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Then if qm > 0 for some m , the Hausdorff and fractal dimensions of X are finite. 29) m (1 + max — - ) . ,U (tl] . We also note that Q m depends on ^i"m''^m ' although the dependence is not explicitly indicated. 1 used above. 1. 30) Q T A every (T where = I . . family (*,. A . . m l is A I the m A * and the data. > m l orthogonal 45 in H , and let . . A I + . . ,$ A ... A * projector m ) in , = |*1 i + I A I A . . linear A I A T of H , A ... A $m|2Tr(T0Q) m' , space by onto H be the Tm the spanned .

22) - 3/2 B(9,u) = 0 . 15)) R4]9|2 + v ||e||2 < ^ | | w | | + 2 + ^||w|| |w| + ^ | 2 |w| R4|ej2 , for o o 2c* R4 rt 0 < —2c'p J - exp (2cl 3 R t ) . Sup |w(s) |. 23) a(e) = The proof is complete. +2c') R" | — — e exp ( ~ -) . 1. 1 actually shows that S(t) is uniformly differentiable on & (S(t)). 20), together with Sup Sup Sup 6 0 , p > 0 , t1 —> 0 | v -u | — > ° ° | | S ( t ) v o - S ( t ) u o - L ( t , U Q ) .

I=l ( ) These computations are elementary ; they will be repeated in more complicated (in infinite space dimensions) situations in the next sections. , P. 36 Let <$>± = ( x i , y i , z i ) Tr(B(u)0Q) = CONSTANTIN , , we 2 I z |Tr(B(u)0Q) | < | x 3 | C. FOIAS, R. 44) we find that for t x 2 Y + 2 + |yJ 3 ^ 1 -< 4 2 X 2 3 Z y 3 2 + | zJ /^7z u(t) . large, b(r + a 4/£(b-l) Tr(B(u)0Q) > 6 >O TEMAM arbitrarily small. Thus |U. (L(t,u )) < 1 , V u , whenever d = 2+s and -s(l+b+cr) + (l-s) (k2-6) < O . 50) 6 is arbitrary small we find the condition k„ d < 2 + l+b+a+k2 With the values of b,a,r (8/3,10,28) of the original paper of Lorenz [33] we find the following bound of the Hausdorff dimension of the Lorenz attractor.

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