Fujita, T.

Ito, S.

Keane, M.S.

Ohtsuki, M.

English

CWI's Institutional Repository

Ergod. Th. & Dynam. Sys. (1996), 16, 1345-1352 Printed in the United Kingdom Copyright© Cambridge University Press On almost everywhere exponential convergence of the modified Jacobi-Perron algorithm: a corrected proof T. FUJITAt, S. ITO:j:, M. KEANE§ and M. OHTSUKI:j: t Department of Mathematics, Hitotubashi University, Kunitachi, Tokyo, Japan · (e-mail: fujita@math.hit-u.ac.jp) :j: Department of Mathematics, Tsuda College, Tsuda-machi, Kodaira, Tokyo, Japan (e-mail: ito@tsuda.ac.jp) (e-mail: ohtsuki@tsuda.ac.jp) § Centrum voor Wiskunde en lnformatica, Kruislaan 413, 1098 SJ Amsterdam, The Netherlands (e-mail: keane@cwi.nl) (Received 17 January 1996 and revised 7 August 1996) 0. Introduction The following theorem was published in [2]. THEOREM. There exists a constant 8 > 0 such that for Lebesgue almost every (et, {3) E X = [O, l] x [O, I], there exists no = no(ct, {J) such that for any n 2:: no I rn I 1 f3 - - < --y:+:;i, qn qn where the integers Pn, qn, rn are provided by the modified Jacobi-Perron algorithm. There is a gap in the proof of this theorem, where Jensen's inequality is used, which has been pointed out by T. Fujita, one of the authors. The gap is cleared by considering the metrics llxll 1 = lxl + IYI and llxlloo = max(lxl, lyl) for x = (x, y) E IR.2 in addition to the Euclidean metric. The first part of the proof is the same as the original proof, but the latter part is improved and becomes simpler by using these new metrics. Therefore, we will present the improved proof here, referring to the original article for the unchanged parts. A complete version is available (electronically) from the authors on request. 1. The modified Jacobi-Perron algorithm This section remains unmodified. 2. An application of the subadditive ergodic theorem This section remains unmodified until the end of the statement of the theorem. The corrected proof now follows. 1346 T. Fujita et al To prove this result, it is sufficient to show ( ~logllD2ll2dµ. < 0. lx 2 In fact, let us denote the infimum in the proposition above by y. We have by formula (**)that are bounded from above by By the proposition, . 1 hm - log l1Dnl12 = y, n-+oo n so that I Pn I I rn I 1 n , 1 <Y. - - R - - < -e y = --' JJ - I+~' qn qn qn qn for sufficiently large n, provided that s' < _r. c with c = - fx logfJ dµ > 0 being the constant of the corollary. Therefore, it is sufficient to show that y < & f x log II D2 lh d µ < 0. 3. Calculus Before calculating the value of the integral, we note the following properties of the norms. Let us define the norms of x = (x1, x2) E JR2: llxll2 = Jx? + x~, llxll1 = L Ix;!, llxlloo = max lx;I. i=l,2 i=l,2 For the corresponding norms of a 2 x 2 matrix A, we use the notation llAll2. llAlli. JIAlloo· Then the following properties hold. LEMMA. Put A=(%), 1 :::: i, j:::: 2. Then: (i) llAll2 =JI where A. is the maximum eigenvalue of 1AA; (ii) 11AII1 = 13]a; ( L laij 1); 1-1-2 i=l,2 (iii) II A lloo = max ( L laij 1); l::;i::;2 j=l,2 (iv) JIAll~:::: llAll1 llAll00· On almost everywhere exponential convergence 1347 Proof The proof of (i), (ii) and (iii) are well known (see [1]). For any norm on JR.2, the following relation (submultiplicativity) is valid: llABll:::: llAllllBll, where llAll := sup llAvll. llvll:S I From (i), let llAll~ =>..and let y be an eigenvector of 1AA w.r.t. >... Then we have 11A11~ =A= 1>..1 = w~:i~ 11 ~ WAA11:::: 11 1A1111A11. In particular, putting II · II = II · 112, from (ii) and (iii), we have llAll~:::: llAll1 llAll00· D Let us define the partitions {Xo, Xi} of X and {Xn,o, Xn, 1 In = 1, 2, ... } of Xo as follows: Xo :={(a, /3) I /3 <a}, X1 :={(a, /3) I a< /3} and Xn,O .- {(a, I l f3 > 1 - na} /3) E Xo I -- < a < - , n + 1 n {(a, I l fJ < I - na}. Xn,I .- /3) E Xo I -- <a< -, n +I n Let us calculate the 2 x 2 matrix D2 (a, /3) explicitly: [ 01 -a ] [ 0 -? ] __ [ -a I - an ] 1 '"" if (a, /3) E Xn,O /3 -- + n -/3 -n/3 a [ ~ 7 l [ -±: n : ] ~ [ l ~~:· n if (a, ~) E X" ' The other two cases are, when (a, /3) E X 1, symmetric formulas in a and {J. Therefore we see that { max(a + (3, I - an+ f3n) if (a, /3) E X 11 .o llD2lli = max(l -an+f3n, I) if(a,(3) E Xn,I· From the fact that I - (I - na + nf3) = n(a - /3) > 0 if (a, /3) E Xn,l• we know that llD2ll1 = l if (a, /3) E Xn,I for every n; also, for (a, /3) E X11 ,o, II D2 II 1 :'.": l if and only if (a, /3) E X 1,0· For the norm II · lloo we have { max(a + l - an, f3 + f3n) llDzlloo = max(l - an, 1 + f3n) if (a, /3) E Xn,o if (a, /3) E Xn,1· 1348 T. Fujita et al Therefore, llD2lloo '.'.:I if and only if (a, fJ) E Xn,I or (a, fJ) E Xn,O and (n + 1),B 2: 1. (*****) For the norm II · 1J 2, the trace and determinant of 1D2 D2 on Xo are given by f rD D { a2 + fJ 2 +(I - an)2 + n2 fJ 2 if (a, /3) E Xn,o t = trace o ? ? = ? ? 2 'f - - l+(l-an)-+n-/3 1 (a, /3)EX,,,1 and . . r { {32 if (a, /3) E Xn,O d = determrnant of D2 Dz = 2 .f ( /3) X (I-an) I a, E n,l· Therefore, the norm II D2 II~ is given as the maximum solution of A 2 - tA. + d = 0: ? t + )t2 - 4d llD?ll; = · - -2 Now, let us begin calculating the integral. By using property (iv), the value is estimated as follows: L, log II Dzll~p(a, /3) da d,B = ~ l .. o log llDzll~p(a, /3) da df3 S fx 1 • 0 logi1D2ll~p(a,f3)dadf3 (Ii) + fx1•1 log II Dzll~p(a, ,B) da d,B (12 ) +; l .. o log llD2ll1P(a, f3)dadf3 (I3) +; l .. , log llD21i1P(a, f3)dadf3 (14 ) + ~ l .. o log II DzllooP(a, /3) da df3 (I5 ) + ~ l .. i log II DzllooP(a, /3) da df3. (16 ) Each of the six values given above is now estimated. The estimation of I1 and h For each E = 0 or 1 we have 1 log llD2ll~p(a, /3) da df3 X1,, S 1 CllD2ll~-l)p(a,f3)dadf3 X1,, = ~ r t(a,f3)p(a,f3)dadf3+~ r Jt 2 -4dp(a,f3)dadf3 lx1,, 2 lx1,, -1 p(a, f3) da df3 X1,, On almost everywhere exponential convergence By using Maple, the values of the integrations can be calculated: r (a2 + f3 2 + (1 - a) 2 + [3 2 )p(a, f3)dadf3 = 0.168378576 ... 1x1.o { J (2a2 - 2a + I+ 2fJ2)2 - 4{32 p(a, [3) da df3 = 0.1036941092 ... 1x1.o r p(a, /3) da df3 = 0.1338263206 ... lxl,O r (1 + (1 - a)2 + {3 2)p(Ot, {3) da df3 = 0.117559369 ... 1x11 r j(l + (1 - a)2 + [32)2 - 4(1 - a)2 p(a, f3)dadf3 = 0.09320701616 ... lxl,I r p(a, {3) da df3 = 0.1008454268 .... 1x11 Therefore, the values 11 and 12 can be estimated as 11 :5 0.00221002185 .. . Ii :5 0.00453776578 ... . 1349 The estimation of l3. We know from ( * * * *) that log II D2II1 :5 0 for all n :=::: 2. For each n, let us divide the integration into two parts: fxn.o log llD2ll1p(a, f3)da df3 1 logllD2ll1P(a,f3)dadf3 Xn,0,1 + 1 log llD2ll1p(a, f3)da df3, Xn,0,2 where Xn.0,1 = Xn,o n {(a, /3) I a + f3 ::'.: 1 - na + nf3) and Xn,0,2 = Xn,o n {(a, /3) I a+ f3 :5 l - na + nf3}. Using the change of variables l a=--n+ y' x {3=-, n+y = r log (-1- + _x_) g(x, y) dx dy 1x0 n1(x,y)IY::'O(l-n)x+ll n + Y n + Y r log (1- _n_ + ~) g(x, y)dxdy lxon/(x,y)ly?:(n-l)x+l) n + y n + y = f1 1" dy [1 log (y + nx) g(x, y) dx (13,3,n) Jo J(l-y)/(n-1) n + Y + f dy 11 log (y + nx) g(x, y) dx, (13,4,n) 1/n y n + y 1350 where T. Fujita et al 2n + 2y + x + 1 g(x, y) = ---------------(n + y)(n + y + l)(n + y + x)(n + y + x + 1) Using Maple, we have 10 2:h.1,n = -0.01334879149 ... n=2 10 Ll3,3,n = -0.006379319311 ... n=2 10 LI3,4,n = -0.007224728418 ... n=2 and so JO I: 13,2.n = -0.013604041129 .... n=2 Therefore we have l3 ::::: -0.026952739219 .... The estimation of4. From the fact that llD2ll 1 = 1 if (a, {3) E Xn.1 for every n, we know that log llD2ll 1 = 0 if (a, {3) E Xn,I· Therefore, we have l4 = 0. The estimation of Is. From(*****), we know that for (a, {3) E Xn.o, llD2 11 00 ~ 1 if and only if (n + I );'l ~ I. Using the change of variables as above, we have Is,n = fx00logllD2llooP(ct,f3)dad;'l :S { logi1D2llooP(a,f3)dad;'i j X,,on((a,f:l)l(n+l),B;::a+l-an} = r 1og((n+l)x)g(x,y)dxdy. J xtJn((x,y)ly:O:(n+l)x-1} n + y Notice that if 3x - l :S y :S (n + 1 )x - 1 and n ~ 2, then Thus, Is,n l (n + l)x 0 og < . n+y -:S 1 log ((n+ l)x) g(x, y)dxdy X0n((x,y)ly::;3x-l} II+ Y 1 1 ((n + l)x) Xon((x,y)ly::;3x-l} og II+ Y x - dxd { 1 1 } (n+y)(n+y+x) (n+y+1)(n+l+y+x) y. On almost everywhere exponential convergence 1351 Therefore, 00 Is = Lls,n n=2 < f, { ((n+l)x) n=2 lxon[(x,y)[ys3x-l} log n + y x Ln + y)(n1 + y + x) - (n + y + l)(n1+ 1 + y + x)} dx dy = ~/{1og(n+l)x -log nx } 1 dxd ~ n+y n-l+y (n+y)(n+y+x) y + f log~ 1 dxdy. lx0n{(x,y)lys3x-I) 2 + Y (2 + y)(2 + y + x) Here, we see that log - log = log 1 - < 0. (n + l)x nx ( 1 - y ) n+y (n-I)+y n(n+y) -Thus, we obtain Is s { log(~) 1 dxdy lxon[(x,y)lys3x-I) 2 + Y (2 + x)(2 + y + x) [112dy11 log(~) 1 dx Jo o+y)/3 2+y (2+x)(2+y+x) +j1 dyf 1 1og(~) 1 dx 1;2 y 2+y (2+x)(2+y+x) = -0.004356831580 ... + (-0.001098840366 .. . ) -0.005455671946 .... The estimation of I6. We know from the definition of II D2 l1 00 that II D2 lloo = 1 + {Jn if (a, fJ) E Xn,I · Thus, 16,n = 1 logllD2ll 00 p(a,fJ)dadfJ Xn.1 = f I ( l R ) ( 2 +a+ fJ) d dfJ Jx •. 1 og + pn (1 + a)(l + fJ)(l +a+ tJ) a 1 ( nx) (2n+2y+x+l) d d = log 1 +-- x y x1 n+y (n+y)(n+y+I)(n+y+x)(n+y+x+I) s r _!!:,!__ (2n + 2y + x + 1) dx dy J x 1 n + y (n + y) (n + y + 1) (n + y + x) (n + y + x + 1) 1 (2n + 2y + x + 1) d d s x x y. x 1 (n + y)(n + y + l)(n + y + x)(n + y + x + 1) Therefore, we obtain that I · ~ ~1 (2n+2y+x+ I) d d 6 = L .. }6n=L..., x x y n=l ' n=l x 1 (n+y)(n+y+l)(n+y+x)(n+y+x+l) 1352 T. Fujita et al "°1 ( 1 1 ) = I: x - dxdy n=2 x 1 (n + y)(n + y + x) (n + 1+y)(n+1 + y +x) l 00 { 1 1 } = l:x - b~ x 1 n=2 (n + y)(n + y + x) (n + 1+y)(n+1 + y + x) = f x 1 dx dy lx1 (2+ y)(2+ y +x) = 0.019707476 . ... Summing up the estimated values, we have ( logllD21!~p(a,/3)dad,8 ~ I1+l2+···+I6 Jxo = -0.00595324 . .. < 0. REFERENCES [1] F. R. Gantmacher. The Theory of Matrices (2 vols). Chelsea, New York, 1964. [2] S. Ito, M. Keane and M. Ohtsuki. Almost everywhere exponential convergence of the modified Jacobi-Perron algorithm. Ergod. Th. & Dynam. Sys. 13 (1993), 319-334. 

On almost everywhere exponential convergence of the modified Jacobi-Perron algorithm: a corrected proof

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