Are the cusps in the plots of f ( a ) a real effect ?

An explicit example where the cusps in the plots of the . f(a ) spectrum appear is given. An analysis is provided which gives the support for the claim that these cusps are not a computer artefact and they are a signal of the breakdown of the scaling law. In recent years progress has been made in describing the strange (fractal) sets occurring in many areas of physics, in particular in the theory of dynamical systems and growth phenomena. It has been recognised that there exist sets which are not strictly self-similar and due to this fact cannot be characterised by the Hausdorff dimension alone. The Renyi dimensions (Renyi 1970) D,, q = 1,2 , . . . were applied to dynamical systems and fractal sets by Grassberger (1983), Hentschel and Procaccia (1983) and Grassberger and Procaccia (1984). This progress culminated in the introduction of the so-called f ( a ) formalism (Benzi er a1 1984, Halsey et al 1986; for a review see Paladin and Vulpiani 1987, Levi 1986). Since that time the f ( a ) formalism has been applied to a variety of phenomena; let us mention only diffusion-limited aggregation (Halsey et al 1986, Amitrano er al 1986, Nittmann er a1 1987), the HCnon attractor (Arneodo et al 1987), and attractors of non-hyperbolic dynamical systems (Politi et al 1988). Quite recently much attention has been paid to the practical limitations imposed on the determination of Dq. In particular the systematic bias and errors caused by the finiteness of the data samples were discussed (Grassberger 1988, Ramsey and Yuan 1989). Smith (1988) derived the necessary bound on the amount of data required for a reliable dimension calculation. Also the problems with the determination of D, for negative q were studied (Arneodo er al 1987, Lee and Stanley 1988, Blumenfeld and Aharony 1989). In this letter we are going to discuss another phenomenon: we claim that the cusps in the plots o f f ( a ) are a real effect. We know (Livi and Politi, private communication) that the cusps in the plots of f ( a ) were observed previously but they were dismissed as a computer artefact. We have encountered these cusps in the plots of f ( a ) for natural numbers (Wolf 1988) and here we will present a more detailed analysis of this phenomenon. Let us consider a measure p with a support A and let { A i } be a covering of A, A E U Ai, such that all Ai are contained in the ball of radius 1. Next, let us form the partition function where m(1) is the number of covering sets and depends on 1. If the moments x,(/) 0305-4470/89/221075 + 06S02.50 @ 1989 IOP Publishing Ltd L1075 L1076 Letter to the Editor behave in some regime of 1 and q like a power of I: then the function T ( q ) characterises the set A. The generalised dimensions are connected to r ( q ) via the definition x,( I) 1"4 ' (2) and it can be proved that for self-similar sets Do is the usual Hausdorff dimension, D , is the information entropy and D2 is the correlation exponent. Halsey et a1 (1986) proposed using, instead of T ( q ) , the Legendre transform of r ( 4 ) : where q is expressed by a via the relation f ( a ) = r ( q ( a ) ) (4) A good exposition of the Legendre transformation can be found in Arnold (1978). In order to invert ( 5 ) the derivative of a ( q ) has to be of constant sign: for positive " ' ( 4 ) the function r ( q ) is termed convex and for negative a ' ( q ) it is termed concave. In Wolf (1988) we have looked for the moments ( 1 ) for subsets A( N ) = { 1,2, . . . , N } of natural numbers. The measure of the interval Ai( I) = { i, i + 1 , . . . , i + I} E A( N ) of length 1 we defined as the number of prime numbers contained in i t divided by the number of prime numbers in A ( N ) : where ~ ( x ) denotes the number of prime numbers smaller than x. The sets A( N ) with the measure defined by ( 6 ) are very well suited for testing the multifractal formalism because the amount of prime numbers within an interval is precisely determined in contrast to, e.g., the Hinon attractor, where the measure can be obtained only approximately due to the finite number of iterations (Ameodo et a! 1987, Grassberger 1988). We have found in appropriate ranges of I and q values the power-like behaviour (2) of the moments x,( 1) for natural numbers. We have calculated f( a ) numerically and we have obtained the cusps in their plots, see figure 1. These cusps appear in the