Yorke[8]andDevaney[2]butsimilartoGlas-
nerandWeiss1993[3],wenowintroduceatypeofchaosstatedasfollows:
1Amotion f(t,x)iscalledPoissonstableifxitselfisanω-limitpointof f(t,x),i.e. forsomesequencetn ↑∞we
have f(tn,x)→x;anditissaidtobeLagrangianstableifO+(f,x)iscompact.
1 INTRODUCTION 5
Definition 2. The continuous-timetopologicaldynamicalsystem (X, f) is called chaotic, pro-
videdthatthefollowingthreetopologicalconditionsaresatisfied:
(1) (Topologicalquasi-transitivity) f istopologicallyquasi-transitiveonX.
(2) (Minimal-setsdensity)theBirkhoffrecurrentpointsof f aredenseinX.
(3) (Non-minimality)thereareatleasttwomotionsof f farawayeachother;orequivalently,
(X, f)isnotminimalitself.
Thisdefinitionis mainlymotivatedby RobertL. Devaney’schaos[2] thatrequiresthe fol-
lowingadditionalcondition:
(4) Fix(f)∪Per(f)= X.
Weshouldnoteherethatcondition(1)togetherwith(2)impliesthat
(1)′ f istopologicallytransitiveonX.
HoweverourdefinitionofchaosisessentiallyweakerthanDevaney’s,becausehereourtopolog-
icalsemi-flow f possiblyhasneitheranyfixedpointsnorperiodicpointsinX. Ifthereholdsthe
unnecessarilyrestrictivecondition(4)asinDevaney’schaos,thenourcondition(2)issatisfied
trivially.
We note here that if X has no isolated points, conditions (1)′ and (4) imply condition (3)
for any discrete-time dynamicalsystem. However this is not the case for our continuous-time
context. For example, if f is the continuous-time rotation of the unit circle T1 as mentioned
before;conditions(1)′,(2)and(4)allthenholdexceptcondition(3).
Fromthe aboveDefinition2, itis easy tosee thatourchaosis preservedundertopological
equivalenceof semi-flows. Thatis to say, if g: R ×Y → Y is anothertopologicaldynamical
+
system and h: X → Y is a homeomorphismsuch thath transportsorbitsof f ontoorbitsof g,
i.e.,h(O (f,x))=O (g,h(x))foreveryx∈ X,then(Y,g)ischaoticifandonlyifsois(X, f).
+ +
1.3. Observablesensitivity
For a continuous transformation T: X → X, the topological transitivity together with the
densityofperiodicpointsofT impliestheinterestingproperty:
• Guckenheimer sensitivity [5]: there exists a positive ǫ such that for all x ∈ X and all
neighborhoodsUofx,thereissomey∈Uandforsomeintegern>0,d(Tn(x),Tn(y))≥ǫ.
That is to say, T is not Lyapunov stable at every point x ∈ X. See, e.g., [1, 10, 3] for
differentproofsofitand[7]foranextensiontoC-semigroupactions.
However,foramechanicalsystem(X, f),oneoftenexpectsthatanimportantphenomenoncan
be observed repeatedly; in other words, it is expected that to the continuous transformation
T: X → X, for“sufficientlymany”initialvaluesy thereare “sufficientlymany”timesn such
k
thatd(Tnk(x),Tnk(y))≥ǫ forallk.
Forthis,wenowintroduceatypeofobservablesensitivity,whichisthemostinterestingpart
ofourchaosdescribedbyDefinition2.
1 INTRODUCTION 6
Definition3. Thetopologicaldynamicalsystem(X, f)issaidtohavesensitive dependenceon
initialdata,providedthatonecanfindasensitiveconstantǫ > 0suchthatforany x ∈ X,there
existsadenseG -setSu(x)inX sothat
δ ǫ
limsupd(ft(x), ft(y))≥ǫ
t→+∞
foreachy∈Su(x).
ǫ
Here Su(x) is a large set from the topologicalpointof view. So accordingto Definition 3,
ǫ
forany∆ > 0andanyy ∈ Su(x)wecanfindmomentst ≥ ∆suchthatd(ft(x), ft(y)) > ǫ. This
ǫ
showsthatoursensitivityisobservablerepeatedly.ThisisjusttheoppositesideoftheLyapunov
asymptoticalstability.
AlthoughourchaosdefinitionlooksweakerthanDevaney’schaos,itstillimpliesthecentral
idea in chaos—sensitive dependence on initial data in the sense of Definition 3. This will be
provedinSection2;seeTheorem4below.
Herewefurtheraskthefollowingquestion: If(X, f)ischaotic,doesthesetofǫ-observable
times,
O(y)= t>0|d(ft(x), ft(y))≥ǫ
foranyǫ,xandyasinDefinition3,havetheLebesquemeasure∞?
(cid:8) (cid:9)
Aswepointedoutbefore,if f(t,x) istheintegratedflowofa boundedC1-vectorfield, this
holdstrivially.Generally,ifXiscompact,thenO(y)hastheLebesquemeasure∞.
SimilarlytoLi-Yorkescrambledset,bySu wemeantheu-scrambledBorelsubsetof(X, f)
suchthat
limsupd(ft(x), ft(y))>0 ∀x,y∈Suwith x,y.
t→+∞
We willprovethatSu isuncountableanddensein X if(X, f) ischaoticinthesense ofDefini-
tion2;seeTheorem5inSection3.
1.4. Completelynon-chaoticsystem
Let Λ be a subset of X; then (Λ, f) is called a subsystem of (X, f) if Λ is an f-invariant
closedsubsetofX.Thesystem(X, f)issaidtobecompletelynon-chaoticifitdoesnothaveany
subsystemsthatarechaoticintheirrightinthesenseofDefinition2. InSection4,wewillshow
thatif(X, f)isnotcompletelynon-chaotic,thenthereisatleastonemaximalchaoticsubsystem;
seeTheorem7below.
1.5. Li-Yorke’schaos
From the interesting work of Huang and Ye 2002 [6], it follows that for any continuous
transformationT fromX intoitself,Devaney’schaosimpliesLi-Yorke’schaos;thatis,onecan
findanuncountablescrambledsetS ⊆ Xinthesensethat
liminfd(Tn(x),Tn(y))=0 and limsupd(Tn(x),Tn(y))>0
n→+∞ n→+∞
foranyx,y∈S withx,y.
Wenowendthisintroductorysectionwiththefollowing
Question1. Ifthecontinuous-timetopologicaldynamicalsystem(X, f)ischaoticinthesense
ofDefinition2,doesitappeartheLi-YorkechaoticphenomenoninX?
InHuangandYe’sproof,theperiodicpointandtopologicaltransitivitybothplayimportant
roles. However, in our context, although f is topologically transitive by (1)′, here f does not
needtohaveanyperiodicpointsorfixedpoints. Sothereneedsanewideaforthisquestion.
2 SENSITIVEDEPENDENCEONINITIALDATA 7
2. Sensitivedependenceoninitialdata
Let f: R ×X →Xbethecontinuous-timetopologicalsemi-flowonthePolishspace(X,d)
+
asbefore.RecallfromDefinition3that(X, f)issaidtohavesensitivedependenceoninitialdata
ifthereisaconstantǫ > 0suchthattoevery x ∈ X thereisadenseG -setSu(x)ofX suchthat
δ ǫ
limsup d(ft(x), ft(y))≥ǫ forally∈Su(x). HerebyaG -setinXwemeanasetwhichcan
t→+∞ ǫ δ
beexpressedastheintersectionofcountablemanyopensetsofX.
Using statistical propertyof a recurrentmotion, importantlythere followsthe sensitive de-
pendenceoninitialdatafromourdefinitionofchaosasfollows.
Theorem4. Let(X, f)bechaoticinthesense ofDefinition2. Thentherefollowsthesensitive
dependenceoninitialdatainthesenseofDefinition3.
Proof. Since X contains at least two motions of f far away each other from condition (3) of
Definition2,onecanfindanumberδ > 0suchthatforall xˆ ∈ X thereexistsacorresponding
0
motion,say f(t,q ),notnecessarilyrecurrentbutdependentofxˆ,suchthat
xˆ
d xˆ,O (f,q ) ≥δ ,
+ xˆ 0
where d(xˆ,A) = inf d(xˆ,a) for any(cid:0)subset A of(cid:1)X. We will show that f has the sensitive
a∈A
dependenceoninitialdatawithsensitivityconstantǫ =δ /12inthesenseofDefinition3.
0
Writesimplyδ=δ /4anddefineforanyx∈ XtheBorelset
0
Cu(x)= y∈X limsupd(ft(x), ft(y))>ǫ .
ǫ
(cid:26) t→+∞ (cid:27)
(cid:12)
NextwewillprovethatCu(x)isdensein(cid:12)Xforeachx∈ X.
ǫ
Forthis,weletx,xˆbetwoarbitrarypointsinX andletU beanarbitraryneighborhoodof xˆ
inX.
Since the Birkhoffrecurrentmotionsof (X, f) are dense in X fromcondition(2)of Defini-
tion 2, there existsa Birkhoffrecurrentpoint p ∈ U ∩ B (xˆ), where B (xˆ) is the openballof
δ/2 r
radiusrcenteredatxˆinX. Aswenotedabove,theremustexistanotherpointq=q ∈ Xwhose
xˆ
orbitO (f,q)isofdistanceatleast4δfromthegivenpointxˆ.
+
Let η > 0 be such that η < δ/2. Then from the Birkhoff recurrenceof the motion f(t,p),
it followsthatone can find a constantT = T(η,p) > 0 such that forany γ ≥ 0, there is some
momentt ∈[γ,γ+T)verifyingthat
γ
d(p, ftγ(p))<η.
Wesimplywrite
V = f−t(B (ft(q))), where f−t(·)= f(t,·)−1: X →X.
δ
t∈[0,2T)
\
Clearlyfromcondition(II)′ ofthetopologicalsemi-flow f, itfollowsthatV isaneighborhood
ofqinXbutnotnecessarilyopen,anditisnonemptysinceq∈V.
Since f istopologicallyquasi-transitiveonXbycondition(1)ofDefinition2,thereexistsat
leastonepointz∈U∩B (xˆ)suchthat fN(z)∈V forsomesufficientlylargenumberN ≫T. Let
δ
N = jT −r, where0≤r<T and j∈N,
2 SENSITIVEDEPENDENCEONINITIALDATA 8
andchoosebyγ= jT
tjT ∈[jT,(j+1)T) suchthatd(p, ftjT(p))<η.
Then0≤t −N <2T.
jT
Bytheaboveconstruction,onehas
ftjT(z)= ftjT−N(fN(z))∈ ftjT−N(V)⊆ Bδ(ftjT−N(q)).
Fromthetriangleinequalityofmetricandd(p, ftjT(p))<η,itfollowsthat
d(ftjT(p), ftjT(z))≥d(p, ftjT(z))−d(p, ftjT(p))
≥d(xˆ, ftjT(z))−d(p,xˆ)−η
≥d xˆ, ftjT−N(q) −d ftjT−N(q), ftjT(z) −d(p,xˆ)−η.
Consequently,sinceη≤δ/2, p∈(cid:0)Bδ/2(xˆ)and (cid:1)ftjT(z)(cid:0)∈ Bδ(ftjT−N(q)),(cid:1)itholdsthat
d ftjT(p), ftjT(z) ≥2δ.
Thereforefromthetriangleinequality(cid:0)ofmetricagain(cid:1),onecanobtaineither
d(ftjT(xˆ), ftjT(z))≥δ
or
d(ftjT(xˆ), ftjT(p))≥δ.
Repeating this argumentfor anotherlikewise N bigger than (j+2)T, one can find a sequence
tn = jnT ↑ +∞asn → +∞suchthateitherd(ftn(xˆ), ftn(z)) ≥ δord(ftn(xˆ), ftn(p)) ≥ δ,forall
n≥1. Thusineithercase,wehavefoundapointyˆ ∈U suchthat
limsupd(ft(xˆ), ft(yˆ))≥δ=3ǫ.
t→+∞
Usingthetriangleinequalityoncemore,weseeeither
limsupd(ft(x), ft(yˆ))>ǫ
t→+∞
or
limsupd(ft(x), ft(xˆ))>ǫ.
t→+∞
Since xˆ,U botharearbitraryandyˆ ∈U,Cu(x)isdenseinX.
ǫ
Finallyforanyintegern≥1andanyx∈X,let
Wu(x,n)= y∈X|d(ft(x), ft(y))>ǫ forsomet∈[n,∞) .
ǫ
Since f(t,x)iscontinuousfro(cid:8)mcondition(II)inSection1.1,Wu(x,n)isope(cid:9)ninX. Nowforany
ǫ
x∈Xwelet
∞ ∞
Su(x)= Wu(x,n) .
ǫ ǫ
!
k=1 n=k
\ [
ThenSu(x)isaG -setinX. Inaddition,itiseasytoseethatCu(x)⊆Su(x). Therefore,Su(x)is
ǫ δ ǫ ǫ ǫ
adenseG -setinX.
δ
ThisthuscompletestheproofofTheorem4.
3 ONTHEU-SCRAMBLEDSET 9
TheaboveproofofTheorem4hasbeenmotivatedbythesurprisingworkofBanksetal.[1]
forDevaney’schaosinthediscrete-timedynamicalsystemcase. However,withoutthedensityof
periodicpointsandthetopologicaltransitivityof f,wehereneedsomeessentialimprovements
oftheproofof[1].
If(X, f)istopologicallytransitive,notsensitiveoninitialconditionsinthesenseofGucken-
heimer,andXhasnoisolatedpoint,thenforeveryε>0thereexistatransitivepointx ∈ Xand
0
aneighborhoodU of x suchthatforally ∈ U,wehaved(ft(x ), ft(y)) ≤ εforeveryt > 0[3,
0 0
Lemma1.1].Thisequi-continuityof(ft) atx isthekeypointforGlasnerandWeissin[3]to
t>0 0
provetheGuckenheimersensitivityofDevaney’schaosusingergodicapproach.However,inour
situation,thereisanobstructionforustoemployGlasnerandWeiss’ ergodicapproach: When
(X, f) is notsensitive in the sense of ourDefinition 3, we cannotobtain the equi-continuityof
(ft) atanytransitivepoints,becauseforeveryy ∈ U wecanonlygetd(ft(x ), ft(y)) ≤ εfor
t>0 0
t≥t(y,ε). Heret(y,ε)isnotnecessarilyuniformwithrespecttoy∈U.
In[3],GlasnerandWeissintroducedaverygeneralmeasure-theoreticcondition:
(5) Thereexistsan f-invariantprobabilitymeasureonX,whichispositiveoneverynon-empty
openset.
Thenfollowingthesameargumentasin[3, Theorem1.3],fromthesyndeticityofreturntimes
wecaneasilyobtainthecontinuous-timeversionofGlasnerandWeiss’theorem:
Theorem(GlasnerandWeiss). Let(X, f) bea continuous-timetopologicalsemi-flow. Ifthere
hold the conditions (1)′,(5) and (3), then (X, f) has the sensitivity on initial conditions in the
sense of Guckenheimer; that is, there exists a constant ǫ > 0 such that for all x ∈ X and all
neighborhoodsU ofx,thereissomey∈U andforsomet>0,d(ft(x), ft(y))≥ǫ.
Recallthatapointx∈ Xiscalledregularforthesemi-flow f ifitisagenericpointofsome
f-invariantBorelprobabilitymeasureµwithµ(U) > 0foreveryopenneighborhoodU of x in
X. Clearly,ifcondition(5)holds,thentheregularpointsaredenseinX. HoweversinceX isnot
necessarilycompact,theconverseisnotnecessarilytrue. Wenowasknaturallythefollowingat
theendofthissection.
Question2. Ifthecontinuous-timesemi-flow(X, f)satisfiesconditions(1)and(3)suchthatthe
regularpointsaredenseinX,doesithavethesensitivedependenceoninitialdatainthesenseof
Definition3orGuckenheimer?
3. Ontheu-scrambledset
Throughoutthissection,let(X, f)beachaoticcontinuous-timetopologicalsemi-flowinthe
senseofDefinition2,wherethestatespaceX isaPolishspaceasinSection1.1. Clearly,X has
noisolatedpointsfromDefinition2.
ForarelationR⊂ X×Xandx∈ X,writeR(x)={y∈ X|(x,y)∈R}. Thenweneedalemma.
Lemma 1(HuangandYe[6, Lemma3.1]). IfRisasymmetric relationwiththepropertythat
thereisadenseG subset Aof X suchthatforeach x ∈ A, R(x)containsadenseG subsetof
δ δ
X, thenthereisadenseuncountablesubset Bof X suchthat(x,y) ∈ Rwhenever x,y ∈ Bwith
x,y.
AsasimpleconsequenceofthestatementsofTheorem4andLemma1above,wecaneasily
obtainthefollowing.
4 MAXIMALCHAOTICSUBSYSTEMS 10
Theorem5. Forthechaoticdynamicalsystem(X, f),itsu-scrambledsetSuisuncountableand
denseinX.
Proof. LetthesymmetricrelationRbedefinedby(x,y)∈Rifandonlyiflimsup d(ft(x), ft(y))>
t→+∞
0. ThenfromTheorem4provedinSection2,itfollowsthatforanyx∈X,R(x)containsadense
G subsetof X sinceSu(x) ⊆ R(x)forsomeǫ > 0. So,thestatementcomesimmediatelyfrom
δ ǫ
Lemma1.
ThiscompletestheproofofTheorem5.
4. Maximalchaoticsubsystems
Inapplications,itoftenappearsthatthecontinuous-timetopologicaldynamicalsystem(X, f)
itselfisnotchaoticinthesenseofDevaneyandevenofDefinition2,butactuallytherearechaotic
subsystemssuchasAxiomAflows. Forthat,wenowintroducethefollowingconcept.
Definition6. LetΛbean f-invariantclosedsubsetof X. If(Λ, f)itselfischaoticinthesense
ofDefinition2andmoreoverthereisnootherlikewisechaoticsubsetthatproperlycontainsΛ,
thenΛiscalledamaximalchaoticsubset/subsystemof(X, f).
Thissectionwillbedevotedtoprovingthefollowingexistencetheoremofmaximalchaotic
subsystem.
Theorem 7. If (X, f) is a continuous-timetopologicalsemi-flow which is not completely non-
chaotic,thentherealwaysexistsamaximalchaoticsubsystemof(X, f).
Proof. Let D be the family of all chaotic subsets of (X, f). It is nonempty, since (X, f) is not
completelynon-chaotic. We nowequipD withapartialorderasfollows: A (cid:22) Bifandonlyif
A⊆ B,foranyA,B∈D. LetC beanygiventotallyorderedchainof(D,(cid:22))andset
F = C.
C∈C
It is easy to see that F as a subspace of X is[a complete separable metric space and that F is
f-invariant.WethenclaimthatF ∈D.
In fact, we need only check that the semi-flow f is topologicallytransitive restricted to F.
Forthat,letU andV betwonon-emptyopensubsetsofF. ThenonecanfindtwoopensetsU′
andV′ofXsuchthat
U =U′∩F and V =V′∩F.
SothereareatleasttwoelementsC ,C ∈C suchthat
1 2
U′∩C ,∅ and V′∩C ,∅.
1 2
Since C is totally ordered under (cid:22), we have either C ⊆ C or C ⊆ C . Without loss of
1 2 2 1
generality,weassumeC ⊆C . Thenfrom(1)′,itfollowsthat
1 2
U′′ :=U′∩C ,∅ and V′′ :=V′∩C ,∅.
2 2
As(C , f)itselfischaoticfromthedefinitionofD,itfollowsthat
2
V′′∩ ft(U′′),∅ forsomet≥1.
ThereforeV ∩ ft(U),∅andso f istopologicallytransitiverestrictedtoF.
ThisthuscompletestheproofofTheorem7.