DissipativeDynamicalSystems
Therearemanydiscrete-timepopulationmodelsgovernedbydifferenceequations(ormaps),andaswementionedinthePreface,thedynamicsofaperiodicdifferentialsystemcanbeinvestigatedviaitsassociatedPoincar´emap. Theaimofthischapteristointroducebasicdefinitionsanddevelopmain toolsinthetheoryofdiscretedynamicalsystems.InSection 1.1 wepresent conceptsoflimitsetsandattractorsandsomefundamentaltheoremssuchas theLaSalleinvarianceprinciple,theasymptoticfixedpointtheorem,andthe globalattractortheorems.
Chaintransitivityhasremarkableconnectionstothestructureofattractors.InSection 1.2 wefirstgivetypicalexamplesandcharacteristicsofchain transitivesets.Thenweshowthatthe Butler–McGeheepropertiesofomega limitsetsaresharedbychaintransitivesetsforadynamicalsystem,which enableustoobtainfurtherimportantpropertiesofchaintransitivesetssuch asstrongattractivityandconvergence,andtoprovetheequivalencebetween acycliccoveringsandMorsedecompositions.
Uniformpersistenceisanimportantconceptinpopulationdynamics,since itcharacterizesthelong-termsurvivalofsomeorallinteractingspeciesinan ecosystem.Lookedatabstractly,itis thenotionthataclosedsubsetofthe statespaceisrepellingforthedynamicsonthecomplementaryset,andthen itgivesauniformestimateforomegalimitsets,whichsometimesisessentialtoobtainamoredetailedglobaldynamics.InSection 1.3 weprovea strongrepellertheoremintermsofchaintransitivesets,whichunifiesearlier resultsonuniformpersistence,andimpliesrobustnessofuniformpersistence. Thenweshowthatadissipative,uniformlypersistent,andasymptotically compactsystemmusthaveatleastonecoexistencesteady state,whichprovidesadynamicapproachtosomestaticproblems(e.g.,existenceofpositive steadystatesandperiodicsolutions). Wealsointroducetheconceptofgeneralizeddistancefunctionsinabstractp ersistencetheoryso thatthepractical persistencecanbeeasilyobtainedforcertaininfinite-dimensionalbiological systems.
InSection 1.4 wediscusspersistenceunderperturbations.Weproveageneralresultontheperturbationofagloballystablesteadystate.Thenweprove uniformpersistenceuniforminparamete rs,whichisveryusefulinestablishingtherobustnessofglobalasymptoticstabilityofanequilibriumsolution.A dissipativeanduniformlypersistentsystemisoftensaidtobepermanent.For aclassofautonomousKolmogorovsystemsofordinarydifferentialequations wealsoobtainarobustpermanencetheorem.
1.1LimitSetsandGlobalAttractors
Let N bethesetofintegersand N+ thesetofnonnegativeintegers.Let X be acompletemetricspacewithmetric d and f : X → X acontinuousmap.For anonemptyinvariantset M (i.e., f (M )= M ),theset W s (M ):= {x ∈ X : limn→∞ d(f n (x),M )=0} iscalledthestablesetof M .Theomegalimitsetof x isdefinedintheusualwayas ω (x)= {y ∈ X : f nk (x) → y, forsome nk → ∞}.Anegativeorbitthrough x = x0 isasequence γ (x)= {xk }0 k=−∞ such that f (xk 1 )= xk forintegers k ≤ 0.Theremaybenonegativeorbitthrough x,andevenifthereisone,itmaynotbeunique.Ofcourse,apointofan invariantsetalwayshasatleastonenegativeorbitcontainedintheinvariant set.Foragivennegativeorbit γ (x)wedefineitsalphalimitsetas α(γ )= {y ∈ X : xnk → y forsome nk →−∞}.If γ + (x)= {f n (x): n ≥ 0} (γ (x)) isprecompact(i.e.,itiscontainedinacompactset),then ω (x)(α(γ ))is nonempty,compact,andinvariant(see,e.g.,[141,Lemma2.1.2]).
Let e ∈ X beafixedpointof f (i.e., f (e)= e).Recallthat e issaidtobe stablefor f : X → X ifforeach > 0thereexists δ> 0suchthatforany x ∈ X with d(x,e) <δ ,wehave d(f n (x),e) < , ∀n ≥ 0.Thefollowingsimple observationisusefulinprovingtheconvergenceofaprecompactpositiveorbit toafixedpoint.
Lemma1.1.1. (Convergence) Let e beastablefixedpointand γ + (x) a precompactpositiveorbitfor f : X → X .If e ∈ ω (x),then ω (x)= {e}
Proof. Let > 0begiven.Bystabilityof e for f : X → X ,thereexists δ> 0 suchthatforany y ∈ X with d(y,e) <δ ,wehave d(f m (y ),e) < , ∀m ≥ 0. Since e ∈ ω (x),thereisasubsequence nk →∞ with f nk (x) → e,andhence anindex k0 suchthat d(f nk0 (x),e) <δ .Thus d(f nk0 +m (x),e) < , ∀m ≥ 0, whichimpliesthat ω (x) ⊂{z ∈ X : d(z,e) ≤ }, ∀ > 0.Letting → 0,we get ω (x)= {e}.
Definition1.1.1. Let G beaclosedsubsetof X .Acontinuousfunction V : G → R issaidtobeaLiapunovfunctionon G ofthemap f : G → G (orthediscretesystem xn+1 = f (xn ),n ≥ 0)if ˙ V (x):= V (f (x)) V (x) ≤ 0 forall x ∈ G.
Theorem1.1.1. (LaSalleinvarianceprinciple) Assumethat V isaLiapunovfunctionon G of f ,andthat γ + (x) isaprecompactorbitof f and
γ + (x) ⊂ G.Then ω (x) ⊂ M ∩ V 1 (c) forsome c = c(x),where M isthe largestinvariantsetin E := {x ∈ G : ˙ V (x)=0},and V 1 (c):= {x ∈ G : V (x)= c}.
Proof. Clearly,thecontinuousfunction V isboundedonthecompactset
γ + (x) ⊂ G.Let xn = f n (x),n ≥ 0.Then
V (xn+1 ) V (xn )= V (f (xn )) V (xn )= V (xn ) ≤ 0, andhence V (xn )isnonincreasingwithrespectto n andisboundedfrom below.Therefore,thereisarealnumber c = c(x)suchthatlimn→∞ V (xn )= c
Forany y ∈ ω (x) ⊂ G,thereisasequence nk →∞ suchthatlimk→∞ xnk = y .
Since V iscontinuous,limk→∞ V (xnk )= V (y )= c,and ω (x) ⊂ V 1 (c).Since ω (x)isinvariant, V (f (y ))= c and ˙ V (y )=0.Therefore, ω (x) ⊂ E ,andhence ω (x) ⊂ M .
Recallthataset U in X issaidtobeaneighborhoodofanotherset V providedthat V iscontainedintheinteriorint(U )of U .Foranysubsets A,B ⊂ X andany > 0,wedefine
d(x,A):=inf y ∈A d(x,y ),δ (B,A):=sup x∈B d(x,A),
N (A, ):= {x ∈ X : d(x,A) < } and N (A, ):= {x ∈ X : d(x,A) ≤ } .
TheKuratowskimeasureofnoncompactness, α,isdefinedby
α(B )=inf {r : B hasafiniteopencoverofdiameter ≤ r },
foranyboundedset B of X .Weset α(B )=+∞, whenever B isunbounded. ForvariouspropertiesofKuratowski’smeasureofnoncompactness,we referto[242, 91]and[304,Lemma22.2].Theproofofthefollowinglemmais straightforward.
Lemma1.1.2. Thefollowingstatementsarevalid:
(a)Let I ⊂ [0, +∞) beunbounded,and {At }t∈I beanonincreasingfamily ofnonemptyclosedsubsets(i.e., t ≤ s implies As ⊂ At ).Assumethat α(At ) → 0, as t → +∞.Then A∞ = t≥0 At isnonemptyandcompact, and δ (At ,A∞ ) → 0, as t → +∞.
(b)Foreach A ⊂ X and B ⊂ X, wehave α(B ) ≤ α(A)+ δ (B,A).
Forasubset B ⊂ X ,let γ + (B ):= m≥0 f m (B )bethepositiveorbitof B for f ,and
ω (B ):= n≥0 m≥n f m (B )
theomegalimitsetof B .Asubset A ⊂ X ispositivelyinvariantfor f if f (A) ⊂ A.Wesaythatasubset A ⊂ X attractsasubset B ⊂ X for f if limn→∞ δ (f n (B ),A)=0.
Itiseasytoseethat B isprecompact(i.e., B iscompact)ifandonlyif α(B )=0.Acontinuousmapping f : X → X issaidtobecompact(completelycontinuous)if f mapsanyboundedsettoaprecompactsetin X . Thetheoryofattractorsisbasedonthefollowingfundamentalresult, whichisrelatedto[141,Lemmas2.1.1and2.1.2].
Lemma1.1.3. Let B beasubsetof X andassumethatthereexistsacompact subset C of X whichattracts B for f .Then ω (B ) isnonempty,compact, invariantfor f andattracts B.
Proof. Let I = N+ ,thesetofallnonnegativeintegers,and An = m≥n
m (B ), ∀n ≥ 0.
Since C attracts B, fromLemma 1.1.2 (b)wededucethat
α(An ) ≤ α(C )+ δ (An ,C )= δ (An ,C ) → 0, as n → +∞.
Sothefamily {An }n≥0 satisfiestheconditionsofassertion(a)inLemma 1.1.2, andwededucethat ω (B )isnonempty,compact,and δ (An ,ω (B )) → 0, as n → +∞. So ω (B )attracts B for f .Moreover,wehave f ⎛ ⎝m≥n f m (B )⎞ ⎠ = m≥n+1 f m (B ), ∀n ≥ 0,
andsince f iscontinuous,weobtain f (An ) ⊂ An+1 , and An+1 ⊂ f (An ), ∀n ≥ 0.
Finally,since δ (An ,ω (B )) → 0, as n → +∞,wehave f (ω (B ))= ω (B )
Definition1.1.2. Acontinuousmapping f : X → X issaidtobepoint (compact,bounded)dissipativeifthereisaboundedset B0 in X suchthat B0 attractseachpoint(compactset,boundedset)in X ; α-condensing(αcontractionoforder k , 0 ≤ k< 1)if f takesboundedsetstoboundedsets and α(f (B )) <α(B ) (α(f (B )) ≤ kα(B ))foranynonemptyclosedbounded set B ⊂ X with α(B ) > 0; α-contractingif lim n→∞ α (f n (B ))=0 forany boundedsubset B ⊂ X ;asymptoticallysmoothifforanynonemptyclosed boundedset B ⊂ X forwhich f (B ) ⊂ B ,thereisacompactset J ⊂ B such that J attracts B
Clearly,acompactmapisan α-contractionoforder0,andan α-contraction oforder k is α-condensing.Itiswellknownthat α-condensingmapsareasymptoticallysmooth(see,e.g.,[141,Lemma2.3.5]).ByLemma 1.1.2,itfollows that f : X → X isasymptoticallysmoothifandonlyiflimn→∞ α (f n (B ))=0
1.1LimitSetsandGlobalAttractors5
foranynonemptyclosedboundedsubset B ⊂ X forwhich f (B ) ⊂ B .This impliesthatany α-contractingmapisasymptoticallysmooth.
Apositivelyinvariantsubset B ⊂ X for f issaidtobestableifforany neighborhood V of B, thereexistsaneighborhood U ⊂ V of B suchthat f n (U ) ⊂ V, ∀n ≥ 0.Wesaythat A isgloballyasymptoticallystablefor f if, inaddition, A attractspointsof X for f .
Bytheproofthat(i)implies(ii)in[141,Theorem2.2.5],wehavethe followingresult.
Lemma1.1.4. Let B ⊂ X becompact,andpositivelyinvariantfor f .If B attractscompactsubsetsofsomeneighborhoodofitself,then B isstable.
Definition1.1.3. Anonempty,compact,andinvariantset A ⊂ X issaidto beanattractorfor f if A attractssomeopenneighborhoodofitself;aglobal attractorfor f if A isanattractorthatattractseverypointin X ;andastrong globalattractorfor f if A attractseveryboundedsubsetof X .
Remark1.1.1. Thenotionofattractorandglobalattractorwasusedin [164, 304].Thestrongglobalattractorwasdefinedasglobalattractorin [141, 358, 286].Inthecasewherethedimensionof X isfinite,itiseasytosee thatbothglobalattractorandstrongglobalattractorareequivalent.Inthe infinite-dimensionalcaseof X ,however,thereexistdiscrete-andcontinuoustimedynamicalsystemsthatadmitglobalattractors,butnostrongglobal attractors,seeExample 1.3.3 and[241,Sections5.1–5.3].
Thefollowingresultisessentiallythesameas[142,Theorem3.2].Note thattheproofofthisresultwasnotprovidedin[142].Forcompleteness,we stateitintermsofglobalattractorsandgiveanelementaryproofbelow.
Theorem1.1.2. (GlobalAttractors) Let f : X → X beacontinuous map.Assumethat
(a) f ispointdissipativeandasymptoticallysmooth; (b)Positiveorbitsofcompactsubsetsof X for f arebounded.
Then f hasaglobalattractor A ⊂ X .Moreover,ifasubset B of X admits thepropertythat γ + (f k (B )) isboundedforsome k ≥ 0,then A attracts B for f .
Proof. Assumethat(a)issatisfied.Since f ispointdissipative,wecanfind aclosedandboundedsubset B0 in X suchthatforeach x ∈ X, thereexists k = k (x) ≥ 0,f n (x) ∈ B0 , ∀n ≥ k. Define
J (B0 ):= {y ∈ B0 : f n (y ) ∈ B0 , ∀n ≥ 0} .
Thus, f (J (B0 )) ⊂ J (B0 ),andforevery x ∈ X ,thereexists k = k (x) ≥ 0such that f k (x) ∈ J (B0 ).Since J (B0 )isclosedandbounded,and f isasymptoticallysmooth,Lemma 1.1.3 impliesthat ω (J (B0 ))iscompactinvariant,and attractspointsof X .
Assume,inaddition,that(b)issatisfied.Weclaimthatthereexistsan ε> 0suchthat γ + (N (ω (J (B0 )),ε))isbounded.Assume,bycontradiction,that γ + N ω (J (B0 )), 1 n+1 isunboundedforeach n> 0.Let z ∈ X befixed.
Thenwecanfindasequence xn ∈ N ω (J (B0 )), 1 n+1 ,andasequenceof integers mn ≥ 0suchthat d(z,f mn (xn )) ≥ n.Since ω (J (B0 ))iscompact, wecanalwaysassumethat xn → x ∈ ω (J (B )),as n → +∞.Since H := {xn : n ≥ 0}∪{x} iscompact,assumption(b)impliesthat γ + (H )isbounded, acontradiction.Let D = γ + (N (ω (J (B0 )),ε)).Then D isclosed,bounded, andpositivelyinvariantfor f .Since ω (J (B0 ))attractspointsof X for f ,and ω (J (B0 )) ⊂ N (ω (J (B0 )),ε) ⊂ int(D ),wededucethatforeach x ∈ X, there exists k = k (x) ≥ 0suchthat f k (x) ∈ int(D ).Itthenfollowsthatforeach compactsubset C of X ,thereexistsaninteger k ≥ 0suchthat f k (C ) ⊂ D . Thus,theset A := ω (D )attractseverycompactsubsetof X .Fixabounded neighborhood V of A.ByLemma 1.1.4,itfollowsthat A isstable,andhence, thereisaneighborhood W of A suchthat f n (W ) ⊂ V, ∀n ≥ 0.Clearly,the set U := ∪n≥0 f n (W )isaboundedneighborhoodof A,and f (U ) ⊂ U .Since f isasymptoticallysmooth,thereisacompactset J ⊂ U suchthat J attracts U .ByLemma 1.1.3, ω (U )isnonempty,compact,invariantfor f ,andattracts U .Since A attracts ω (U ),wehave ω (U ) ⊂ A.Thus, A isaglobalattractor for f
Toprovethelastpartofthetheorem,withoutlossofgeneralityweassume that B isaboundedsubsetof X and γ + (B )isbounded.Weset K = γ + (B ). Then f (K ) ⊂ K .Since K isboundedand f isasymptoticallysmooth,there existsacompact C whichattracts K for f .Notethat f k (B ) ⊂ f k (γ + (B )) ⊂ f k (K ) , ∀k ≥ 0.Thus, C attracts B for f .ByLemma 1.1.3,wededucethat ω (B )isnonempty,compact,invariantfor f andattracts B .Since A isa globalattractorfor f ,itfollowsthat A attractscompactsubsetsof X .Bythe invarianceof ω (B )for f ,wededucethat ω (B ) ⊂ A,andhence, A attracts B for f .
Remark1.1.2. FromthefirstpartoftheproofofTheorem 1.1.2,itiseasyto seethatif f ispointdissipativeandasymptoticallysmooth,thenthereexists anonempty,compact,andinvariantsubset C of X for f suchthat C attracts everypointin X for f .
Thefollowinglemmaprovidessufficientconditionsforthepositiveorbit ofacompactsettobebounded.
Lemma1.1.5. Assumethat f ispointdissipative.If C isacompactsubset of X withthepropertythatforeveryboundedsequence {xn }n≥0 in γ + (C ), {xn }n≥0 or {f (xn )}n≥0 hasaconvergentsubsequence,then γ + (C ) isbounded in X
Proof. Since f ispointdissipative,wecanchooseaboundedandopensubset V of X suchthatforeach x ∈ X thereexists n0 = n0 (x) ≥ 0suchthat
f n (x) ∈ V, ∀n ≥ n0 .Bythecontinuityof f andthecompactnessof C ,it followsthatthereexistsapositiveinteger r = r (C )suchthatforany x ∈ C , thereexistsaninteger k = k (x) ≤ r suchthat f k (x) ∈ V .Let z ∈ X be fixed.Assume,bycontradiction,that γ + (C )isunbounded.Thenthereexists asequence {xp } in γ + (C )suchthat
xp = f mp (zp ),zp ∈ C, andlim p→∞ d(z,xp )= ∞.
Since f iscontinuousand C iscompact,withoutlossofgeneralitywecan assumethat
lim p→∞ mp = ∞, and mp >r,xp / ∈ V, ∀p ≥ 1.
Foreach zp ∈ C ,thereexistsaninteger kp ≤ r suchthat f kp (zp ) ∈ V .Since xp = f mp (zp ) / ∈ V ,thereexistsaninteger np ∈ [kp ,mp )suchthat yp = f np (zp ) ∈ V, and f l (yp ) / ∈ V,
Clearly, xp = f lp (yp ), ∀p ≥ 1,and {yp } isaboundedsequencein γ + (C ).
Weonlyconsiderthecasewhere {yp } hasaconvergentsubsequencesince theproofforthecasewhere {f (yp )} hasaconvergentsubsequenceissimilar. Thus,withoutlossofgeneralitywecanassumethatlimp→∞ yp = y ∈ V Inthecasewherethesequence {lp } isbounded,thereexistaninteger ˆ l and sequence pk →∞ suchthat lpk = ˆ l, ∀k ≥ 1,andhence,
d(z,f ˆ l (y ))=lim k→∞ d(z,f ˆ l (ypk ))=lim k→∞ d(z,xpk )= ∞,
whichisimpossible.Inthecasewherethesequence {lp } isunbounded,there existsasubsequence lpk →∞ as k →∞.Thenforeachfixed m ≥ 1,there existsaninteger km suchthat m ≤ lpk , ∀k ≥ km ,andhence, f m (ypk ) ∈ X \ V, ∀k ≥ km .
Letting k →∞,weobtain
f m (y ) ∈ X \ V, ∀m ≥ 1, whichcontradictsthedefinitionof V .
Thefollowingresultontheexistenceofstrongglobalattractorsisimplied by[142,Theorems3.1and3.4].Sincetheproofofthisresultwasnotprovided in[142],weincludeasimpleproofofit.
Theorem1.1.3. (StrongGlobalAttractors) Let f : X → X bea continuousmap.Assumethat f ispointdissipativeon X ,andoneofthe followingconditionsholds:
(a) f n0 iscompactforsomeinteger n0 ≥ 1,or (b) f isasymptoticallysmooth,andforeachboundedset B ⊂ X ,thereexists k = k (B ) ≥ 0 suchthat γ + (f k (B )) isbounded.
Thenthereisastrongglobalattractor A for f .
Proof. Theconclusionincase(b)isanimmediateconsequenceofTheorem 1.1.2.Inthecaseof(a),since f n0 iscompactforsomeinteger n0 ≥ 1,it sufficestoshowthatforeachcompactsubset C ⊂ X , n≥0 f n (C )isbounded.
ByapplyingLemma 1.1.5 to f = f n0 ,wededucethatforeachcompactsubset C ⊂ X, n≥0 f n (C )isbounded.SoTheorem 1.1.2 impliesthat f hasaglobal
attractor A ⊂ X .Weset B = 0≤k≤n0 1 f k A .Bythecontinuityof f ,it thenfollowsthat B iscompactandattracts everycompactsubsetof X for f , andhence,theresultfollowsfromTheorem 1.1.2.
Remark1.1.3. Itiseasytoseethatametricspace(X,d)iscompleteifand onlyifforanysubset B of X , α(B )=0impliesthat B iscompact.However,wecanprovethatLemmas 1.1.3 and 1.1.4 alsoholdfornon-complete metricspacesbyemployingtheequivalencebetweenthecompactnessandthe sequentialcompactnessformetricspaces.ItthenfollowsthatTheorems 1.1.2 and 1.1.3 arestillvalidforanymetricspace.Wereferto[64, 286]forthe existenceofstrongglobalattractorsofcontinuous-timesemiflowsonametric space.
Clearly,iftheglobalattractorisasingleton {e},then e isaglobally attractivefixedpoint.Let A betheglobalattractorclaimedinTheorem 1.1.2 with X beingaBanachspaceandwith“asymptoticallysmooth”replacedby “α-condensing.”Thefollowingasymptoticfixedpointtheoremimpliesthat thereisatleastonefixedpointin A.Foraproofofit,wereferto[257, 143] or[141,Section2.6].
Theorem1.1.4. (Asymptoticfixedpointtheorem) Suppose E isaBanachspace.If f : E → E is α-condensingandcompactdissipative,then f has afixedpoint.
Let Λ beametricspace.Thefamilyofcontinuousmappings fλ : X → X,λ ∈ Λ,issaidtobecollectivelyasymptoticallysmoothifforanynonemptyclosed boundedset B ⊂ X forwhich fλ (B ) ⊂ B,λ ∈ Λ,thereisacompactset Jλ = J (λ,B ) ⊂ B suchthat Jλ attracts B under fλ and ∪λ∈Λ Jλ isprecompact in X .Wethenhavethefollowingresultontheuppersemicontinuityofglobal attractors.Foraproof,wereferto[141,Theorem2.5.3].
Theorem1.1.5. Let f : Λ × X → X becontinuous, fλ =: f (λ, ·),andsuppose thereisaboundedset B thatattractspointsof X under fλ foreach λ ∈ Λ,and foranyboundedset U ,theset V = ∪λ∈Λ ∪n≥0 f n λ (U ) isbounded.Ifthefamily
{fλ : λ ∈ Λ} iscollectivelyasymptoticallysmooth,thentheglobalattractor Aλ of fλ isuppersemicontinuousinthesensethat limλ→λ0 supx∈Aλ d(x,Aλ0 )=0 foreach λ0 ∈ Λ.
1.2ChainTransitivityandAttractivity
Inthissectionwecontinuetoassumethat X isametricspacewithmetric d, andthat f : X → X isacontinuousmap.
1.2.1ChainTransitiveSets
Definition1.2.1. Apoint x ∈ X issaidtobechainrecurrentifforany > 0,thereisafinitesequenceofpoints x1 ,...,xm in X (m> 1)with x1 = x = xm suchthat d(f (xi ),xi+1 ) < forall 1 ≤ i ≤ m 1.Thesetofall chainrecurrentpointsfor f : X → X isdenotedby R (X,f ).Let A ⊂ X bea nonemptyinvariantset.Wecall A internallychainrecurrentif R (A,f )= A, andinternallychaintransitiveifthefollowingstrongerconditionholds:For any a,b ∈ A andany > 0,thereisafinitesequence x1 ,...,xm in A with x1 = a,xm = b suchthat d(f (xi ),xi+1 ) < , 1 ≤ i ≤ m 1.Thesequence {x1 ,...,xm } iscalledan -chainin A connecting a and b
FollowingLaSalle[212],wecallacompactinvariantset A invariantlyconnectedifitcannotbedecomposedintotwodisjointclosednonemptyinvariant sets.Aninternallychainrecurrentsetneednothavethisproperty,e.g.,apair offixedpoints.However,itiseasytosee thateveryinternallychaintransitive setisinvariantlyconnected.
Wegivesomeexamplesofinternallychaintransitivesets.
Lemma1.2.1. Let f : X → X beacontinuousmap.Thentheomega(alpha)limitsetofanyprecompactpositive(negative)orbitisinternallychain transitive.
Proof. Let x ∈ X andset xn = f n (x).Assumethat x hasaprecompact orbit γ = {xn },anddenoteitsomegalimitsetby ω .Then ω isnonempty, compact,andinvariant,andlimn→∞ d(xn ,ω )=0.Let > 0begiven.By thecontinuityof f andcompactnessof ω ,thereexists δ ∈ (0, 3 )withthe followingproperty:If u,v arepointsintheopen δ -neighborhood U of ω with d(u,v ) <δ ,then d(f (u),f (v )) < 3 .Since xn approaches ω as n →∞,there exists N> 0suchthat xn ∈ U forall n ≥ N .
Let a,b ∈ ω bearbitrary.Thereexist k>m ≥ N suchthat d(xm ,f (a)) < 3 and d(xk ,b) < 3 .Thesequence
{y0 = a,y1 = xm ,...,yk m = xk 1 ,yk m+1 = b}
isan 3 -chainin X connecting a and b.Sinceforeach yi ∈ U ,1 ≤ i ≤ k m, wecanchoose zi ∈ ω suchthat d(zi ,yi ) <δ .Let z0 = a and zk m+1 = b. Thenfor i =0, 1,...,k m wehave
(f (zi ),zi+1 ) ≤
(yi+1 ,zi+1 ) < /3+ /3+ /3.
Thusthesequence z0 ,z1 ,...,zk m ,zk m+1 isan -chainin ω connecting a and b.Therefore, ω isinternallychaintransitive.Byasimilarargument,we canprovetheinternalchaintransitivityofalphalimitsetsofprecompact negativeorbits.
Let {Sn : X → X }n≥0 beasequenceofcontinuousmaps.Thediscrete dynamicalprocess(orprocessforshort)generatedby {Sn } isthesequence {Tn : X → X }n≥0 definedby T0 = I =theidentitymapof X and Tn = Sn 1
Theorbitof x ∈ X underthisprocessistheset γ + (x)= {Tn (x): n ≥ 0}, anditsomegalimitsetis
ω (x)= y ∈ X : ∃nk →∞ suchthatlim k→∞ Tnk (x)= y .
Ifthereisacontinuousmap S on X suchthat Sn = S, ∀n ≥ 0,sothat Tn isthe nthiterate S n ,then {Tn } isaspecialkindofprocesscalledthediscrete semiflowgeneratedby S .Byanabuseoflanguagewemayrefertothemap S asadiscretesemiflow.
Definition1.2.2. Theprocess {Tn : X → X } isasymptoticallyautonomous ifthereexistsacontinuousmap S : X → X suchthat nj →∞,xj → x ⇒ lim j →∞ Snj (xj )= S (x).
Wealsosaythat {Tn } isasymptoticto S
Itiseasytoseefromthetriangleinequalitythatiflimn→∞ Sn = S uniformlyoncompactsets,thentheprocessgeneratedby {Sn } isasymptotic to S .
Lemma1.2.2. Let Tn : X → X , n ≥ 0,beanasymptoticallyautonomous discreteprocesswithlimit S : X → X .Thentheomegalimitsetofany precompactorbitof {Tn } isinternallychaintransitivefor S .
Proof. Let N+ = N+ ∪{∞}.Foranygivenstrictlyincreasingcontinuous function φ :[0, ∞) → [0, 1)with φ(0)=0and φ(∞)=1(e.g., φ(s)= s 1+s ), wecandefineametric ρ on N+ as ρ(m1 ,m2 )= |φ(m1 ) φ(m2 )|,forany m1 ,m2 ∈ N+ ,andthen N+ iscompactified.Let X := N+ × X .Definea mapping S : X → X by
S (m,x)=(1+ m,Sm (x)), S (∞,x)=(∞,S (x)), ∀m ∈ N+ ,x ∈ X.
ByDefinition 1.2.2, S : X → X iscontinuous.Let γ + (x)beaprecompact orbitof Tn .Since
S n ((0,x))=(n,Sn 1 ◦ Sn 2 ◦···◦ S1 ◦ S0 (x))=(n,Tn (x)), ∀n ≥ 0,
and N+ iscompact,itfollowsthattheorbit γ + ((0,x))of S n isprecompact and {∞}× ω (x)= ω (0,x),where ω (0,x)istheomegalimitsetof(0,x)for S n .ByLemma 1.2.1, ω (0,x)isinvariantandinternallychaintransitivefor S ,which,togetherwiththedefinitionof S ,impliesthat ω (x)isinvariantand internallychaintransitivefor S
Definition1.2.3. Let S : X → X beacontinuousmap.Asequence {xn } in X isanasymptoticpseudo-orbitof S if lim n→∞ d(S (xn ),xn+1 )=0.
Theomegalimitsetof {xn } isthesetoflimitsofsubsequences.
Let {Tn } beadiscreteprocessin X generatedbyasequenceofcontinuous maps Sn thatconvergestoacontinuousmap S : X → X uniformlyon compactsubsetsof X .Itiseasytoseethateveryprecompactorbitof Tn : X → X , n ≥ 0,isanasymptoticpseudo-orbitof S .
Example1.2.1. Considerthenonautonomousdifferenceequation xn+1 = f (n,xn ),n ≥ 0, onthemetricspace X .Ifwedefine Sn = f (n, ·): X → X,n ≥ 0,andlet
T0 = I,Tn = S
then xn = Tn (x0 ),and {xn : n ≥ 0} isanorbitofthediscreteprocess Tn .If f (n, ·) → f : X → X uniformlyoncompactsubsetsof X ,then Tn isasymptoticallyautonomouswithlimit f .Furthermore,inthiscaseany precompactorbitofthedifferenceequationisanasymptoticpseudo-orbitof f ,since d(f (xn ),xn+1 )= d(f (xn ),f (n,xn )) → 0.
Lemma1.2.3. Theomegalimitsetofanyprecompactasymptoticpseudoorbitofacontinuousmap S : X → X isnonempty,compact,invariant,and internallychaintransitive.
Proof. Let(N + ,ρ)bethecompactmetricspacedefinedintheproofof Lemma 1.2.2.Let {xn : n ≥ 0} beaprecompactasymptoticpseudo-orbit of S : X → X ,anddenoteitscompactomegalimitsetby ω .Defineametric space
Y =({∞}× X ) ∪{(n,xn ): n ≥ 0}
and g : Y → Y,g (n,xn )=(n +1,xn+1 ),g (∞,x)=(∞,S (x)).
ByDefinition 1.2.3 andthefactthat d(xn+1 ,S (x)) ≤ d(xn+1 ,S (xn ))+ d(S (xn ),S (x))for x ∈ X,n ≥ 0,iteasilyfollowsthat g : Y → Y iscontinuous.Let γ + (0,x0 )= {(n,xn ): n ≥ 0} bethepositiveorbitof(0,x0 )for thediscretesemiflow g n : Y → Y,n ≥ 0.Then γ + (0,x0 )isprecompactin Y , anditsomegalimit ω (0,x0 )is {∞}× ω ,whichbyLemma 1.2.1 isinvariant andinternallychaintransitivefor g .Applyingthedefinitionof g ,weseethat ω isinvariantandinternallychaintransitivefor S .
Let A and B betwononemptycompactsubsetsof X .Recallthatthe Hausdorffdistancebetween A and B isdefinedby
dH (A,B ):=max(sup{d(x,B ): x ∈ A}, sup{d(x,A): x ∈ B })
Wethenhavethefollowingresult.
Lemma1.2.4. Let S,Sn : X → X , ∀n ≥ 1,becontinuous.Let {Dn } bea sequenceofnonemptycompactsubsetsof X with lim n→∞ dH (Dn ,D )=0 for somecompactsubset D of X .Assumethatforeach n ≥ 1, Dn isinvariant andinternallychaintransitivefor Sn .If Sn → S uniformlyon D ∪ (∪n≥1 Dn ), then D isinvariantandinternallychaintransitivefor S
Proof. Observethattheset K = D (∪n≥1 Dn )iscompact.Indeed,since anopencoverof K alsocovers D ,afinitesubcoverprovidesaneighborhood of D thatmustalsocontain Dn foralllarge n.If x ∈ D ,thenthereexist xn ∈ Dn suchthat xn → x.Since Sn (xn ) ∈ Dn and Sn (xn ) → S (x),wesee that S (x) ∈ D .Thus S (D ) ⊂ D .Ontheotherhand,thereexist yn ∈ Dn such that Sn (yn )= xn .Since dH (Dni ,D ) → 0,wecanassumethat yni → y ∈ D forsomesubsequence yni .Then xni = Sni (yni ) → S (y )= x,showingthat S (D )= D
Byuniformcontinuityanduniformconvergence,forany > 0thereexist δ ∈ (0, /3)andanaturalnumber N suchthatfor n ≥ N and u,v ∈ K with d(u,v ) <δ ,wehave
d(Sn (u),S (v )) ≤ d(Sn (u),S (u))+ d(S (u),S (v )) < /3
Fix n>N suchthat dH (Dn ,D ) <δ .Forany a,b ∈ D ,therearepoints x,y ∈ Dn suchthat d(x,a) <δ and d(y,b) <δ .Since Dn isinternallychain transitivefor Sn ,thereisa δ -chain {z1 = x,z2 ,...,zm+1 = y } in Dn for Sn connecting x to y .Foreach i =2,...,m wecanfind wi ∈ D with d(wi ,zi ) <δ , since Dn iscontainedinthe δ -neighborhoodof D .Let w1 = a,wm+1 = b.We thenhave
d(S (wi ),wi+1 ) ≤ d(S (wi ),Sn (zi ))+ d(Sn (zi ),zi+1 )+ d(zi+1 ,wi+1 ) < /3+ δ + δ<
1.2ChainTransitivityandAttractivity13
for i =1,...,m.Thus {w1 = a,w2 ,...,wm+1 = b} isan -chainfor S in D connecting a to b.
Let Φ(t): X → X , t ≥ 0,beacontinuous-timesemiflow.Thatis,(x,t) → Φ(t)x iscontinuous, Φ(0)=Iand Φ(t) ◦ Φ(s)= Φ(t + s)for t,s ≥ 0.A nonemptyinvariantset A ⊂ X for Φ(t)(i.e., Φ(t)A = A, ∀t ≥ 0)issaidtobe internallychaintransitiveifforany a,b ∈ A andany > 0,t0 > 0,thereisa finitesequence {x1 = a,x2 ,...,xm 1 ,xm = b; t1 ,...,tm 1 } with xi ∈ A and ti ≥ t0 , 1 ≤ i ≤ m 1,suchthat d(Φ(ti ,xi ),xi+1 ) < forall1 ≤ i ≤ m 1.The sequence {x1 ,...,xm ; t1 ,...,tm 1 } iscalledan( ,t0 )-chainin A connecting a and b.Wethenhavethefollowingresult.
Lemma 1.2.1 Let Φ(t): X → X , t ≥ 0,beacontinuous-timesemiflow. Thentheomega(alpha)limitsetofanyprecompactpositive(negative)orbit isinternallychaintransitive.
Proof. Let ω = ω (x)betheomegalimitsetofaprecompactorbit γ (x)= {Φ(t)x : t ≥ 0} in X .Then ω isnonempty,compact,invariantand limt→∞ d(Φ(t)x,ω )=0.Let > 0and t0 > 0begiven.Bytheuniform continuityof Φ(t)x for(t,x)inthecompactset[t0 , 2t0 ] × ω ,thereisa δ = δ ( ,t0 ) ∈ (0, 3 )suchthatforany t ∈ [t0 , 2t0 ]and u and v intheopen δ -neighborhood U of ω with d(u,v ) <δ ,wehave d(Φ(t)u,Φ(t)v ) < 3 .It thenfollowsthatthereexistsasufficientlylarge T0 = T0 (δ ) > 0suchthat Φ(t)x ∈ U ,forall t ≥ T0 .Forany a,b ∈ ω ,thereexist T1 >T0 and T2 >T0 with T2 >T1 + t0 suchthat d(Φ(T1 )x,Φ(t0 )a) < 3 and d(Φ(T2 )x,b) < 3 .Let m bethegreatestintegerthatisnotgreaterthan T2 T1 t0 .Then m ≥ 1.Set
y1 = a,yi = Φ(T1 +(i 2)t0 )x,i =2,...,m +1,ym+2 = b, and ti = t0 for i =1,...,m; tm+1 = T2 T1 (m 1)t0
Then tm+1 ∈ [t0 , 2t0 ).Itfollowsthat d(Φ(ti )yi ,yi+1 ) < 3 forall i =1,...,m + 1.Thusthesequence
{y1 = a,y2 ,...,ym+1 ,ym+2 = b; t1 ,t2 ,...,tm+1 }
isan( 3 ,t0 )-chainin X connecting a and b.Since yi ∈ U for i =2,...,m +1, wecanchoose zi ∈ ω suchthat d(zi ,yi ) <δ .Let z1 = a and zm+2 = b.It thenfollowsthat d(Φ(ti )zi ,zi+1 ) ≤ d(Φ(ti )zi ,Φ(ti )yi )+ d(Φ(ti )yi ,yi+1 )+ d(yi+1 ,zi+1 ) < /3+ /3+ /3,i =1,...,m +1.
Thisprovesthatthesequence {z1 = a,z2 ,...,zm+1 ,zm+2 = b; t1 ,t2 ,...,tm+1 } isan( ,t0 )-chainin ω connecting a and b.Therefore, ω isinternallychaintransitive.Byasimilarargumentwecanprovetheinternalchaintransitivityof alphalimitsetsofprecompactnegativeorbits.
WithLemma 1.2.1 itiseasytoseethatthereareanaloguesofLemmas 1.2.2 and 1.2.3 forcontinuous-timesemiflows.Thefollowingresultisan analogueofLemma 1.2.4 forcontinuous-timesemiflows.
Lemma 1.2.4 Let Φ and Φn becontinuous-timesemiflowson X for n ≥ 1
Let {Dn } beasequenceofnonemptycompactsubsetsof X with limn→∞ dH (Dn ,D )=0 forsomecompactsubset D of X .Assumethatfor each n ≥ 1, Dn isinvariantandinternallychaintransitivefor Φn .Ifforeach T> 0, Φn → Φ uniformlyfor (x,t) ∈ (D ∪ (∪n≥1 Dn )) × [0,T ],then D is invariantandinternallychaintransitivefor Φ.
Proof. Itiseasytoseethat K = D (∪n≥1 Dn )iscompactand D isinvariant for Φ.Byuniformcontinuityanduniformconvergence,forany > 0and t0 > 0thereexists δ ∈ (0, /3)andanaturalnumber N suchthatfor n ≥ N , t ∈ [0, 2t0 ],and u,v ∈ K with d(u,v ) <δ ,wehave d(Φn t (u),Φt (v )) ≤ d(Φn t (u),Φt (u))+ d(Φt (u),Φt (v )) < /3.Fix n>N suchthat dH (Dn ,D ) <δ
Forany a,b ∈ D ,therearepoints x,y ∈ Dn suchthat d(x,a) <δ and d(y,b) <δ .Since Dn ischaintransitivefor Φn ,thereisa(δ,t0 )-chain {z1 = x,z2 ,...,zm+1 = y ; t1 ,...,tm } in Dn for Φn ,with t0 ≤ ti < 2t0 connecting x to y .Foreach i =2,...,m wecanfind wi ∈ D with d(wi ,zi ) <δ ,since Dn is containedinthe δ -neighborhoodof D .Let w1 = a,wm+1 = b.Wethenhave
d(Φti (wi ),wi+1 ) ≤ d(Φti (wi ),Φn ti (zi ))+ d(Φn ti (zi ),zi+1 )+ d(zi+1 ,wi+1 ) < /3+ δ + δ< for i =1,...,m.Thus {w1 = a,w2 ,...,wm+1 = b; t1 ,...,tm } isan( ,t0 )chainfor Φ in D connecting a to b
Example1.2.2. NotethatifinLemma 1.2.4 Dn isanomegalimitsetfor Φn (andthereforeinternallychaintransitivebyLemma 1.2.1 ),theset D neednotbeanomegalimitsetforthelimitsemiflow Φ,althoughitmustbe chaintransitive.Easyexamplesareconstructedwith Φn = Φ, ∀n ≥ 1.For example,considertheflowgeneratedbytheplanarvectorfieldgiveninpolar coordinatesby r =0,θ =1 r.
Theunitcircle D = {r =1},consistingofequilibria,ischaintransitivebutis notanomegalimitsetofanypoint,yet D istheHausdorfflimitoftheomega limitsets Dn = {r =1+ 1 n }.
Lemma1.2.5. Anonemptycompactinvariantset M isinternallychaintransitiveifandonlyif M istheomegalimitsetofsomeasymptoticpseudo-orbit of f in M
Proof. ThesufficiencyfollowsfromLemma 1.2.3.Toprovethenecessity, wecanchooseapoint x ∈ M since M isnonempty.Forany > 0, thecompactnessof M impliesthatthereisafinitesequenceofpoints
{x1 = x,x2 ,...,xm ,xm+1 = x} in M suchthatits -netin X covers M ; i.e., M ⊂∪m i=1 B (xi , ),where B (xi , ):= {y ∈ X : d(y,xi ) < }.Foreach 1 ≤ i ≤ m,since M isinternallychaintransitive,thereisafinite -chain {y i 1 = xi ,y i 2 ,...,y i ni ,y i ni +1 = xi+1 } in M connecting xi and xi+1 .Thenthe sequence {y 1 1 ,...,y 1 n1 ,y 2 1 ,...,y 2 n2 ,...,y m 1 ,...,y m nm ,y m nm +1 } isafinite -chain in M connecting x and x,andits -netin X covers M . Foreachinteger k ,letting = 1 k intheaboveclaim,wehaveafinite 1 kchain {z k 1 = x,z k 2 ,...,z k lk ,z k lk +1 = x} in M whose 1 k -netin X covers M .It theneasilyfollowsthattheinfinitesequenceofpoints
{z 1 1 ,...,z 1 l1 ,z 2 1 ,...,z 2 l2 ,...,z k 1 ,...,z k lk ,... }
isanasymptoticpseudo-orbitof f in M anditsomegalimitsetis M
Block–FrankeLemma ([33], TheoremA) Let K beacompactmetric spaceand f : K → K acontinuousmap.Then x ∈ R (K,f ) ifandonly ifthereexistsanattractor A ⊂ K suchthat x ∈ W s (A) \ A.
Lemma1.2.6. Anonemptycompactinvariantset M isinternallychaintransitiveifandonlyif f |M : M → M hasnoproperattractor.
Proof. Necessity. Assumethatthereisaproperattractor A for f |M : M → M .Then A = ∅ and M \ A = ∅.Since A isanattractor,thereisan 0 > 0such that A attractstheopen 0 -neighborhood U of A in M .Choose a ∈ M \ A and b ∈ A andlet {x1 = a,x2 ,...,xm = b} bean 0 -chainin M connecting a and b.Let k =min{i :1 ≤ i ≤ m,xi ∈ A}.Since b ∈ A and a ∈ A,we have2 ≤ k ≤ m.Since d(f (xk 1 ),xk ) < 0 ,wehave f (xk 1 ) ∈ U andhence xk 1 ∈ W s (A) \ A.BytheBlock–Frankelemma, xk 1 ∈ R (M,f ),which provesthat M isnotinternallychainrecurrent,andafortiorinotinternally chaintransitive.
Sufficiency. Foranysubset B ⊂ X wedefine ω (B )tobethesetoflimits ofsequencesoftheform {f nk (xk )},where nk →∞ and xk ∈ B .Since f |M : M → M hasnoproperattractor,theBlock–Frankelemmaimpliesthat M is internallychainrecurrent.Given a,b ∈ M and > 0,let V bethesetofall points x in M forwhichthereisan -chainin M connecting a to x;thisset contains a.Forany z ∈ V ,let {z1 = a,z2 ,...,zm 1 ,zm = z } bean -chainin M connecting a to z .Since
lim x→z d(f (zm 1 ),x)= d(f (zm 1 ),z ) < , thereisanopenneighborhood U of z in M suchthatforany x ∈ U , d(f (zm 1 ),x) < .Then {z1 = a,z2 ,...,zm 1 ,x} isan -chainin M connecting a and x,andhence U ⊂ V .Thus V isanopensetin M .We furtherclaimthat f (V ) ⊂ V .Indeed,forany z ∈ V ,bythecontinuityof f at z ,wecanchoose y ∈ V suchthat d(f (y ),f (z )) < .Let {y1 = a,y2 ,...,ym 1 ,ym = y } bean -chainin M connecting a and y .It
thenfollowsthat {y1 = a,y2 ,...,ym 1 ,ym = y,ym+1 = f (z )} isan -chain in M connecting a and f (z ),andhence f (z ) ∈ V .Bythecompactnessof M and[141,Lemma2.1.2]appliedto f : M → M ,itthenfollowsthat ω (V ) isnonempty,compact,invariant,and ω (V )attracts V .Since f (V ) ⊂ V ,we have ω (V ) ⊂ V andhence ω (V )= f (ω (V )) ⊂ V .Then ω (V )isanattractor in M .Nowthenonexistenceofaproperattractorfor f : M → M implies that ω (V )= M andhence V = M .Clearly, b ∈ M = V ,andhence,bythe definitionof V ,thereisan -chainin M connecting a and b.Therefore, M is internallychaintransitive.
1.2.2AttractivityandMorseDecompositions
Recallthatanonemptyinvariantsubset M of X issaidtobeisolatedfor f : X → X ifitisthemaximalinvariantsetinsomeneighborhoodofitself.
Lemma1.2.7. (Butler–McGehee-typelemma) Let M beanisolatedinvariantsetand L acompactinternallychaintransitivesetfor f : X → X . Assumethat L ∩ M = ∅ and L ⊂ M .Then
(a)thereexists u ∈ L \ M suchthat ω (u) ⊂ M ; (b)thereexist w ∈ L \ M andanegativeorbit γ (w ) ⊂ L suchthatits α-limit setsatisfies α(w ) ⊂ M
Proof. Since M isanisolatedinvariantset,thereexistsan > 0suchthat M isthemaximalinvariantsetintheclosed -neighborhoodof M .Bythe assumption,wecanchoose a ∈ L ∩ M and b ∈ L with d(b,M ) > .For anyinteger k ≥ 1,bytheinternalchaintransitivityof L,thereexistsa 1 k -chain {y k 1 = a,...,y k lk +1 = b} in L connecting a and b,anda 1 k -chain {z k 1 = b,...,z k mk +1 = a} in L connecting b and a.Defineasequenceofpoints by {xn : n ≥ 0} := {y 1 1 ,...,y 1 l1 ,z 1 1 ,...,z 1 m1 ,...,y k 1 ,...,y k lk ,z k 1 ,...,z k mk ,... }
Thenforany k> 0andforall n ≥ N (k ):= k j =1 (lj + mj ),wehave d(f (xn ),xn+1 ) < 1 k+1 ,andhencelim n→∞ d(f (xn ),xn+1 )=0.Thus {xn }n≥0 ⊂ L isaprecompactasymptoticpseudo-orbitof f : X → X .Thenthereare twosubsequences xmj and xrj suchthat xmj = a and xrj = b forall j ≥ 1. Notethat d(xsj +1 ,f (x)) ≤ d(xsj +1 ,f (xsj ))+ d(f (xsj ),f (x)).Byinduction,it thenfollowsthatforanyconvergentsubsequence xsj → x ∈ X,j →∞, wehavelimj →∞ xsj +n = f n (x)foranyinteger n ≥ 0.Wecanfurther choosetwosequences lj and nj with lj <mj <nj andlimj →∞ lj = ∞ suchthat d(xlj ,M ) > , d(xnj ,M ) > ,and d(xk ,M ) ≤ foranyinteger k ∈ (lj ,nj ),j ≥ 1.Since {xn : n ≥ 0} isasubsetofthecompactset L,wecan assumethatupontakingaconvergentsubsequence, xlj → u ∈ L as j →∞. Clearly, d(u,M ) ≥ andhence u ∈ L \ M .Since u ∈ L and L isacompactinvariantset,wehave ω (u) ⊂ L.Wefurtherclaimthat ω (u) ⊂ M ,whichproves
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other counter jumping namby-pamby, goody-goodies of the Howells stripe, including his own weary history of himself, and the “Books Which Most Influence Him,” the baleful effects of which are legitimately and plainly perceptible in his works. There are shams in literature more dreadful than Mr. Howells, who is a turgid fact and no sham. For instance;
I know of one evanescently popular young creature who chronically contributes to the magazines, whose mother it is said, writes his tales which, she being a clever woman and he an uncommonly stupid man, appears credible to say the least; and there is another “man” I am told of whose sister is said to write his poems and modestly efface herself, and as the stories are good and the poems fairly readable, it should be the part of T P to disclose to the world the real authors and chastise these and other shams, for shams are the hardest hurdles in the steeplechase which Truth has to make in this world, since they substitute the false for the real and crown the fool with the laurels of the genius.
How much more might be said of the tasks you have to accomplish, brave P with your brawny arm and your good naked sword! So much that the very thought of it fatigues one and that, hailing you as the latest and best contestant in the tourney of Knighthood and yet, considering you as a publication in an embryonic stage, I am compelled to quote these lovely lines of Longfellow:
“Oh, little feet that such long years Must wander through this vale of tears, I, nearer to the wayside inn Where travail ends and rest begins, Grow weary thinking of your road.”
M S. H .
It is a land of free speech, Philistia, and if one of us chooses to make remarks concerning the work of the others no sense of modesty keeps us quiet. It is because we cannot say what we would in the periodicals which are now issued in a dignified, manner in various places, that we have made this book. In the afore-mentioned periodicals divers men chatter with great fluency, startling regularity and “damnable complacency,” each through his individual bonnet. Edward W. Bok, evidently assisted by Mrs. Lydia Pinkham and W. L. Douglas, of Brocton, Massachusetts, prints the innermost secrets of dead women told by their living male relatives for six dollars a column. Thereby the authors are furnished with the price of a week’s board, and those of us who may have left some little sense of decency, wonder what manner of man it may be who sells his wife’s heart to the readers of Bok. But the “unspeakable Bok” is “successful.” His magazine flourishes like a green bay tree. Many readers write him upon subjects of deportment and other matters in which he is accomplished. So, the gods give us joy! Let him drive on, and may his Home Journal have five million readers before the year is out—God help them!
Mr Gilder dishes up monthly beautifully printed articles which nobody cares about, but which everybody buys, because The Century looks well on the library table.
Mr. Howells maunders weekly in a column called “Life and Letters” in Harper’s journal of civilization. This “Life and Letters” reminds me
of the Peterkin’s famous picnic at Strawberry Nook. “There weren’t any strawberries and there wasn’t any nook, but there was a good place to tie the horses.”
So it goes through the whole list. There are people, however, who believe that Romance is not dead, and that there is literature to be made which is neither inane nor yet smells of the kitchen sink. This is a great big merry world, says Mr. Dana, and there’s much good to be got out of it, so toward those who believe as we do—we of Philistia—this paper starts upon its great and perilous voyage at one dollar a year.
It was Balzac, or some one else, who used to tell of a flea that lived on a mangy lion and boasted to all the rank outside fleas that he met: I have in me the blood of the King of Beasts.
It is a comforting thought that somewhere, at some time, every good thing on earth is brought to an accounting of itself. Thereby are the children of men saved from much tyranny. For the good things of earth are your true oppressors.
For such an accounting are Philistines born in every age. By their audit are men perpetually set free from trammels self-woven.
Earnest men have marvelled in all times that convention has imputed to husks and symbols the potency of the things they outwardly stand for. Many also have protested, and these, in reproach, have been called Philistines. And yet they have done no more than show forth that in all things the vital purpose is more than the form that shrines it. The inspirations of to-day are the shams of to-morrow—for the purpose has departed and only the dead form of custom remains. “Is not the body more than raiment”—and is not life more than the formulæ that hedge it in?
Wherefore men who do their own thinking, and eke women betimes, take honor rather than disparagement in the name which is meant to typify remorseless commonplace. They hesitate not to question custom, whether there be reason in it. They ask “Why?” when one makes proclamation:
“Lo! Columbus discovered America four hundred years ago! Let us give a dance.” There have been teachers who sought to persuade mankind that use alone is beauty—and these too have done violence to the fitness of things. On such ideals is the civilization of Cathay founded. Neither in the grossness of material things nor in the false refinements that “divorce the feeling from its mate the deed” is the core and essence of living.
It is the business of the true Philistine to rescue from the environment of custom and ostentation the beauty and the goodness cribbed therein. And so the Philistines of these days, whose prime type is the Knight of La Mancha, go tilting at windmills and other fortresses—often on sorry nags and with shaky lances, and yet on heroic errand bent. And to such merry joust and fielding all lovers of chivalry are bidden: to look on—perhaps to laugh, it may be to grieve at a woeful belittling of lofty enterprise. Come, such of you as have patience with such warriors. It is Sancho Panza who invites you.
The Chip-Munk has a bright reference in the issue of May 15 to Coventry, Patmore, Pater and Meredith. These are four great men, as The Chip-Munk boldly states.
The Chip-Munk further announces that the Only Original LynxEyed Proof Reader has not gone on a journey Really, I supposed of course he had been gone these many moons!
I wonder if Carman is still upon a diet of Mellin’s Food that he imagines people do not know that this poem
LITTLE LYRICS OF JOY—V.
Lord of the vasty tent of Heaven, Who hast to thy saints and sages given A thousand nights with their thousand stars, And the star of faith for a thousand years.
Grant me, only a foolish rover, All thy beautiful wide world over, A thousand loves in a thousand days, And one great love for a thousand years.
B C in The Chap Book, May, 1895.
was written years and years ago as follows:
The night has a thousand eyes, And the day but one; Yet the light of the bright world dies With the dying sun.
The mind has a thousand eyes, And the heart but one; Yet the light of a whole life dies When love is done.
—F W. B .
I desire to swipe him after this manner:
LITTLE DELIRICS OF BLISS.
.
Lord of the wires that tangle Heaven, Who hast to thy brake-persuaders given, The longest of days to ring and grind, And no least screen from the winter’s wind.
Grant me, only, a summer lover, Sunshiny days the long year over, A thousand whirls and a thousand fares, And one long whirl of a thousand hours.
- - .
White and rose are the colors of strife, What care I for the crimson and blue? Greater than football the battle of life
J T .
And tragic as aught the gods may view, The clutch and the gripe of inward ills; Pallid the People and Pink the Pills.
J C .
Mark Twain says he is writing “Joan of Arc” anonymously in Harper’s because he is convinced if he signed it the people would insist the stuff was funny. Mr. Twain is worried unnecessarily. It has been a long time since any one insisted the matter he turns out so voluminously was or is funny.
The amusing William Dean Howells writes that he is so bothered by autograph seekers that he will hereafter refuse to send his signature “with a sentiment” unless the applicant for his favor produces satisfactory evidence he has read all of his works, “now some thirty or forty in number.” When this proof has been sent if Mr. Howells does not return his autograph on the bottom of a check for a large amount, he deserves to be arrested for cruelty to his fellows.
There is no doubt that a teacher once committed to a certain line of thought will cling to that line long after all others have deserted it. In trying to persuade others he convinces himself. This is especially so if he is opposed. Opposition evolves in his mind a maternal affection for the product of his brain and he defends it blindly to the death. Thus we see why institutions are so conservative. Like the coral insect they secrete osseous matter; and when a preacher preaches he himself always goes forward to the mourners’ bench and accepts all of the dogmas that have just been so ably stated.
Literature is the noblest of all the arts. Music dies on the air, or at best exists only as a memory; oratory ceases with the effort; the painter’s colors fade and the canvas rots; the marble is dragged from its pedestal and is broken into fragments; but the Index Expurgatorius is as naught, and the books burned by the fires of the auto da fe still live. Literature is reproduced ten thousand times ten thousand and lodges its appeal with posterity. It dedicates itself to Time.
The action of various theatrical managers in cutting from their programmes the name of the author of the plays running at their houses and the similar action of numerous librarians in withdrawing his books from their shelves is simply another proof of the marvellous powers of stultification possessed by the humans of the present time. These managers, having the scattering wits of birds, do not seem to appreciate that, whatever the character of the author, the plays he has written were as bad before they were produced as they are now that he has been so effectually extinguished; and these librarians cannot comprehend, evidently, that his books were fully as immoral as they are now when they were first put on the shelves. Would it not be a refreshing thing to find a theatrical manager who managed a theater because he had an honest purpose of elevating, perpetuating, purifying and strengthening the drama, instead of speculating in it as a Jew speculates in old clothes? And would it not be a marvel to discover librarian who knew something about books?
Buffalo, New York, is getting to be very classic in some things. It tolerated the nude with great equanimity in the recent Art Exhibition and exhibits the female embodiment of everything ideal, from the German muse of song to the still more German muse of barley products, at the great variety of fests, more or less related to beer, that follow in swift succession in that town. But the classic climax was reached on Good Friday of this year, when the Venus of Milo, mounted on a Bock beer pedestal, was the center piece of an Easter symbol picture in a Hebrew clothing advertisement. The limit of Buffalo congruity seems to have been reached.
The Chip-Munk for May has a bit of folk-lore about a man who advised another to join a conspiracy of silence. This item appeared in 1893 and during 1894 was published by actual count in one hundred and forty-nine newspapers. The editors of The Chip-Munk are a bit slow in reading their exchanges.
The Two Orphans at the Kate Claxton Building, Chicago Stockyards, have a motto on their letter heads that reads, “We are the people and wisdom will die with us.”
The editor of The Baseburner, who claims to be a veritist, states that it is not true that the Garland stoves were named after Ham Garland of Chicago Stockyards; but the fact is Garland named himself after the stoves.
Current Literature recently had a long article on Louise Imogene Guinly. Doubtless the spelling of the name was a typographical error, as the editor probably refers to Miss Louisa Imogene Quinney, who is postmistress at Auburn, New York, and daughter of Richard Quinney, manufacturer of the famous Quinney Mineral Water.
Judge Robert Grant has in preparation a series of articles called “How to Live on a Million a Minute and Have Money to Burn.”
I hear the voice of the editors of The Chip-Munk complaining that Little Journeys, The Bibelot, Chips and other publications are base, would-be imitators of their own chaste periodical. Why, you sweet things, did you know that many hundred years ago a great printer made a book which was printed in black inside with a cover in red and black. I believe this is the thing which you claim is original with yourselves. So far as the rest of the periodicals are concerned I have no means of knowing whether they are imitations or not, but Little Journeys was in type and printed long before The Chip-Munk came out of its hole.
Messrs. Copeland & Day of Boston recently published for Mr. Stephen Crane a book which he called “The Black Riders.” I don’t know why; the riders might have as easily been green or yellow or baby-blue for all the book tells about them, and I think the title, “The Pink Rooters,” would have been better, but it doesn’t matter. My friend, The Onlooker, of Town Topics, quotes one of the verses and says this, which I heartily endorse:
I saw a man pursuing the horizon; Round and round they sped.
I was disturbed at this; I accosted the man.
“It is futile,” I said,
“You can never”——
“You lie,” he cried. And ran on.
This was Mr. Howells proving that Ibsen is valuable and interesting. It is to be hoped that Mr. Crane will write another poem about him after his legs have been worn off.
I was moved to read Mr. Hermann Sudermann’s diverting novel, “The Wish,” upon observing an extended notice in the “Sub Rosa” column of the Buffalo Courier. The writer therein alleged that the novel taught a great moral lesson, and desiring to be taught a great moral lesson I bought the book. It treats of the wish of a girl for her sister’s death in order that she might marry the husband. I suppose the great moral truth is that one should not wish for such things, but I supposed that had been taught in one of the Commandments, which tells of coveting thy neighbor’s wife, and my Sunday School teacher used to tell me that it referred equally to husbands. I was evidently mistaken, and Hermann Sudermann is hereby hailed as a teacher of morals. I should think, from the style of the “Sub Rosa” article, that the writer is a woman. If she is, I’ll bet her feet are cold if she enjoys such things as this:
When Old Hellinger entered the gable room he saw a sight which froze the blood in his veins. His son’s body lay stretched on the ground. As he fell he must have clutched the supports of the bier on which the dead girl had been placed, and dragged down the whole erection with him; for on the top of him, between the broken planks, lay the corpse, in its long, white shroud, its motionless face upon his face, its bared arms thrown over his head. At this moment he regained consciousness, and started up. The dead girl’s head sank down from his and bumped on the floor.
This cheerful book is translated from the German by Lily Henkel and published by the Appletons. I commend it to Mr. Bliss Carman and his shroud washers.
Mr. Thomas B. Mosher, of Portland, Maine, deserves the thanks of the reading public for the issuing of The Bibelot. Each month this dainty periodical comes like a dash of salt water on a hot day, and is as refreshing. After reading the longings and the heartburnings of the various degenerates who inflict their stuff on us these days, Mr. Mosher’s “Sappho” comes and makes us really believe that there is a man up on the coast of Maine who has the salt of the sea and the breath of the pines in him, and is willing to think that there are other people who care for purity and sweetness, rather than such literature as “Vistas” and the plays of Maeterlinck.
When in five consecutive stories, printed in the same periodical, the hero or heroine has ended the narrative by shooting himself or herself, is it not about time to hire somebody to invent some other denouement?
Many a man’s reputation would not know his character if they met on the street.
To be stupid when inclined and dull when you wish is a boon that only goes with high friendship.
Every man has moments when he doubts his ability. So does every woman at times doubt her wit and beauty and long to see them mirrored in a masculine eye. This is why flattery is acceptable. A woman will doubt everything you say except it be compliments to herself—here she believes you truthful and mentally admires you for your discernment.
