An Improved Multifractal Formalism and Self Affine Measures
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To characterize the geometry of a probability measure fi with bounded support, its so-called spectrum / has been introduced recently. A mathematically precise definition has been given in [Falc4]: /(a) = UmHJ°6M« + e)-"*(«-£)) V ' e\0 610 log 8 whenever this limes exists. Thereby n{(a) is the number of boxes B = n[Wi {h+i)8[ in TRd with integers Ik, such that p(B) > 8". As will be shown, this definition is unsatisfactory for reasons of convergence as well as of undesired sensitivity to the particular choice of coordinates. A new definition F of the spectrum is introduced, which is based on box-counting too, but which carries relevant information about /i. The essential modification is that nj is replaced by the number of boxes Ng(a) with fi((B)i) > Sa, where {B)\ is the box of size 3<5 concentric to B. In addition, the lims_o is replaced by the limsup{_,0 for obvious reasons. The adaptation of the well known singularity exponents to this concept reads: log(EMQB)i)*) TM=*>*£*—n^— This notion renders exponents T(q), which are invariant under bi-lipschitzian co¬ ordinate transformations and for which the limit behaviour can be extracted from considering any sequence 8n such that 8n > 6n+i > i/8n with constant v. The important relation T(q) — sup (F(a) — qa) is valid for q ^ 0 and in the case of multiplicative cascades also for q = 0. Conse¬ quently T(q) is convex. On the other hand, F(a) need not be concave, as examples prove. In other words F may provide more detailed information than the Legendre transform of T. However, if T(q) equals lim^_o • and is differentiable on all of R, then F(a) , Mmlin>W(«+ ')-*'(«-')) = inf (T(,) + ,«) * ' ^0 S\0 -log 8 9IR for all a. Invariant measures play a crucial role in multifractal theory. They satisfy an invariance condition p ~Y1[ Pi">,./* with positive numbers p, such that pi + ... + pr = 1.