Diffeomorphisms of Rn with oscillatory jacobians

Publicacions Ma emá iques, Vol 37 (1993), 255-269 . DIFFEOMORPHISMS OF Rn WITH OSCILLATORY JACOBIANS WALDYR M . OLIVA * , NELSON M . KUHL AND LUIz T . MAGALHAES A bs ac The pape p esen s, mainly, wo esul s : a new p oo o he spec- al p ope ies o oscilla o y ma ices and a ans e sali y heo em o di eomo phisms o R' wi h oscilla o y jacobian a e e y poin and such ha Nm( (x) -- (y)) < NM(x-y) o all x, y E R', whe e Nm(x) - 1 deno es he maximum numbe o sign changes in he componen e zi o zE R", whe e all zi a e non ze o and z a ies in a small neighbo hood o x . An applica ion o a semi- implici disc e iza ion o he scala hea equa ion wi h Di ichle bounda y condi ions is aleo made . I . In oduc ion The p esen pape deals wi h di eomo phisms : Rn -> R'z such ha (x) is an oscilla o y ma ix o all x E R' . Oscilla o y ma ices we e s udied ex ensi ely by Gan mache and K ein (see [G K ]) and p esen e y in e es ing spec al p ope ies . They belong o he class o ma ices ha a e a ia ion-diminishing, Le ., ans o ma ions ha dec ease he numbe o sign changes in he componen s o a ec o . These ma ices ha e applica ions in mechanical sys ems, in app oxima ion heo y and in p obabili y . Many esul s abou oscilla o y ma ices as well as his o ical no es and a long lis o e e en es appea in he book "To al Posi i i y", by S . Ka lin (see [Ka]) . In Sec ion II we decided o p esen ano he p oo o he spec al p op- e ies o oscilla o y ma ices, summa ized in Theo em 2 .12 ; ou app oach is based on some ideas appea ing in [FO-1] and [FO-2], whe e o he kind o ma ices we e also s udied . Theo em 2 .13 shows ha di eomo phisms wi h oscilla o y jacobians sa is ying an addi ional hypo hesis ha e he *This esea ch was suppo ed in pa by he "P oje o BID-USP" and by FAPESP, p os . n° 90/3918-5 . 256  W . M . OLIVA, N . M . KUHL, L . T . MAGALHAES p ope y ha gi en wo hype bolic ixed poin s, he co esponding un- s able and s able mani olds a e ans e sal . The p oo ollows some echniques and, as a ma e o ac , was mo i a ed by he main esul o [FO-1] whe e i is s udied a class o o dina y di e en ial equa ions whose low map a a ixed ime is a di eomo phism wi h oscilla o y jacobian . In Sec ion III, a e a semi-implici double disc e iza ion o he hea equa ion wi h Di ichle bounda y condi ion, we cons uc ed a class o Mo se-Smale di eomo phisms o R' wi h oscilla o y jacobians . The i s au ho wan s o hank Jaume Llib e and Ca les Simó by he in i a ion o isi he "Cen e de Rece ca Ma emá ica" o he "Ins i u d'Es udis Ca alans" whe e in Oc obe 1990 he de eloped his con ibu ion o he p esen wo k . II . De ini ions . Basic esul s . Main Theo em All he ma ices we will use a e eal . 2 .1 . No a ion . Fo a m x n ma ix A, le 's use he no a ions 1 < < min(m, n) ; 2 .2 . De ini ion . o all 1 < < min(m, n), a22 , , aj . J 1<21<i2< . . .<i  <m - ,1< ,2< . . .< <n . A m xn ma ix A is called o ally posi i e (s ic ly o ally posi i e) i A ( il  i2  . . .  a ) > 0  ( esp . > 0) ,1 ice Le ., i all i s mino s a e non-nega i e ( esp . posi i e) . A squa e ma ix A is called oscilla o y i i is o ally posi i e and i some o i s powe AX (X a posi i e in ege ) is a s ic ly o ally posi i e ma ix . 2 .3 . The p oduc C o wo o ally posi i e ma ices A and B is a o ally posi i e ma ix . The p oduc C o wo squa e ma ices A and B, whe e one is s ic ly o ally posi i e and he o he is o ally posi i e and non-singula is a s ic ly o ally posi i e ma ix . i2 . . . i aZaK1 ai2 -2 . . . ) - de K2 . . . K La ¡ , . ., ai  w2 . . . DIFFEOMORPHISMS WITH OSCILLATORY JACOBIANS  257 2 .4 . The p oduc C = AB o wo oscilla o y ma ices is an oscilla o y ma ix . The p oduc A = A j A 2 . . . A  ,, o m >_ n - 1 n x n oscilla o y ma ices is a s ic ly o ally posi i e ma ix . Le A be a o ally posi i e squa e ma ix . Then, A is oscilla o y i , and only i , A is non-singula and aij > 0 o 1 i - j1 = 1 . 2 .5 . Fo a ma ix A, le A* deno e he ma ix de ined by a*, = Then, i A is an oscilla o y ma ix, he same is ue o (_1)i+ia ij . (A*) -1 . 2 .6 . Le J= al b1  0 cl a2 b2 bn-1 0 C n -1 an be annxn Jacobi ma ix . Then, J is ocilla o y i , and only i , he mino s j(1 2 . . . ) , 1 < < n, aj , 1 < j < n, and he coe icien s bi and ci, 1 < i <n- 1, a e posi i e (> 0) . Following Fusco and Oli a, le us de ine now he disc e e unc ional al eady conside ed in [FO-1] . (See also [K-L-S]) . Fo x = (x 1 , . . . , xn) E Rn such ha x i =,A 0, 1 _< i _< n, le N(x) - 1 be he numbe o sign changes in he sequence x1,x2' . . .' x n . Fo an a bi a y x E Rn, de ine . ,,(x) and Nm(x) as he minimum and maximum alue o N(x'), o x' a ying in a small neighbou hood o x wi h (x') i 0 0, 1 _< i_< n . Ex end N o he se N = {x E Rn N,,(x) = Nm(x)} making N(x) = N,,(x) = Nm(x) o all x E N . N is he unc ional used in [FO-1] . The nex wo esul s can be ound in [Ka], and o comple eness we ep oduce he p oo s . 2 .7 . P oposi ion . Le A be an n x m ma ix (m < n) such ha all he mino s A (i1 2 . . . m a e nonze o and Na e he same sign, . . . ) and le xE R m , x =,,~ 0 . Then y =Ax sa is ies NM (y) < m . P oo . . Suppose, a guing by con adic ion, ha he e is a nonze o ec o x E Rm such ha Nm(y) >_ m + 1 (y = Ax) . Le y = ( y 1 , y 2 . . . . , yn) . Then he e exis indices il < i2 < . . . < in, .+1 wi h he p ope y ha o = 1, 2, . . .,m+ 1 he quan i y (- 1)'y'- is o cons an sign, e en in 258  W . M . OLIVA, N . M . KUHL, L . T . MAGALHE E S he possible case whe e se e al o he e numbe s a e ze o . Now sople o he y'-'s a e nonze o ; o he wise, i y i1 , y'2, . . . , y'- a e all ze o, hen he sys em o homogeneous equa ions has a non i ial solu ion, which implies in con adic ion wi h he hypo hesis . Conside now he de e minan P oo . ai,7x7 =0  =1,2, . . .,m A 21 i2 . . . i n , __ (  0 1 2 . . . m) This de e minan is ze o, since yi  = elemen s o he las colunln, we ob ain 1 . ( - 1)m+1+ y i  A 21 i2 . . . 1 2 . . . m1 a? .jx3 . Expanding by he -1 i +1 . . . %m+1 . Now, since he y's al e na e in sign, some o hem being nonze o, and since he A's all Na e he same p ope sign, he igh hand side canno be ze o . This con adic ion comple es he p oo . 2 .8 . Lemma . Le A be an n x m s ic ly o ally posi i e ma iz and le x E R"1, x :7~ 0 . Then y = Ax sa is ies NM(y) < Nm(x) . Le p + 1 = Nm(x) . Then he componen s o x can be di ided in o p + 1 g oups, 1 2 l +1 +2 p  y +1 p + 2 m (x ,x ,. . . . x  ), (x"  ,XVI  , . . .,x  ), . . .,(x  ,x  ,. . .,x  ) whe e each componen in he i h g oup, say, ei he is ze o o has a sign (-1)i+1 . F~z he mo e, he e mus be a leas one nonze o componen in ai11 aá12 . . . ailm y i 1 ai 2 1 ai22 ... ai2m y i2 ai_+1 1 aim+12 . . ai~, +1 m .  Zm+1 DIFFEOMORPHISMS WITH OSCILLATORY JACOBIANS  259 each g oup . Unless n > p + 1, which we assume o be he case, he e is no hing o p o e . We se o = 0, p+1 = m, and cons uc he ec o s and o em . k = j= k-i+l V (i,  i2  . . .  ip+1 1  2  . . .  p+ 1 ) _ji 1 Ixi laiij Qi=1 22= 1+1 whe e aj is he j h column ec o o A . Then _", ~1 IxjIaln+lj  . . .  Ej= m ~,+1 Ixj I a in+ij 21 22 . . . 2p+1 V (1 2 . . . p+1) k=1,2, . . .,p+1 p+1 ( - 1)k+1  Ixjlaj =(-1)k+l k j=1 k=1 j= k-1+1 k= m j= p + 1 I x J Iaáij l x e1 1  X121 . . . ix Q + 1 JA ~ zli2 . . . p+1  )~ L  Q1~2 . . . 2P+1= n+i  ~p+1 whe e V is he ma ix whose k h column ec o is k . By he na u e o he cons uc ion o he blocks weknow ha he e is a leas one selec ion (~1, ~a, . . . , Qp+1) o which Ixel 1 X121 . . . IxeP 1 > 0 . Thus all he mino s a e posi i e . Now y = l : P Ci(-1)k+l k and since V sa is ies he hy- po hesis o P oposi ion 2 .7, we ob ain NM(y) < p-1- 1 = N,, (x) . Lemma 2 .8 will be c ucial in he p oo o he spec al heo em o oscilla o y ma ices . Be o e we go o his spec al heo em, le us p o e ano he change-sign esul , Lemma 2 .9, use ul o he ans e sali y he- 2 .9 . Lemma . Le A be a nonsingula o ally posi i e n x n ma ix . Then NM(Ax) < Nm(x) o any x E Rn . 260  W . M . OLIVA, N . M . KUHL, L . T . MAGALHAES 0 and le K 1 be he se 0 1 E 0 J(E) =  E  1  E  . E (0 E1 By con inui y and by 2 .6 he e exis s Eo > 0 such ha J(E) is oscilla o y o any 0 < E < Eo . Hence, by 2 .4 B(E) = J(E)n -1 is a s ic ly o ally posi i e ma ix o any 0 < E < Eo and using 2 .3 and con inui y we conclude ha C(E) = B(E)A is s ic ly o ally posi i e, 0 < E < Eo, and lim,-o C(E) = A . Thus, o E su icien ly small, N,,,(Ax) < N . .(C(E)x) < NM(C(E)x) and by Lemma 2 .8, applied o C(E), we ob ain o x SA 0 N .(Ax) < ,. .x) . Now, since A is nonsingula , he e exis s a neighbou hood U o x such ha N ,(x) =  max N(x') and NM(Ax) =  max N(y) . x~EuniV  yEAUnN Thus NM(Ax) = N(A .T) o some xE U and by (*) N(A . ) < N,,~(x) < Nm(x)- Since x E U, Nm(x) < Nm(x) and he lemma is p o ed . Following [FO-1] we in oduce now a amily o " ones" in Rn which plays an impo an ole in he spec al and ans e sali y heo ems . 2 .10 . De ini ion . Fo any gi en in ege 1 < i < n le K i be he se Kz={XERn :NM(x)<i} K1 = {0} U in Ki = {0} U {x E Rn : Nm(x) < i} . Rema k . No e ha by Lemma 2 .9, i A is o ally posi i e n x n ma ix, he se s K i and {0} U (Rn K¡) a e in a ian unde A and A-1, espec- i ely . Also, K1 1{0}U(R n Ki)) _ {0} and C$[KiU{ 0 }UR n Ki)] = Rn . Aswe will see, he e in a ian and ans e sali y p ope ies will be use ul o p o e he ans e sali y heo em . We s a e now, wi hou p oo , he gene aliza ion o Pe on's Theo em, wi h he e sion due o C . Fusco and W . M . Oli a, which, oge he wi h Lemma 2 .9, will be he ools o p o e he spec al heo em o oscilla o y ma ices . P oo .. Le J(E) be he n x n Jacobi ma ix DIFFEOMORPHISMS WITH OSCILLATORY JACOBIANS  261 2 .11 . Theo em ([FO-2]) . Le E be an n-dimensional eal ec o s space, le K C E be a closed se wi h non emp y in e io and le T be a linea ans o ma ion o E in o E . Assume ha hl) xEK,aER=>axEK ; h2) max{dim W : W a subspace, W CK}= d, 1 <_ d < n ; and h3) T(K {0}) C in K . Then he e exis (unique) subspaces Wl, W 2 such ha 1) W l n W 2 = {0}, dim W l = d, dim W 2 = n- d ; 2) TW j C W j , j = 1, 2 ; and 3) W l C {0} U in K, W 2 n K = {0} . Mo eo e , i oI(T), Q2(T) a e he spec a o T es ic ed o Wl, W2, hen, be ween al (T) and Q2 (T) he e is a gap A E u1(T), E 0 `2(T) ==> JAJ > Iwl . 2 .12 . Theo em ([G K ]) . Le A be an n x n oscilla o y ma ix . Then, (a) all he eigen alues o A a e eal, simple and posi i e ; (b) I A1 > A2 > . . . > A n > 0 deno e he eigen alues o A and uá is an eigen ec o co esponding o ai, hen ui E V and N(ui) = i ; and (c) I w = j :k j=h Ciu7, whe e 1 < h < k < n, hen w 7É 0 implies h < Nm(w) < N M (w) < k . P oo . I su ices o p o e he heo em o a s ic ly o ally posi i e ma ix since by 2 .4 A'' -i and A' a e s ic ly o ally posi i e ma ices . We e e he eade o [FO-2, Theo em 2] o he p oo , including he ac s ha he se s Ki, 1 < i < n, sa is y hl) and h2), wi h d = i . Lemma 2 .8 implies ha A(K i {0}) C in Ki, 1 <_ i < n, since i x E Ki {0} hen N,,,,(x) < i and hence NM(Ax) < N,,(x) < i ( emem- be ha in K i = {x E Rn : NM(x) < i}) . The e o e Theo em 2 .11 implies ha , o each 1 < i < n, he e is an A-in a ian ¡-dimensional 0 subspace Wi C Ki and an A-in a ian (n - i)-dimensional subspace W2, WZ n Ki = {0} wi h co esponding spec al gap .  Clea ly (wi h W1 =W2 = Rn) we ha e Wi C Wi +l , WZ C W2-1 , 1 <_ i <n . Fo each 1 < i <_ n le Vi = Wi n W2 -1 . V is an A-in a ian sub- space and dim Vi = 1 because Wi n W2i = {0} . I is also clea ha span{Vl, . . . , V n } = R n , he e o e he spec um o A is he se o he 262  W . M . OLIVA, N . M . KUHL, L . T . MAGALHAES ( eal) eigen alues {A1, A2, . . . . A2} co esponding o V l ,. . . . V  , . Since A is s ic ly o ally posi i e, o each 1 <_ <_ n he h ex e io powe o A is a ma ix wi h posi i e coe icien s and by he classical Pe - on Theo em (which is a pa icula case o Theo em 2 .11) one easily concludes ha all he ai's a e posi i e . By he spec al gap we lla e Al > A2 > . . . > An > 0 . Le u i E V i be a nonze o ec o , hen ui E in KZ and he e o e NM(ui) _< i . Also, ui 1 Ki_1 and he e o e N, (u ¡ ) >d- 1 . I ollows ha ui E N and N(ui) = i . The las s a e- men is p o ed simila ly, jus emembe ing ha w = _ k j=h Cjuj 0 0 belongs o he subspaces Wi n WZ -1 . Rema k . Le A be an nx m ma ix . We say ha A is s ic ly sign egula (see zl . . . i  Q [Ka]) i o each 1 < < min(m, n), all he mino s A ~ ki . . . k, a e nonze o and lla e he same sign . Lemma 2 .8 is ue o hese ma ices and hence we lla e he same spec al heo em o s ic ly sign egula squa e ma ices i we dis ega d he posi i eness o he eigen alues and conside hem in dec easing o de o hei moduli . We now p o e he ans e sali y heo em : 2 .13 . Theo em . Le : Rn --, Rn be a Ck di eomo phism, k >_ 1, such ha : (H1) Fo all x, yE Rn, Nm( (x) - (y)) < NM(x - y) ; and (H2) Fo all x E Rn, , (x) is an oscilla o y ma ix . Then, i e - and e+ a e wo hype bolic ixed poin s o , he uns able mani old W''(e - ) and he s able mani old Ws(e+) a e ans e sal . P oo£ Le x E W'u(e - ) n Ws(e+) . Conside he sequence By (H 1), NM(u + ) < NM(u-) . k+l(x)__  k(x) uk  11 k+i(x) _ k(x)I1 Taking a subsequence i necessa y we lla e : lim uk = U+E T e +W s (e + ) k- .+oo li m  uk = u- E T e -W s (e - ) . k -~ _oo k E Z . DIFFEOMORPHISMS WITH OSCILLATORY JACOBIANS  263 Le {u }i=1, . . .,n be a basis o med by eigen ec o s o '(e ) as in Theo em 2.12 and le m : ~ be he dimension o W'(e : ~) . Then, u - á u i c,uz F om Theo em 2 .12 NM(u+) >_ m+ + 1 and Nm(u - ) <_ m - ; since NM(u+)< Nm(u - ), we ha e m + <m - -l . Now we p oceed as in he p oo o he main esul o [F0-11 . In ac , he las inequali y implies ha he n-m - eigen ec o s u .- +15 . . . )un o '(e+) belong e T e +W'(e+) and by Theo em 2.12 we conclude ha E = span{u+m-+1, . . . , uñ } is con ained in {0} U (Rn K m - ) . Since R n K, .- is an open se and Ws(e+) is a smoo h mani old, he e exis s an in ege no > 0 such ha T n o (,,) W' (e+) con ains a (n - m - )-dimensional ec o subspace E con ained in {0} U (Rn K, ,-) . Le E o be he ec o space E ° _ ( -no), .o (x)) E . Then, dim Eo = n- m - and by (H2) we ha e Eo C T,, :W-1(e+) n ({0} U (Rn K m -» . A simila a gumen shows ha 0  0 T ' WU(e - ) C K, n- . Since dim(T S Wu(e - )) = m - and since K m - n ({0} U (Rn K,  ,-)) = {0} we ha e Eo ®T,,Wu(e+) = Rn and he e o e he heo em is p o ed . 2 .14 . Example . Le A be a s ic ly o ally posi i e ma ix and A be an eigen alue o A wi h eigen ec o u . De ine g : Rn __> Rn by 1 xi  xñ g(x) = 2A ~u2 + x1, . . . , u2 + xn 1  n whe e x = (x1, . . . , x n ) and u = (u1, . . . , un) . g is a C°°-di eomo phism wi h de i a i e 3 CUI z + 1 J o allxER' .