DSpace at VNU: The extreme value of local dimension of convolution of the cantor measure

DSpace at VNU: The extreme value of local dimension of convolution of the cantor measure tài liệu, giáo án, bài giảng ,...

V N U Journal o f S c ie n c e , M athem atics - P hysics 25 (2 0 ) -6 The extreme value o f local dim ension o f convolution o f the cantor measure Vu Thi Hong Thanh^’*, Nguyen Ngoc Quynh^, Le Xuan Son^ ^ D ep a rtm en t o f M a th e m a tics, Vĩnh U n ive rsity ^ D e p a rtm e n t o f F u n d a m en ta l S cience, Vietnam A c a d e m y o f T d itio n a l M e d ic in e Received March 2009 Abstract Let /i be the m —fold convolution of the standard Cantor measure and be the lower extreme value of the local dimension of the measure ịx The values of g t ^ for ~ 2, 3, were showed in [4] and [5] In this paper, we show that 427 log i ^ ( v / l c o s ( — ^ V a*: — ) + 5)1 ^ yj ^ log This values was estimated bv p Shinerkin in [5], but it has not been proved K e y w o rd s: L(x:al dimension, probability meaiiure, standard Cantor measure 2000 AMS Mathematics Subject Classification: Primary 28A80; Secondary 42BI0 Introduction Let He contractive similitvules on and {pỹ}Ị"(0 < Pj < 1, Pj ~ j= \ probabilitv w eights Then, there cxivsts a unique probability measure /i satisfying m(^ ) j= i for all Borcl measurable sets Á (see [1]) Wc call /X a self-sim ila r m easure and a system iterated functions Syn iirc similarities with equal contraction ratio p € ( , ) on R , i.e., S j { x ) ~ When p {x 6j ) , 6j G R for j = I j m , the self-sim ilar measure /X can be seen as follows; Let Xo, X \^ be a scqucncc o f independent identically distributed random variables each taking real values i , b m oo with pn)bability P \ , Pm respectively Wc define a random variable s = Y l P^Xu then the probability i= i measure Ịip induccd by 5*; = P{UJ : S{uj) G A ) is callcd a fr a c ta l m easure and ỊẤp = fj, (see [ ]) Let u be the standard Cantor measure, then I/ can be considered to be generated b y the tw o maps Si { x) = + I?!, Í = ,1 with weight ị on each Si CorrcKponding author Eĩ*mail: vu_hong_thanh@ yahoo.com 57 Then the attractor o f this system 58 VT.ỈỈ Thanh et a i / w Jou fn al o f Science, M athem atics - P h ysics 25 (2009) -6 u c r d ic d lu n c liu n s IS th e sia n d a r u c a n t o r s e t o , I.e., c = ) u L ci ^ ~ * * / / t)c ÍÌÌC m —fold convolution o f the standard Cantor measure For m > 3, this measure docs not satisfy the open set condition (see [2 ]), so the studying the local dim ension o f this measure in this case is vcr> difficult Another convenient w ay to look at /X is as the distribution o f the random sum, i.e., Ị-1 can be obtained in the follow ing way: Let X be a random variable taking values { , , m } with probality Pi — P { X — i) — = , , m and let be a sequence o f independent random variable ^ n witfi the same distribution as X Lets ~ Sn — anJ be th j= i measurc o f , Sn respectively It is w ell known that /X is either singular or absolutely continuous (see [2 ])Recall that let /i be a probability measure on R For G su pp fi, the local dim ension o f Ị.Ì at s is denoted by a { s ) and defined by / ^ a {s) = lim - -^—7 /1^ 0+ log h \ o g f i { B h { s ) ) i f the lim it exists, oth erw ise, let a ( ) and a { s ) denote the upper and low er dim ension by taking the upper and lower limits respectively Let E — { « ( ) : € su pp /x} be the set o f the attainable local dim ensions o f the measure fi and for each m — , , put = inf{a(s) : e supp n}\ ăm = s u p { ă (s ) : s G su pp /x} It is showed in [4] that ã m = is an isolated point o f E for all m = 2, 3, and log log a„, = log 0.63093 if m = 2; - pí; 0.89278 if m = or This results were proved b y using combinatoric, it depends on som e careful counting o f the multiple oo representations o f 6’ = 3~^Xj , Xj — , rn, and the associated probability After that, in [5], Pablo j= i Shmerkin showed the for rn = 2, ,4 by the other way He used the spectral radius o f matrixes to define his results He said that the identifying formulae for and he only estimated the values o f for m > was a difficult problem, for ^ m ^ 10 N ow , in this paper, w e are interested Ú1 the identifying for m = and w e show that our result coincides with Pablo Shmerkin’s estimate We have Main result Main Theorem be the 5—fold convolution o f the standard Cantor measure, then the lower extreme value o f the local dimension o f n is Le( log [ ^ ( v ^ c o s ( ^ " ^ 3^ log ) +5) I 972638 V.T.ĨỈ Thanh et a l / VNƯ Journal o f Science, M athem atics - Physics 25 (2009) 57-68 59 The proof o f our Maim Theorem is divided in to tw o steps In Section 2.1 w e w ill give some notations and primal*)^ results The Main Theorem is proved in Section 2.2 2.1 N o ta tio n s an d P rim a ry Results Let be the standard Cantor measure and /X — z/ * ♦ Ỉ/ (m —fold) Then, b y similar proof as the Lemma 4.4 in [5], w e have P rop osition Let u be the standard C antor measure, i.e., u is induced b y the two maps S i { x ) = - , wUh weigh ^ on each Si ITĩen its m —f o ld convolution 4- ị ỉ with weight hv S i { x ) — ^ on with Si fo r i = Ư^ u is gen erated =0,1 , P rop osition ( [ | ) L e t n i > 2, then aÍA*) — lim p ro vid ed that the lim it exists Otherwise, we can replace a(,s) b v a ( 6‘) and a { s ) and sider the u pper and the low er limits Put D ^ {Oj 1, , } and for cach n € N w e denote : Xi e D } D ^ ^ { ( x i , X , ) : a;i G D } - {(xi, = { ( 2/ , ,yn) e D - : ((x,, i=:\ if ( 3^], € ((x'l, Xfị) und then w c denote (zi , \ \ , J Zifi) (xVt-f , ( 21 , i= l ~ ( xi , Clearly that if (zi, ~ Xm) then 2^ ) ~ ( xj , 1) ( Wc denote {{Xu ,Xnyx)) = {(ỉ/l, ,yn,x) : € ((xi, ,xj)} Thu follcnving lemma w ill be used tVen\iently in this paper Lemm a Lcl s„ = s'j - - j-i -'a;' b e tw o p o in ts in su pp ịi,i I f Sn = S’J, then X ,1 = j-i (mod 3) Proposition Let X = ( X] , x-2 , ) = ( 2, 3, , , ) e D °°, w e have i) I f n IS even then ( y i , (?71 , 2/,i) e ( ( T i , a ; „ ) ) = ( ( , , , , ) ) i f f Vn) € ((X], it) I f n is odd Ihen (yi , { y \ , - , y , t ) G {(.Ti, or (yi, ,yn) e ( ( x i , x „ _ 2, o:„_2 , )) e { ( z i , x „ ) ) = ( ( , , , , , ) ) if f or (yi, ,y„) e ( ( xi , x„_ , a:„-2 , 5)) Proof ;) The ease n is even If ( ỉ / i , - , ỉ /,0 € ((aJi, ,a;n)) = ( ( 2, , , 2, 3) ) then w e have (y, - ) " -' + {V2 - )3 " -2 + + Therefore, a) If _ )3 + (y„ - 3) = - 3= (mod == then 3) Since yn e D , w e have Vn = ^ or n -l -3 = 0.By (2) w e have ^ 3~^yj = j= l ((xi, % (1) w c have ( 2/ , € { ( x i , X „ _ 1, 3)) (2) yn = n -1 Y , Hence, t=i 60 V.T.H Thanh et al / VNƯ Jou rn al o f Science, M athem atics b) If t/n = then (yi - 2)T-'^ - P h ysics 25 (2009) 57-6 ~ = —3 B y (2) w e have + ÌV2 - 3)3"-^ + + (2/„_2 - 3)3 + (2/„_i - 3) = e ( ( , , , , ) ) = ( ( x i , , x „ _ 2, x „ _ 2)) B y ( l ) w e h a v e (yi, Hence, Conveserly, if ( 2/ , , y„) G { So we consider ứie case (yi, ( x i , e 3) ) , then w e have G ( ( x i , a ; „ _ , x„_2 , 0)) Then we have 2/n = and ( y i , y „ _ ] ) ( ( ( 2, 3, , , , ) ) We w ill show that { y i , , y n ) € ( ( xj , ( ( 2, 3, , , , ) ) , b y Lemma w e have t/n- — = (mod 3) In fact, since ( 2/ , y u- i ) e This im plies that y n - i = or 2/n -l “ - a) If t/„_i = then — — and (y i - ) " - + (^2 - )3 " -^ + f (yn - - )3 + (y „_2 - 3) = Therefore, ( 2/ , y n - 2) ~ ( , , , ) = { x i , , X n - 2) and (y „ -i,2 /n ) = ( , ) ( , ) , b y ( ) w e have ( 2/ , •••, 2/n) e ( ( x i , x „ ) ) b) If 1/„_1 = then from (yi , ~ ( , , , , , ) w e get ( 2/1 - )3 ” -2 + _ ) 3n - + Since ( , ) ~ - )3 - = Hence, (yi - 2) 3” - + (y2 - 3) 3” -^ + + (y„_3 - 2)3 + Therefore, y „_2 - V n - - = (3) = (mod 3) Since Tjn- e D , w e have y „_2 = or y „_2 = 1- We consider the tw o follow ing cases C a s e have y „_2 = 4, then ( 2/ „ - , 2/ n - i , 2/n) = ( , , ) and y „_2 - = By (3) we e ((2, 3, 2, 3, )) Sincc ( , , ) ~ ( , , ) , by (1) w c Viuvc (yi, y,i) - (2/ ) •••) 2/n- ) 4,0, 0)G ((2,3, ,2,3)) = ((x i, , ®n)) • C a s e y,i _2 = 1, then y „_2 - = - and (y „ _ , y n - , 2/n) = (4, , ) From (3), w e get ( 7/1 - 2)3'*-^ + ÌV2 - 3)3'^-^ + + (y„_4 - )3 + 2/„_3 - = (4) Therefore, ( y i , Vn-s) e ( ( , , , , ) ) By similar argument, we get T/ „_ = or y„ - = +) I f y „_3 = then { y n - , y n - , y n - u y n ) = ( , , , ) and from (4) w e get ( y u - , y n - ) e ( ( , , , ) ) Since ( , , , ) ~ (2, 3, , ) , w e get (t/i, yn) € ((2, , , ) ) = { ( x \ , X n ) ) +) If y„_3 = then the form (4) is sim ilar to the fom i (3) Thus, b y repeating about argument w e get the proof o f the proposition in this case o f n a) The case n is odd Assume that G ( ( x i , a : „ ) ) = ( ( , , , , ) ) then (t/i - ) " -i + (y - )3 " -2 + + (y „ _ i - )3 + y„ - = This implies 2/n “ or = a) If 1/n = then from (5), w e have {yii This means •” 2/n-i) ^ ((2j , , ) ) = (5) V.T.ỈỈ Thanh et aỉ / VNƯ Journal o f Science, M athem atics - P hysics 25 (2009) 57-68 61 b) I f ijri = then from (5), w e have ( 2/1 - )3 " -2 + ( 2/2 - 3)3" -^ + + (ỉ /„_2 - )3 + Vn-X - - y„_i ) ~ ( , , , , , , ) = (a:i, ,a :„ _ 2,a ;„ - )- This im plies Therefore, ( y i , {y\ Ĩ x „ _ 2, Vn) Ễ ( ( x ] , 5)) Comversely, if (yi, G ( ( x i , X „ _ 1, 2)) then w e have inunediatelythat (j/ , y „ ) G { {xi , ,Xn)) So ’ive consider the follow ing case (z/i, •••,2/n) e {(®1 , ,x„_2,x„_2,5)) then w c have y„ = and (yi I ■••) V n - 1) s ((ajj, X r i — , ^ n - 2) ) — ((2,3, ,2 ,3 ,2 ,2 )) We will prove that ( 2/ , , y„) e ( ( x i , a:„)) In fact, since ( 1/ , y„_i ) e ((2, , , , , ) ) , w e have ( 2/1 - 2)3'^-^ + (t/ - 3)3"-=' + + (y „ _ - 2)3 + y „ - i - = (6) Therefore, y„_i - or Un-X = a) If ỉ/n- i = then from ( ), we have ( 2/ , ( ) Since (2, 5) ^ (3, ), by (1) w e have (t/ i , y „ _ 2) e ( ( , 3, , , , ) ) and = (( , , , , ) ) - ( ( x i , x j ) € b) If Ijn-I = then from ( ), w e have f iv2 - + + (jy, - Therefore, = or Vn- = b l) li'yn - = then from (7), w e have ( y i , ( ) Sincc ( , , ) - (2, 3, 2), by (1) w e have (j/„ - m + v„ - = n (7) y„_a) G ( ( , , , 2, 3) ) and (ỉ/„_ , ! / n - i , y n ) = ( y i , - - - , 2/u) e ((2, , , , , ) ) = b2) If y „ - = then from (7), w e have ( 7/ , y „ _ ) e ((2, , , , , ) } and ( y „ - , y „ - i , Vn) = ( ) Sincc ( , , ) ~ ( , , ) , by (1) w e have ( i J u - , y n ) e ( ( , , , , , , ) ) Therefore, by repeating above aigumcnt for the case yn - — and i y u - , y n ^ ) ( ( x , , , x „ _ 2, x „ _ )) = {( , , , , , , )) Wc have the assertion o f the proposition From Proposition w e have the follow ing corollary C orrolary Let X = { X\ , X , ) = ( , , , , ) G D °° F or each n G N, p u t Sn = ~ ‘xj and t=i = E where (x 'l, ụ-ìisi) = o = { x x , X „ _ 1, x „ _ i) Then we have ^ ( 52) = ^ ) M2(-S2) = 62 V.T,H Thanh et al / VNƯ Jou rn al o f Science, M ath em atics - P h ysics (2009) 57-68 Proof, i) For n = w e have {(oTi)) = ((Xj)) = { ( ) } T herefore, ịi , {s^) = ^ , { s \ ) = P { X , = ) = Ệ For n = we have { { X\ , X )) — { ( , ) , ( , ) } and { { x \ , x )) = { ( , ) , ( , ) } T herefore, 10 10 10 _ 110 — ^' ^ ^ ^ ' ^ ~ ^“ ’ , - 10 10 _ 105 25 '25 ’25 “ 210’ a) By Proposition 3, we have a) If n is even th en ((a;i, = ((a:i, ,x„_i,3))U((xi, ,a;^_i,0)) b) If n is odd then ( ( x i , a : „ ) ) = { ( x i , Z „ _ , 2)) u { { x \ , < _ , 5)) T h erefore, for all n € N w e have /x„(s„) = P { Xn = )/i„_i(s„_,) + P { X n = _ 10 , - , The corollary is proved □ To have the recuưence formula of ịJLn{sn)y we need the following pn^osition P rop osition L et X ^ (X ,X 2, ) = ( , , , , ) € { x u , x „ - i , x n - ỉ ị TTien w e have F o r each n € N, p u t ^ i) I f n is even then (2/ , ,y„) G ( (x 'i, ,0 ) = ((2,3, ,2,3,2,2)) (yi, V r ) e ( ( xi , ) ) i i ) I f n i s o d d then ( y i , , y « ) e {{ x\ , (2/ ) Vn) € ((a:i, ) u { ( a ; i , x „ _ , , 5)> u ((x' j, , 5)) = ((2,3, ,2,3,3)) ) u (( xi , x„_ , 4,0)) u ( ( x j, x ^ _ 2i 1) o))' Proof, i ) The case n is even a) If ( ĩ / i , , yn) e ( ( x i , , X„ _ , )) th en y„ = and ( y i , y „ _ i ) € ( ( x i , T h e r e f o r e , by ( ) w e have ( y i , - , y n ) e ( ( , , , , , , ) ) - ( ( x '„ ,x ; ) ) b) If (ĩ/i, ,yn) e ((xi, ,a:„_ , 1,5)) then y„ = 5, y„-i = and (ỉ/i) •••1 y n - ) ẽ ((^^I1•••) ^n-2 )) = ((2) 3, J 2,3)) Since (1,5) ~ (2,2), by (1) we have (yi,-,yn) e ((2,3, ,2,3,2,2)) = c) If (yi, ,y„) e ((a:i, ,x„_ , , )) = ((2,3, ,2,3,2,2,4,5)) then by (1) we have (yi, ,y„) € ((2,3, ,2,3 ,2 ,2) ) = since (2 , ,4,5) ~ (2,3, , ) V.T.ỈỈ Thanh et a i / VNƯ Journal o f Science, M athem atics - P hysics 25 (2009) 57-68 C onversely, if € ((2, , , , 2, 2)) th en we have (y, - 2)3"-> + H ence, — or — a) I f yj^ ~ th en yrt, {V2 - 3)3" + + (y„-i - 2)3 + — — Hence, T h erefore, ( 7/ , i/„) e ( ( x i , - = (9) from (9), we get (y i, •••,ỉ/n -i) e ((2 ,3 , ,2 ,3 ,2 ) ) = b) I f 63 ((x i, X„ _ 1, 2)) = th en yn — ~ z Hence, from (9), we get (y, - 2)3"-2 + (2/2 - 3)3"-^ + + (t/ „ _ - 3)3 + 2/„-i - = (10) TÌÙS i m p l i e s / n - i “ o r Un-I ~ b l ) I f yn-\ = t h e n f r o m ( ) w e h a v e ivi) y r i - 2) ^ ( ( j , , T h e r e f o r e , { y] , , ĩ j n) G ( ( x i , x „ _ 2, ) ) = ( ( x i , Xn~2)}’ 1, ) ) b ) I f yn-\ = t h e n f r o m ( ) w e h a v e { y u : , y n - ) e ( ( , , , , , , ) ) = ( ( x ; , , x ; _ 2) ) Therefore, ( y i , , y „ ) € ( ( x i , a ; „ _ 2, , 5) ) n ) T h e c a s e n is o d d a) C le a r ly t h a t if ( y i , ,y „ ) G ( ( x i , X „ _ 1, ) ) t h e n {yi, ,yn) e ((2,3, ,2,3,3)) = t>) If { y \ , - , yn) e ( ( x i , ,x„-_ 2, , )) = ( ( , , , 2, 3, , , ) ) th en by ( ) w e have ( 2/ , Vn) e {(2, , , , ) ) = { { x \ , Since (4, 0) ~ (3, c) If (i/ , ,yn) e ((a;'), , < ^ 2- I.O)) ( ( , , , , , , , 0)) th en by ( ) we have { y u - , y n ) {(2,3, ,2,3,3)) = ( ( x ' , , ) , sin ce (3, 1,0) ~ ( , , ) , C o n v e r s e l y , i f { y \ , , y n ) € { { x\ , = ( ( , , , , ) ) , then w e have (yi - ) " -' + { y - 3)3" -" + + Hencc, = or a ) I f Vn "= t h e n - )3 -f 2/„ - = (11) = 2/ri — — H en ce,from (1 ) we have e ((2 ,3 , ,2 ,3 )) = ((x i, T h e r e f o r e , b y ( ) w e h a v e ( 2/ , , y „ ) G ( ( X ] , X „ _ , ) ) b) If yri = th en yn — = —1 Hence, from (11) we have (y, - 2)3'^-" + T h is im p lies y „ _ ] = {V2 - 3)3"-" + + (y„_2 - 2)3 + - = 0.(12) o r y „ _ i = b l) If yn~ \ = th en 2/„_i - = - H ence, (2/1 , T h is im p lies ( y i, ,y „ ) G V n - 2) from (12) we have G ( (2 ,3 ,2 ,3 ,3 ) ) = x ^ _ 2, , ) ) { { x \ , x;_ 2)>- V.T.ĨĨ Thanh et a ỉ 64 / l ^ Journal ( f Science, M athetnatics - P hysics 25 (2009) 57-68 b2) If ĩ/ri-i “ th en ĩjn-\ —4 = llen ce, from (12) w e have (yi 5•'-ỉ ỉ/n-2) ^ (('•^) T \ủ s im plies (t/i, , 2/n) € ( ( x i , ***í 2, 3, 2)) = ((xị J X n - )) • 4, 0)) T h e prop osition is proved □ From Proposition 4, w e have the follow ing corollary, w hich will be used to establish the rc:currence formula o f //n(5n) for n € N C orrolary Lei X = — (2, 3, 2, 3, ) G where F or each n € N, p u t Sn = = (xu ,Xr,-uXn-x), have i=\ = ^ / i u - l ( n - l ) + ^ ( / i n - ( n - ) + M n - « - 2) ) ‘ Proof, By Proposition 4, w e have a) I f n is even then Therefore, / / 10 / , !^n\^n) ~ T^Mn( ^n?r^Mu-2(^n-2) r ' n- ilvn- ily) • 7^ •7^FMn-2 (^ n -2/ "'t" 7^ ^*• ^ X , , , 10 / \ / / \ ” T^Mn -1 V-^n-l ) +« 210 M n-2{5n-2) + Mn- (^ n - / 25A-n-i b) If n is odij then ( ( X i , = {(a^i, , a ; „ - i , ) ) u ( ( x i , , a : „ _ , , ) ) u ( ( x 'l , , ) ) Therefore, 10 , ~ , ^ 2TÕ /^^~2 (^ n - ) / / \ f^ri ~2 \^ri~'2 ) * Hence, ~ ^ M a -1 + ĨÕ ^ ^ri~ Ì^ n - ) ) ' The corollary is proved From Corollaries and 2, w e have C orrolary Lei X ~ { X] , X , ) = (2, 3, 2j 3, ) € we have / X ị^nK^n) — /*roỡ/ 10 / X F or each n e N, p u i Sri ~ Ĩ 3~^Xi ITien i=\ 15 / X 45 , , By Corollaries and 2, vve have Mn(Sn) = ^ / / „ - l ( s „ _ i ) + H n - l ự n - i ) = - ^ ịlj i- {Sn-2) + ^ (13) (^ « -3 (-Sn-s) + M n -3 (s0 -3 )) /^71—2(■Sn-2) ~ ^ M n —s i^ n -s ) ^ M n -sC ^ n -s)' V.T.H Thanh et al / VNƯ Journal o f Science, M ath em atics - Physics 25 (2009) 57-68 65 Frorm (Ì3), (14j and (15), the assertion o f the corollary follow s 2.2 7Tie p r o o f o f the main theorem Lem im a Let X — { x\ , X , ) (2, 3, 2, 3, ) G F or each n € N, p u t Sn = “*Xj Then i=i we /have /x,,(5n) > Mn(in) f o r a ll u, e supp fin, Wc will prove the lemma bv induction For n — w e have = 2) = = ị for -all t\ € supp fi\ A ssum e that the lemma is true for n = k, i.e., ụ-kịsk) > ụ-h{tk) for all tk € su pp Hkn We will show that the lemma is true f o r n — Ả: + For any y — ( 2/ , 2/ ) •••) € D ^ , put in — XI i=i k+\ for each n € N, then tk-ị-i — Y1 consider the follow ing cases o f yk+\C a :se If yk+ \ — (or 4), then by Lemma 1, tk+i = tk + has at m ost tw o representations = + Therefore, by induction hypothesis, w e have /Ífcfi(ífct 1) = ịJk{t k)ỉ *{^k+ì = w 5 , ) + Hk{t ' k)P{^k+i = ' 1) 10 , , By Corollan (ii), w c have ị í k + \ { s k + ì ) > ^ ị i k i s k ) > ụ-k+i{tk+ĩ)- C a se If Uk+I - (or 3), then by Lemma 1, tk+i has at m ost tw o representations tk+i = tk + a) Ifxjk = (or 3), then ( 2/fc,i/fc+i) G { ( , ) , ( , ) } Therefore, b y Lemma w e have ( i / , ( ( y i , - , y f c + i ) > iff (y)^.2/Ui)e{(0,0), (3,0), (2,3), (5,3), (0,3), (1,0), (3,3), (4,0),} G 66 V.T.H Thanh et a i / w Jou fn al o f Science, M athem atics - P hysics 25 (2009) 57-68 By induction h>pothcsis, w e have ^Ik+i{tk+i) ^ f i k - i { s k ~ i ) ị P { X k = ) P { X k + i = ) + P i X k = ) P { X k + i = 0) + P { X k = ) P ( X k + : = 3) + P { X k = ) P ( X f c H = 3) + P ( X k = ) P { X k + ị = 3) + P { X k = ) P ( X k + i = 3) + P { X k = l ) P { X k + : = 0) + P { X k = ) P { X k + , = ) 1 10 10 10 10 — Mfc-I (■^fc-i)V25 '25 2^ ' ^ ^ ® ' ^ ^ ^ ’ ^ 10 10 10 5 25 '25 25 ■25 ^' ^ ^ ^ ’ ^ 241 By hypothesis induction and Corollary (ii), w e have / X 10 Mfc+i(sfc+i) > / X 241 , , , > ^ / J k ~ i { s k - i ) = fj'k+iitk+i)- b) If yfc = (or 1), then {vk, rjk+i) e { ( , ) , ( , ) } Therefore, by Lemma w e have (y'l, {{y\,-;yk+i)) e iff (yl, yfc+i) e { ( , ) , ( 5, 0) , ( , ) , ( , ) , ( , ) , ( , ) , ( 3, 3) , ( , ) , } By induction hypothesis, w e have /iMiíítM) < Ilk ,(ífc-i)[PÍXfc = 2)P(Xtn + P { X k = 1) P { X m = ) + PfXt = r))P(Xtn =0) = 3) + P { X k - )P (;rfcH - 3) + P { X k = 0) P{Xk+i = 3) + P { Xk = l)P(Xjt+i =0 ) + P ( X , = ) P { X k + i = 3) + P { X k = 10 1 10 10 — Mfc-I (s/c + ^ ’25 ®' ^ ^ ^ ' ''^ 10 10 10 + ^ ' + 25'25 2^ ' 2^ 4)P(Xm , = 0)1 By hypothesis induction and Corollary (ii), w e have 10 fik+\(sk+\) > ^fj-kisk) > 231 ^ ^ k - \ { s k - i ) > fJ■k+ì{tk+\)■ c ) ỉ f y k = (or 5), then ( y k , y k + i ) e { ( , ) , ( , ) } Therefore, by Lemma w e have ( y ' l , / f c +i ) ^ {{y\, ,yk+\)) iff e ( ( , ) , ( 3, ) , ( , ) , ( , ) , ( , ) , ( , ) , ( 2, ) , ( , ) , } V.T.H Thanh e t al / VNU Journal o f Science M ath em atics - Physics 25 (2009) 57-68 67 By iinduction hypothesis, \vc have ^ Hk-i{tk-i)[P{Xk = ) P { X k + i = )+ P { X k = ) P i X k + i = ) + P{Xk = 2)P(Xk+i = 3) + P { X k = b ) P { X M = 3) \ P{ Xk = 2)P(Xk+i = 0) + P { X k = ) P { X k + i = 0) + P { X k = l ) P { X k + i = 3) + P { X k = ) P { X k + i 1 10 10 10 10 — Pk u - u i ) ( ' »' 25 ^ ' ® ^ 2^ ' ^ = 3)1 10 10 ”*”2 ‘ 25 - 231 , 10 I K ^ 2^ 2® ^ 2^ 2^ Theircfore, by Corollary (ii), w e have 10 231 > ^ M f c - i ( « f c - i ) > /^fc+iiifc+i)- Mfc+i(sfc+i) > C ase Ifyk^\ (or 5) This case is proved similarly to theCase Thcuifore,the lemm a is proved By resolving Fibonacci recurrence formula o f /Xn(^n) Corollary 3, w e have the follow ing corollary C orrolary Let X = (a:i,X 2, ) “ (2, , , , ) € D ^ F or each n € N, p u t S n = 3~^Xị Then i=\ we /have fln{^n) = a i X Ĩ - ^ a X Ĩ + a s X Ĩ fo r ,Vl arccos 497 y ^ ị / Ĩ Bros( - _59xm5) ^ o z _ n, ss n s 5 o X = : f ậ r Ị / Ĩ cos( Ĩ ) -|- | ~ ,0 9 7 _o _ arccos " -7 ^ — X = ^ [ / l c o s ( 1 ^ _ !L) + j ~ -0 ,0 8 and a \, 02, 03 are ro o ts o f the fo llo w in g system o f three equations H\ ( s i ) = a ] X j + 02^2 + 03^3 /