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1 The inequalities fh g fl are always true (see below) Thus, it would be more accurate to say that the Legendre spectrum is an approximation (by excess) of the large deviation spectrum, rather than that of the Hausdorff spectrum
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Since the sets E ( ) decrease with , we may de ne: E = lim E ( ) = {x/ n (x) n }
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DEFINITION 15 Let d be any dimension We de ne the following spectra: fd ( ) = d(E ) = d lim sup E ( , N )
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0 N lim fd ( ) = lim d E ( ) = lim d sup E ( , N ) 0 0 N lim fd sup ( ) = lim sup d E ( , N ) 0 N
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When d is the Hausdorff dimension, then fd is just the Hausdorff spectrum fh Let D = Im (A) be the closure of the image of A Then D is the support of the spectra Indeed, for any D, E ( , N ) = , and thus fd ( ) = lim lim fd ( ) = fd sup ( ) = (obviously this also applies to fg ) The following inequalities are easily proved: fd
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lim fd sup lim fd lim fd
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lim There is no relation in general between fd and fd sup However, if d is -stable:
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fd ( )
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lim lim fd ( ) = fd sup ( )
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(133)
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Besides, if d1 and d2 have two dimensions such that, for any E [0, 1]: d1 (E) then: fd1 (E)
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lim fd2 (E), fd1 (E) lim lim fd2 (E), fd1 sup (E) lim fd2 sup (E)
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It is not hard to prove the below sequence of inequalities PROPOSITION 14 For any A: fh ( )
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lim lim fh sup ( ) = fh ( ) lim f sup ( ) lim min f ( ), fg ( )
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(134)
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This is an improvement on the traditional result fh ( ) fg ( ) In particular, lim as soon as fh ( ) = fg ( ), all the above spectra coincide, except perhaps f If, lim sup on the contrary, fh is smaller than fg , then we may hope that f ( ) will be a better approximation of fh than fg The important point here is that the calculation of lim f sup ( ) only involves box dimensions and that it is of the same order of complexity lim as that of fg The spectrum f sup ( ) is thus a good substitute when we want to lim numerically estimate fh For example, the practical calculation of f sup ( ) on a multinomial measure (see section 146) yields good results Since d([0, 1]) = 1, all the spectra have a maximum lower than or equal to 1 In lim lim certain cases, a more precise result concerning fd sup , fd and fg is available: PROPOSITION 15 Let K be the set of x in [0, 1] such that the sequence ( n (x)) converges Let us suppose that |K| > 0 Then, there exists 0 in D such that:
lim lim fd sup ( 0 ) = fd ( 0 ) = fg ( 0 ) = 1
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Let us note that such a constraint is typically not satis ed by fh This shows that fh contains, in general, more information A more precise and more general result may be found in [LEV 04b]
lim lim The three spectra fd sup , fd and fg also obey a structural constraint, expressed by the following proposition lim lim PROPOSITION 16 The functions fg , fd and fd sup are upper semi-continuous
NOTE 14 Recall that a function f : D R R is upper semi-continuous (USC) if, for any x D and for any sequence (xn ) of D converging to x: lim sup f (xn )
f (x)
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