Morse decomposition of global attractors with infinite components

Manusc ip submi ed o Websi e: h p://AIMsciences.o g AIMS’ Jou nals Volume xx, Numbe 0x, xxxxx xxxx pp. 1–xx MORSE DECOMPOSITION OF GLOBAL ATTRACTORS WITH INFINITE COMPONENTS Tom´ as Ca aballo, Juan C. Ja a, Jos´ e A. Langa Dp o. Ecuaciones Di e enciales y An´alisis Num´e ico, Uni e sidad de Se illa, Apdo. de Co eos 1160, 41080-Se illa, Spain Jos´ e Vale o Cen o de In es igaci´on Ope a i a, Uni e sidad Miguel He n´andez, A da. de la Uni e sidad, s/n, 03202-Elche, Spain (Communica ed by Associa e Edi o ) Abs ac . In his pape we desc ibe some dynamical p ope ies o a Mo se decomposi ion wi h a coun able numbe o se s. In pa icula , we a e able o p o e ha he g adien dynamics on Mo se se s oge he wi h a sepa a ion assump ion is equi alen o he exis ence o an o de ed Lyapuno unc ion associa ed o he Mo se se s and also o he exis ence o a Mo se decomposi ion - ha is, he global a ac o can be desc ibed as an inc easing amily o local a ac o s and hei associa ed epelle s. 1. In oduc ion. The asymp o ic beha iou o a sys em o (o dina y o pa ial) di e en ial equa ions modeling eal phenomena om di e en a eas o Science is usually desc ibed by he analysis o hei global a ac o s, a compac in a ian se o he associa ed semig oups a ac ing (uni o mly) bounded se s o wa ds in ime. This subjec has ecei ed much a en ion h oughou he las decades (see, o ins ance, [4], [9], [12], [16], [19], [18] o [20]). We ecall now he de ini ion o global a ac o associa ed o a semig oup. Fi s , le Xbe a me ic space wi h me ic d:X×X→R+, whe e R+= [0,∞), and deno e by C(X) he se o con inuous maps om Xin o X. Gi en a subse A⊂ X, he -neighbo hood o Ais he se O(A) := {x∈X:d(x, a)<  o some a∈ A}. De ini ion 1.1. A amily {T( ) : ≥0} ⊂ C(X)is a semig oup in a comple e me ic space Xi : •T(0) = IX, wi h IXbeing he iden i y map in X, •T( +s) = T( )T(s), o all , s ∈R+, •R+×X3( , x)7→ T( )x∈Xis con inuous. 2000 Ma hema ics Subjec Classi ica ion. 37B25, 37L99, 35B40, 35B41. Key wo ds and ph ases. Mo se decomposi ion; in ini e componen s; g adien dynamics; Lya- puno unc ion; g adien -like semig oup. Pa ially suppo ed by FEDER and Minis e io de Econom´ıa y Compe i i idad (Spain) unde g an s MTM2011-22411 and MTM2012-31698, and by Jun a de Andaluc´ıa unde P oyec o de Excelencia P12-FQM-1492. 1 2 T. CARABALLO, J.C. JARA, J.A. LANGA, J. VALERO The no ion o in a iance plays a undamen al ole in he s udy o he asymp o ic beha io o semig oups. De ini ion 1.2. A subse Ao Xis said in a ian unde he semig oup {T( ) : ≥ 0}i T( )A=A o all ≥0. Gi en A, B ⊂X, he Hausdo semidis ance om A o Bis gi en by d(A, B) := sup a∈A in b∈Bd(a, b). De ini ion 1.3. Gi en wo subse s A, B o Xwe say ha Aa ac s Bunde he ac ion o he semig oup {T( ) : ≥0}i d(T( )B, A) →∞ −→ 0. We a e now in a posi ion o de ine global a ac o s. De ini ion 1.4. A subse Ao Xis a global a ac o o a semig oup {T( ) : ≥0}i i is compac , in a ian unde he ac ion o {T( ) : ≥0}and o e e y bounded subse Bo Xwe ha e ha Aa ac s Bunde he ac ion o {T( ) : ≥0}. De ini ion 1.5. The semig oup {T( ) : ≥0}is e en ually dissipa i e i o any bounded se B he e exis s ∗= ∗(B)≥0such ha ∪ ≥ ∗T( )Bis bounded. Rema k 1.6. I is ob ious ha i T( )possesses a global a ac o , hen i is e en- ually dissipa i e. One o he main p ope ies in he s udy o a ac o s is e e ed o he desc ip ion o hei geome ical in e nal s uc u e. Gene ically, a global a ac o is cha ac e - ized by a ( ini e o in ini e) numbe o isola ed in a ian se s and he connec ing o bi s among hem. This ac leads o a Mo se decomposi ion o he global a ac o in e ms o a amily o a ac ing- epelle pai s (see [8,17,11,14,15]). We now in oduce his concep . De ini ion 1.7. Le {T( ) : ≥0}be a semig oup on X. We say ha an in a ian se E⊂X o he semig oup {T( ) : ≥0}is an isola ed in a ian se i he e is an  > 0such ha Eis he maximal in a ian subse o O(E). De ini ion 1.8. A disjoin amily o isola ed in a ian se s is a amily {M1,· · · , Mn} o isola ed in a ian se s wi h he p ope y ha O(Mi)∩ O(Mj) = ∅,1≤i<j≤n, o some  > 0. De ini ion 1.9. A global solu ion o a semig oup {T( ) : ≥0}is a con inuous unc ion ξ:R→Xwi h he p ope y ha T( )ξ(s) = ξ( +s) o all s∈Rand o all ∈R+. We say ha ξ:R→Xis a global solu ion h ough x∈Xi i is a global solu ion wi h ξ(0) = x. I is also well known ha he global a ac o is he union o all bounded comple e global solu ions o he semig oup T. De ini ion 1.10. Le {T( ) : ≥0}be a semig oup which possesses a disjoin amily o isola ed in a ian se s M={M1,· · · , Mn}. A homoclinic s uc u e asso- cia ed o Mis a subse {Mk1,· · · , Mkp}o M(p≤n) oge he wi h a se o global solu ions {ξ1,· · · , ξp}such ha Mkj →−∞ ←− ξj( ) →∞ −→ Mkj+1 ,1≤j≤p, whe e Mkp+1 := Mk1. MORSE DECOMPOSITION WITH INFINITE COMPONENTS 3 Rema k 1.11. He e, ξ( ) →±∞ −→ Mmeans ha d(ξ( ), M)→0as → ±∞. We will s udy he dynamics o he semig oup inside he global a ac o A. We now de ine gene alized dynamically g adien semig oups (see [6,5]). De ini ion 1.12. Le {T( ) : ≥0}be a semig oup wi h a global a ac o Aand a disjoin amily o isola ed in a ian se s M={M1,· · · , Mn}in A. We say ha {T( ) : ≥0}is a gene alized dynamically g adien semig oup ela i e o Mi : a) Fo any global solu ion ξ:R→ A he e a e 1≤i, j ≤nsuch ha Mi →−∞ ←− ξ( ) →∞ −→ Mj. b) The e is no homoclinic s uc u e associa ed o M. Rema k 1.13. The concep o gene alized dynamically g adien semig oup is he same as he concep o g adien -like semig oup as gi en in [1],[5]. To in oduce he no ion o a Mo se decomposi ion o he a ac o Ao a semi- g oup {T( ) : ≥0}(see [8], [17] o [18]) we p e iously need he no ion o a ac o - epelle pai . We ecall ha he omega-limi se o B⊂Xis de ined by ω(B) = ∩ ≥0∪s≥ T(s)B. De ini ion 1.14. Le {T( ) : ≥0}be a semig oup wi h a global a ac o A. We say ha a non-emp y subse Ao Ais a local a ac o i he e is an  > 0such ha ω(O(A)) = A. The epelle A∗associa ed o a local a ac o Ais he se de ined by A∗:= {x∈ A :ω(x)∩A=∅}. The pai (A, A∗)is called an a ac o - epelle pai o {T( ) : ≥0}. No e ha i Ais a local a ac o , hen A∗is closed and in a ian . De ini ion 1.15. Gi en an inc easing amily ∅=A0⊂A1⊂ · · · ⊂ An=A, o n+1 local a ac o s, o j= 1,· · · , n, de ine Mj:= Aj∩A∗ j−1.The o de ed n- uple M:= {M1, M2,· · · , Mn}is called a Mo se decomposi ion o A. De ini ion 1.16. We will say ha a semig oup {T( ) : ≥0}wi h a global a ac o Aand a disjoin amily o isola ed in a ian se s M={M1,· · · , Mn}in Ais a g adien semig oup wi h espec o M, i he e exis s a con inuous unc ion V: X→Rsuch ha [0,∞)3 7→ V(T( )x)∈Ris non-inc easing o each x∈X M, Vis cons an in Mi o each 1≤i≤n, and V(T( )x) = V(x) o all ≥0i and only i x∈ n S i=1 Mi. Vis called a Lyapuno unc ion ela ed o M. I has been p o ed in [1] ha gi en a disjoin amily o isola ed in a ian se s on he global a ac o M={M1,· · · , Mn} o a semig oup T( ), he dynamical p ope y o being gene alized dynamically g adien , he exis ence o an associa ed o de ed amily o local a ac o - epelle s, and he exis ence o a Lyapuno unc- ional ela ed o M, a e equi alen p ope ies. Many o he a gumen s in [1] make a p ecise use o he ac ha he numbe o Mo se se s is ini e. The aim o his pape is o gene alize his esul o he case o a coun able numbe o Mo se se s. Indeed, he gene al heo y o Mo se decomposi ion o in a ian se s is gene ically adap ed o he exis ence o a ini e numbe o isola ed Mo se se s. Howe e , i is no 4 T. CARABALLO, J.C. JARA, J.A. LANGA, J. VALERO unusual o ha e an in ini e numbe o in a ian s in a global a ac o . Fo ins ance, conside he scala di e en ial equa ion dy d = (y) wi h (y) =      −y, i y≤0, (1 −e−y)sin π y,i 0 < y ≤1, 1−y, i y≥1. No e ha he equa ion possesses he ollowing ixed poin s: y1= 1, y2=1 2, y3=1 3, ..., yk=1 k, ..., y∞= 0, wi h hei espec i e associa ed uns able mani olds (see De ini ion 4.2) Wu(1) = 1, Wu1 2= [1 2,1), ..., Wu1 k= [1 k,1 k−1), ..., Wu(0) = 0, and as global a ac o A= [0,1].In [3] he au ho s s udy a mul i alued e sion o he well-known Cha ee-In an e equa ion, also leading o a global a ac o wi h an in ini e numbe o equilib ia, which ac ually has mo i a ed he necessi y o de el- oping he heo y in his pape . We will conside his applica ion in a subsequen wo k. In Sec ion 2we ecall some esul s on he dynamics ela ed o an a ac o - epelle pai . In Sec ion 3we will gene alize De ini ions 1.12,1.15 and 1.16 o he case o an in ini e numbe o disjoin isola ed in a ian se s M∞={Mi}∞ i=1 ∪M∞inside he global a ac o . In Sec ions 4,5and 6we p o e he main esul o his pape , he equi alence be ween a gene alized dynamically g adien semig oup e e ed o M∞wi h a sui able sepa a ion assump ion, he exis ence o an o de ed Lyapuno unc ion associa ed o M∞, and he exis ence o a Mo se decomposi ion on he global a ac o . This is done in se e al s eps: i s , we p o e ha he p ope y o he semig oup o being gene alized dynamically g adien oge he wi h a sepa a ion assump ion implies ha a Mo se decomposi ion can be cons uc ed; hen we p o e ha om a Mo se decomposi ion ela ed o M∞an o de ed Lyapuno unc ion can be de ined; inally, we check ha he exis ence o an o de ed Lyapuno unc ion implies ha he semig oup is gene alized dynamically g adien semig oup e e ed o M∞and ha he sepa a ion assump ion holds. 2. P elimina y esul s on a ac o - epelle pai s. The ollowing esul s on he dynamics on a ac o - epelle pai s a e aken om [1]. We ecall ha local a ac ion o Ain Ais equi alen o local a ac ion in X, o which we i s ly need he ollowing esul . Lemma 2.1. Le {T( ) : ≥0}be a semig oup in Xwi h a global a ac o A. I A⊂ A is a compac in a ian se o {T( ) : ≥0}and he e is an  > 0such ha Aa ac s O(A)∩ A, hen gi en δ∈(0, ε) he e is a δ0∈(0, δ)such ha γ+(Oδ0(A)) ⊂ Oδ(A), whe e γ+(Oδ0(A)) = S x∈Oδ0(A)S ≥0 {T( )x}. The nex esul gene alizes o semig oups a known esul o g oups gi en in [8] and shows ha ou de ini ion o local a ac o is equi alen o ha one in [8,17]. MORSE DECOMPOSITION WITH INFINITE COMPONENTS 5 Lemma 2.2. I {T( ) : ≥0}is a semig oup in Xwi h a global a ac o Aand S( ) := T( )|A, clea ly {S( ) : ≥0}is a semig oup in he me ic space A. I Ais a local a ac o o {S( ) : ≥0}in he me ic space A( ha is, he e exis s ε > 0 wi h ω(O(A)∩ A) = A), and Kis a compac subse o Asuch ha K∩A∗=∅, hen Aa ac s K. Fu he mo e Ais a local a ac o o {T( ) : ≥0}in X. We now desc ibe he dynamics on an a ac o - epelle pai . Lemma 2.3. Le {T( ) : ≥0}be a semig oup in Xwi h a global a ac o Aand (A, A∗)an a ac o - epelle o {T( ) : ≥0}. Then: (i)I ξ:R→Xis a global bounded solu ion o {T( ) : ≥0} h ough x /∈A∪A∗, hen ξ( ) →∞ −→ Aand ξ( ) →−∞ −→ A∗. (ii)A global solu ion ξ:R→Xo {T( ) : ≥0}wi h he p ope y ha ξ( )∈ Oδ(A∗) o all ≤0 o some δ > 0such ha Oδ(A∗)∩A=∅mus sa is y d(ξ( ), A∗) →−∞ −→ 0. (iii)I x∈X A, hen T( )x →∞ −→ A∪A∗. Pa (i) o he p e ious lemma is p o ed in Theo em 1.4 in [17]. Pa s (ii) and (iii) can be ound in [1]. 3. Gene alized dynamically g adien semig oups. In his sec ion we will in- oduce he concep s o gene alized dynamically g adien semig oups and Mo se decomposi ion o a coun able se o isola ed in a ian se s. De ini ion 3.1. A disjoin (coun able) amily o in a ian se s is a amily M∞= {Mi}∞ i=1 ∪M∞o in a ian se s wi h he p ope y ha , gi en j∈N, he e exis s δj such ha Oδj(Mj)∩ Oδj(Mi) = ∅, o all i6=j,i∈N∪ {∞}.(3.1) De ini ion 3.2. Le {T( ) : ≥0}be a semig oup which possesses a disjoin amily o in a ian se s M∞={Mi}∞ i=1 ∪M∞wi h Mjisola ed o each j∈N. A homoclinic s uc u e associa ed o M∞is a ini e subse {Mk1,· · · , Mkp}o M∞ oge he wi h a se o global solu ions {ξ1,· · · , ξp}such ha Mkj →−∞ ←− ξj( ) →∞ −→ Mkj+1 ,1≤j≤p, whe e Mkp+1 := Mk1. Rema k 3.3. The se M∞is no assumed o be isola ed. The eason is ha ypically in applica ions M∞is an accumula ion se o he sequence Mnas n→ ∞. Hence, i is no isola ed. This is he case in he example gi en in he in oduc ion, and also, o ins ance, in he applica ion o mul i alued semi lows in [3]. De ini ion 3.4. Le {T( ) : ≥0}be a semig oup wi h a global a ac o Aand a disjoin amily o in a ian se s M∞={Mi}∞ i=1 ∪M∞in Awi h Mjisola ed o each j∈N. We say ha {T( ) : ≥0}is a gene alized dynamically g adien semig oup ela i e o M∞i o any global solu ion ξ:R→ A such ha ξ( 0)6∈ Mk, o some 0∈Rand any k∈N∪ ∞, i holds ha Mj →−∞ ←− ξ( ) →∞ −→ Mi, o 1≤i<j≤ ∞.(3.2) Rema k 3.5. I is ob ious ha condi ion (3.2) implies he ollowing p ope ies: •Fo any global solu ion ξ:R→ A he e a e 1≤i, j ≤ ∞ such ha Mj →−∞ ←− ξ( ) →∞ −→ Mi. 6 T. CARABALLO, J.C. JARA, J.A. LANGA, J. VALERO •The e is no homoclinic s uc u e associa ed o M∞. When he numbe o se s Miis ini e, hen i is p o ed in [2] ha hese las wo p ope ies imply (3.2) o a sui able ea angemen o he se s. In ac , De ini ion 3.4 is he way in which i is de ined a Mo se decomposi ion o a global a ac o in [18]. No e ha , in pa icula , (3.2) implies ha he e is no global solu ion ξ( ) : R→ A wi h ξ( 0)6∈ M1 o some 0∈Rsuch ha lim →−∞ d(ξ( ), M1)=0. The ollowing lemma implies ha an isola ed in a ian se inside a global a ac- o is compac . Lemma 3.6. Le Mbe an isola ed in a ian se which is ela i ely compac . Then Mis compac . P oo . We need o p o e ha Mis closed. Le yn→y, whe e yn∈M. By he con inui y o Twe ha e ha T( )yn→T( )y o any > 0. Hence, T( )y∈M. Thus, T( )M⊂M o all ≥0. On he o he hand, as Mis in a ian , o any > 0 he e exis s zn∈Msuch ha T( )zn=yn. Since Mis ela i ely compac , passing o a subsequence we ha e zn→z∈M, and hen T( )z=y. The e o e, M⊂T( )M o all > 0. I ollows ha Mis in a ian . As Mis an isola ed in a ian se , we ge M=M. As a consequence o he i s s a emen in Lemma 2.3 we ob ain he ollowing. Co olla y 3.7. I {T( ) : ≥0}is a semig oup in Xwi h a global a ac o A and (A, A∗)is an a ac o - epelle pai o {T( ) : ≥0}, hen {T( ) : ≥0} is a gene alized dynamically g adien semig oup associa ed o he disjoin amily o isola ed in a ian se s {A, A∗}. No e ha (3.1) implies Mi∩M∞=∅, o each i∈N.(3.3) Lemma 3.8. Condi ion (3.2) implies ha he e is no global solu ion ξ:R→ A wi h ξ( 0)∈ A M∞ o some 0∈Rsuch ha lim →+∞d(ξ( ), M∞)=0.(3.4) P oo . I is ob ious, as in (3.2), ha he index icanno be ∞. Lemma 3.9. Le M∞={Mi}∞ i=1 ∪M∞be compac in a ian se s such ha Mj∩ Mi=∅ o i6=j,i, j ∈N∪ ∞, and also suppose ha he in a ian compac se M∞⊂ A is such ha lim i→∞ d(Mi, M∞)=0.(3.5) Then M∞is a disjoin amily o in a ian se s. P oo . Take j∈Na bi a y. We ha e o check (3.1). The e exis s δ1>0 such ha Oδ1(Mj)∩ Oδ1(M∞) = ∅. In iew o (3.5) he e is N > j such ha Mi⊂ Oδ1 2 (M∞) i i > N. MORSE DECOMPOSITION WITH INFINITE COMPONENTS 7 Hence, Oδ1(Mj)∩ Oδ1 2 (Mi) = ∅i i > N. Ob iously, he e exis s δ2>0 o which Oδ2(Mj)∩ Oδ2(Mi) = ∅ o 1 ≤i≤N,i6=j. Then he esul ollows o δj= min{δ1/2, δ2}. We can now in oduce he concep o a Mo se decomposi ion e e ed o M∞. De ini ion 3.10. Gi en an inc easing amily ∅=A0⊂A1⊂ · · · ⊂ An⊂ · · · A∞= Ao local a ac o s, o j∈Nde ine Mj:= Aj∩A∗ j−1,M∞=∩∞ j=0A∗ j.The o de ed coun able se M∞:= {Mi}∞ i=1 ∪M∞is called a Mo se decomposi ion o A. The ollowing p ope ies o he se s Mj ollow. Lemma 3.11. M∞∩Aj=∅ o any j∈N.Hence, M∞⊂A∞ ∪∞ j=1 Ajand M∞∩Mj=∅ o all j∈N. P oo . Le y∈M∞. Then y∈A∗ j, o any j∈N, implies y6∈ Aj o all j∈N. Lemma 3.12. The se s Mj,j∈N∪ ∞, a e compac . P oo . Since Mj⊂ A, hey a e ela i ely compac . Also, as Mja e he in e sec ion o closed se s, hey a e closed. We can also gi e he ollowing cha ac e iza ion. P oposi ion 3.13. Le {T( ) : ≥0}be a semig oup wi h he global a ac o A and M∞={Mi}∞ i=1 ∪M∞a Mo se decomposi ion o Awi h he amily ∅=A0⊂ A1⊂ · · · ⊂ A∞=Ao local a ac o s. Then, ∞ j=0 (Aj∪A∗ j)=( ∞ [ j=1 Mj)∪M∞. P oo . I z∈ ∞ S j=1 Mj, le k∈Nbe such ha z∈Mk=Ak∩A∗ k−1. Hence z∈Ak⊂Ak+1 ⊂ · · · ⊂ A∞and z∈A∗ k−1⊂A∗ k−2⊂ · · · ⊂ A∗ 0. Thus z∈( ∞ j=k Aj)∩( k−1 j=0 A∗ j)⊂  ∞ j=k (Aj∪A∗ j) ∩  k−1 j=0 (Aj∪A∗ j) = ∞ j=0 (Aj∪A∗ j), p o ing ha ∞ S j=1 Mj⊂ ∞ T j=0 (Aj∪A∗ j).I z∈M∞, hen z∈ ∞ T j=0 A∗ j⊂ ∞ T j=0 (Aj∪A∗ j). Con e sely, we ake z∈ ∞ T j=0 (Aj∪A∗ j).I z∈ ∞ T j=0 A∗ j, hen z∈M∞. O he wise, z∈Aj o some j∈N. Deno e I:= {i1, i2,· · · , ik, . . . }and J:= {j1, j2,· · · , jl, .. . } such ha I∪J=Z+wi h I∩J=∅and z∈Ai o all i∈Iand z∈A∗ j o all j∈J. Clea ly, i i:= min I, necessa ily I={j≥i}and J={0,1,· · · , i−1}, consequen ly z∈Aiand z∈A∗ i−1. So, z∈Ai∩A∗ i−1=Mi, om which ∞ T j=0 (Aj∪A∗ j)⊂ ∞ S j=1 Mj. 8 T. CARABALLO, J.C. JARA, J.A. LANGA, J. VALERO 4. Cons uc ion o a Mo se decomposi ion om he dynamics on M∞. In his sec ion we desc ibe he cons uc ion o a Mo se decomposi ion o he global a ac o A ela i e o he disjoin amily o in a ian se s M∞={Mi}∞ i=1 ∪M∞ in Asuch ha Mjis isola ed i j∈Nand sa is ying (3.2). By Lemma 3.8 we ha e ha (3.4) does no hold. The ollowing lemma will play an impo an ole in wha ollows. Lemma 4.1. Le {T( ) : ≥0}be a semig oup wi h a global a ac o Aand he disjoin amily o in a ian se s M∞={Mi}∞ i=1 ∪M∞={M1, . . . , Mn, . . . ;M∞}in Abe such ha Mja e isola ed o j∈N. Assume ha Tis gene alized dynamically g adien ela i e o M∞. Then, M1is a local a ac o o {T( ) : ≥0}. P oo . We i s ly p o e ha o all δ∈(0, δ1) he e exis s δ0∈(0, δ) such ha γ+(Oδ0(M1)) ⊂ Oδ(M1), whe e δ1sa is ies Oδ1(M1)∩ Oδ1(Mi) = ∅ o i > 1 o i=∞. I no , he e exis 0 < δ < δ1and sequences { k}k∈No posi i e imes and {xk}k∈N o poin s in Xsuch ha o all k d(xk, M1)<1 k, d(T( k)xk, M1) = δ and d(T( )xk, M1)< δ o ∈[0, k). Thus, i we de ine, o each k, ξk( ) := T( + k)xk o ∈[− k,∞),as k→ k→∞ ∞, we conclude ha he e exis s a global solu ion ξ:R→X o T(·) such ha ξk→ k→∞ ξ uni o mly in compac se s o imes (see [7, Lemma 3.1]). Then, d(ξk( ), M1)≤δ o ∈[− k,0] implies d(ξ( ), M1)≤δ < δ1 o ≤0. Bu by (3.2) we ha e ξ( )→Mj,wi h j > 1, as → −∞, a con adic ion. M1is he maximal in a ian se in Oε(M1) o some ε > 0. Thus, o δ < min{ε, δ1} ake δ0∈(0, δ) such ha γ+(Oδ0(M1)) ⊂ Oδ(M1), so ha ω(Oδ0(M1)) ⊂ Oδ(M1)⊂ Oε(M1), and hen, as ω(Oδ0(M1)) is in a ian , ω(Oδ0(M1)) ⊂M1. The o he inclusion is i ial, so ha M1is a local a ac o . Fo M1a local a ac o , le M∗ 1={x∈ A :ω(x)∩M1=∅}be i s associa ed epelle , so each Mi, wi h i≥2,is con ained in M∗ 1and mo e gene ally he o bi ξ(R) o any global solu ion ξ:R→ A ha con e ges o Mi,i≥2, when →+∞,is con ained in M∗ 1. Conside ing he es ic ion {T1( ) : ≥0}o {T( ) : ≥0} o M∗ 1 we ha e ha {T1( ) : ≥0}sa is ies (3.2) in he space M∗ 1wi h he in a ian se s {Mi}∞ i=2 ∪M∞and we may assume, by he las lemma, ha M2is a local a ac o o he semig oup {T1( ) : ≥0}in M∗ 1. I M∗ 2,1is he epelle associa ed o he local a ac o M2 o {T1( ) : ≥0}in M∗ 1we may p oceed and conside he es ic ion MORSE DECOMPOSITION WITH INFINITE COMPONENTS 9 {T2( ) : ≥0}o he semig oup {T1( ) : ≥0} o M∗ 2,1and hen {T2( ) : ≥0} sa is ies (3.2) in M∗ 2,1wi h he associa ed in a ian se s {Mi}∞ i=3 ∪M∞. Se ing A=: M∗ 0,−1and M∗ 1,0:= M∗ 1, o j≥1 we ha e ha Mjis a local a ac o o he es ic ion o {T( ) : ≥0} o M∗ j−1,j−2whose epelle will be indica ed by M∗ j,j−1. De ini ion 4.2. Le {T( ) : ≥0}be a semig oup. The uns able se o an in a ian se Mis de ined by Wu(M) := {z∈X: he e is a global solu ion ξ:R→X such ha ξ(0) = zand lim →−∞ d(ξ( ), M)=0}. De ine A0:= ∅,A1:= M1and o j= 2,3,· · · , Aj:= Aj−1∪Wu(Mj) = j [ i=1 Wu(Mi).(4.1) Also, A∞=A. I is clea ha A=∪∞ i=1Wu(Mi)∪Wu(M∞). Lemma 4.3. Assume he condi ions o Lemma 4.1. Then M∞=∩∞ j=0A∗ j. P oo . Le z∈M∞. Then as M∞is in a ian , ω(z)⊂M∞. Then zcanno be in Wu(Mj) o j∈N, as in such a case by (3.2) we would ha e ω(z)∩Mi6=∅ o some i≤j, a con adic ion. Thus, by (4.1) we ha e ha z6∈ Aj o j∈N. Hence, ω(z)∩Aj=∅, so ha z∈ ∩∞ j=0A∗ j. Con e sely, le z∈ ∩∞ j=0A∗ j. Then ω(z)∩Aj=∅ o all j∈N. I z6∈ M∞, we ake a global solu ion ξ(·) such ha ξ(0) = z. Then by condi ion (3.2) we ha e ha ξ( )→Mias →+∞ o some i∈N. Bu hen ω(z)∩Ai6=∅, a con adic ion. Lemma 4.4. Assume he condi ions o Lemma 4.1. Then he se s Mj,j∈N∪ ∞, a e compac . P oo . We no e ha Mj⊂ A implies by Lemma 3.6 ha he se s Mja e compac i j∈N. Also, Lemma 4.3 implies ha M∞is closed, and hen M∞⊂ A implies ha i is compac . Lemma 4.5. Assume he condi ions o Lemma 4.1. Suppose ha , gi en j∈N, he e exis s δjsuch ha Wu(Mj)∩ Oδj( ∞ [ i=j+1 Mi∪M∞)) = ∅.(4.2) Then, Aj∩ Oδj(( ∞ [ i=j+1 Mi)∪M∞) = ∅.(4.3) P oo . Fo j= 1 he esul ollows since A1=M1=Wu(M1). Suppose (4.3) is ue o j−1 and we will show i o j. I no , he e exis s a sequence {xk}k∈Nin Ajsuch ha o all k d(xk,( ∞ [ i=j+1 Mi)∪M∞)<1 k. As Aj:= Aj−1∪Wu(Mj) and we ha e (4.3) o j−1, hen xk∈Wu(Mj), om which, by hypo hesis, we ge a con adic ion. 16 T. CARABALLO, J.C. JARA, J.A. LANGA, J. VALERO We can now conclude ou main heo em. Theo em 6.7. Le {T( ) : ≥0}be a semig oup wi h global a ac o Aand conside a disjoin amily o in a ian se s M∞={Mi}∞ i=1 ∪M∞in Asuch ha Mja e isola ed o j∈N. Then, he ollowing condi ions a e equi alen : 1. {T( ) : ≥0}is a gene alized g adien semig oup wi h espec o M∞in he sense o he De ini ion 5.1 and M∞is o de ed wi h espec o he espec i e Lyapuno unc ion. 2. {T( ) : ≥0}is a gene alized dynamically g adien semig oup wi h espec o M∞(as in De ini ion 3.4) sa is ying (4.2). 3. M∞is a Mo se decomposi ion o A. P oo . I is a s aigh o wa d consequence o Theo em 4.8 and P oposi ions 5.3,6.3 and 6.6. Co olla y 6.8. Le {T( ) : ≥0}be a semig oup wi h global a ac o Aand conside a disjoin amily o in a ian se s M∞={Mi}∞ i=1 ∪M∞in Asuch ha Mja e isola ed o j∈N. Assume ha M∞is a Mo se decomposi ion o A. Then A=∪∞ j=1Wu(Mj)∪Wu(M∞). P oo . In iew o Theo em 6.7,{T( ) : ≥0}is a gene alized g adien semig oup wi h espec o M∞in he sense o he De ini ion 5.1 and M∞is o de ed wi h espec o he Lyapuno unc ion. Hence, he esul ollows om P oposi ion 6.5. Acknowledgemen . 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