Solving the Inverse Fractal Problem from Wavelet Analysis

We report on a wavelet-based technique for solving the inverse fractal problem. We show that one can uncover a dynamical system which leaves invariant a given fractal object from the space scale arrangement of its wavelet transform modulus maxima. Our purpose is illustrated on Bernoulli invariant measures of linear as well as non-linear "cookie-cutters". Application to period-doubling dynamical systems at the onset of chaos is reported.

Fractal and multifractal concepts[l-31 have proved to be very ihitful to describe physical situations where scale-invariant objects are observed [4,5]. The so-called multifractal formalism [3] which accounts for the statistical scaling properties of singular measures, is now widely used in many fields [4,5] such as dynamical-system theory, fully developed turbulence, transport processes in disordered systems and interfacial growth phenomena. Recently, this formalism has been generalized to singular distributions [6] (including measures and signals) using the wavelet transform [7,8]. However, the mul*actal formalism is a statistical description that only provides macroscopic>^ information about the self-similarity properties of fradal objects through the determination of <<thermodynamical>> functions such as the generalized fractal dimensions D, and thef(a) singularity spectrum. It is then natural to try to get deeper insight into the complexity of such objects and eventually to extract some <<microscopic>> information about their underlying hierarchical structure. The inverse pmblem consists in expressing these properties in terms of a dynamical system which leaves the fractal object invariant. It has been previously approached within the theory of Iterated Function Systems [9,101 (IFS). However, the methods developed in this context are based on the search of a .best fit>> within a prescribed class of IFS attractors (mainly linear homogeneous attractors). In that sense, they approximate the self-similarity properties more than they reveal them. In this letter, we show that the space scale representation of the wavelet transform of a fractal object can be used to extract some dynamical system which accounts for its construction process.
The fractal objects we will consider are the invariant measures of <(cookie-cutters,>. A cookie-cutter [ll] is a map on A = [ 0,1] which is hyperbolic ( I T' I > 1) and such that T -1 ( A ) is a f i t e union of s disjoint subintervals (AJl k of A. For each k, Tk = TI Ak is a one-to-one map on A. An invariant measure p associated to T is a measure which satisfies p o T -' = p. We will suppose that the weights are multiplicatively distributed on A, ie.
1 where z p k = 1. These self-similar measures are also referred to as Bernoulli invariant measures of expanding Markov maps[3b]. They have been widely used for modelizing a large variety of highly irregular physical distributions [4,11].
The wavelet transform is a space scale analysis [71 which has already proved to be particularly well adapted for studying the hierarchical structure of fractal objects [12,13]. The wavelet transform of a measure p according to the analysing wavelet + is defined as [13] where a E R' * is the scale parameter and b E R is the space parameter. Usually + is chosen to have some vanishing moments up to a certain order so that it is orthogonal to possible regular ( i e . polynomial) behaviour of ,U. In the case of invariant measures of cookie-cutters there is no such behaviour so we will use a simple <<smoothing wavelet. $(x) = exp ( -x2>. By combining eqs. (1) and (2), a straightforward calculation at the first order in a (a<<l) leads to the following relation [6b, 141: (3) where connected curves usually referred to as maxima lines [6,8c]. We call bifircation point any point in the space scale plane located at a scale where a maxima line appears and which is equidistant to this line and to the closest longer line. The bifurcation points at coarse scales are displayed in fig. l b ) using the symbols (0). They lie on a binary tree whose root is the bifurcation point at the coarsest scale. Each bifurcation point defies naturally a subtree which can be associated to a rectangle in the space scale plane. Thus the root corresponds to the original rectangle [ 0,1] X I 0, u,, I, whereas its two sons correspond to reduced copies delimited by the dashed lines. The self-similarity relation (eq. (4)) amounts to matching the <<root rectangle. with one of the ason rectangles., i e . the whole tree with one of the subtrees (fig. lb). More generally, this relation associates any bifurcation point (x, , a,) associated to an order-n subtree to its hierarchical homologous (x, -1, a, -) of an order-(n -1) subtree. It follows from eq. (4) that x, = rkx, -+ s k and a, = rku, -1. Thus by plotting x, -vs. x,, one can expect to recover the initial cookie-cutter T. This reconstructed 1D map is displayed in fig. IC). As one can see, the two branches Tl and T2 of the cookie-cutter T provide a remarkable fit of the numerical data. Let us point out that the non-uniform repartition of the data points on the theoretical curve results from the lacunarity of the measure induced by the <<hole. between the two branches Tl and T2. In fig. Id), we show the histogram of the (contraeting) scale ratio values r = U, /U, -1. As expected, it displays two peaks corresponding to the two slopes rl = 3/5 and r2 = 1/5 of T; l and TC', respectively. Note that the peak corresponding to the smallest value of r is lower than the other one; this is a direct consequence of the finite cut-off we use in our wavelet transform calculation at small scales (the so-computed histogram can be artificially corrected by plotting N(r)ln(l/r) instead of N(T)). Figure le) displays the histogram of amplitude ratio values p = I T+ [,u](x,, a, ) I / I T+ [,uI(x, - 1, a, -) I. This distribution appears to be a Dirac at p = 0.5 which indicates (eq. (4)) that the weights p , and p, (defined in eq. (1)) are equal, Le. the measure is uniformly distributed on the cookie-cutter Cantor set. Let us mention that the distribution N(T) of scale ratios is in a way redundant with the 1D map, since it is basically made of two Diracs located at the inverse of the slopes of the two branches of this piecewise linear map. On the contrary, the distribution N( p ) of amplitude ratios brings a very important piece of information which is not present in the 1D map: the repartition of the weights at each construction step. In the case this repartition is not uniform, N ( p ) no longer reduces to a single point p = 1/2 and one can furthermore study the joint law of p with T in order to find out the specific <<rules. for associating a p with a T.
Since eq. (3) generally holds for hyperbolic maps, one can apply exactly the same technique to non-linear cookie-cutters. Figure 2a) displays the 1D map extracted from the wavelet transform modulus maxima skeleton of the uniform Bernoulli measure associated to a non-linear cookie-cutter made of two inverse hyperbolic tangent branches. Once again, the numerical results match perfectly the theoretical curve. In this case, the histogram of amplitude ratios is still concentrated at a single point p = 1/2. The histogram of scale ratios, however, involves more than simply two scale ratios, as before, since the non-linearity of the map implies that new scale ratios are actually operating at each construction step.
As a first application of our technique to a physical problem, let us analyse the natural measure associated to the iteration of quadratic unimodal maps at the accumulation point of period doubling.
It is well known [16] that the discrete-time dynamical system : xi + = fR (xi) = 1 -Rxt exhibits, as the parameter R is increased, an infinite sequence of subharmonic bifurcations which accumulate at R , where the system possesses a 2" orbit. Beyond this critical value, the dynamics becomes chaotic. As originally discovered independently by Feigenbaum [17a] and by Coullet and Tresser defined on this Cantor set as the visiting probability of the orbit of x = 0. In fig. 2b), we show the results of our wavelet transform analysis of this measure. It reveals a well-defined map with two distinct hyperbolic branches ( h e r resolution computations would reveal that the left-hand branch is linear whereas the right-hand one is not). One can compute the amplitude ratio histogram and find that the weights associated to these two branches are equal (pl = p2 = 1/2). The perioddoubling natural measure can thus be seen as the invariant measure of the cookiecutter displayed in fig. 2b) with uniform probability distribution. Ledrappier and Misiurewicz 1181 have proved that the invariant measure of f*(x) is actually the same as the one of the cookie-cutter defined by T ( x ) = x/f*(l) on [ f * ( l ) , (f*(1))2] and T ( x ) =f*(x)/f*(l) on [f * (f * (l)), 11. This map is represented in fig. 2b) by a solid line. Our numerical data are in remarkable agreement with the theoretical prediction.
In the case where s is no longer equal to 2, one can easily adapt the technique by trying to match not only the root bifurcation point on its sons but also on its grandsons and so on ... . For instance, in the case s = 3, we will match the root with one of its sons and with each of the two sons of its other son. The general algorithm uses a <<best matching,, procedure, so that it automatically performs the matching which is the most consistent (e.g., so that the different derivatives of T, [,u] follow the same self-similarity relations as T+ [,U]). Thus the algorithm is not looking for a given number s of branches that the user would have guessed a priori, it automatically comes up with the <<best>, value of s. In fig. 3 we show the 1D map and the histograms of scale and amplitude ratios obtained in the linear case where s = 3, pl = p2 = p3 = 1/3 and rl = 08, r2 = 0.3, r, = 0.5. All these values are very accurately recovered by our algorithm. Let us notice that we have considered in this letter only measures which do not involve any <<memory>> effect in their hierarchical structure, i e . the successive iterations always consist in applying the same dynamical system T , independently of the previous iterations. However, in a certain way, a memory component can be accounted for by increasing the number s of branches of a mo-memory* map T. As illustrated in fig. 3, this class of dynamical systems is directly amenable to our wavelet transform algorithmic procedure. Nevertheless, it is important to emphasize that because of finite-size effects, it is meaningless to look for dynamical systems with a rather high number of branches; generally, there would not be enough scales in the data in order to ensure the theoretical validity of the outcoming discrete map.
In this letter, we have elaborated on a wavelet-based method for extracting a dynamical system associated to a given fractal measure. Its robustness and accuracy have been illustrated on various systems including non-linear ones. As demonstrated for the 2 cycle of period-doubling dynamical systems, our method is undoubtedly a powerful tool for analysing multifractal measures arising in a wide variety of physical systems. Furthermore, the ability of the wavelet transform to get rid of smooth behaviours [6,8] allows a natural generalization of our approach to fractal signals that possess an underlying multiplicative structure. More details of this unified framework for solving the inverse-fractal problem, including stochastic processes, will be reported elsewhere.
Applications of this method to diffusion-limited aggregates, turbulent velocity signals and DNA <<walks>, nucleotide sequences are currently in progress.