Wavelet Analysis Of Fractals: from the Mathematical Concepts to Experimental Reality

Abstract Fractal and multifractal concepts (Mandelbrot 1977, 1982; Halsey et al. 1986; Paladin & Vulpiani 1987) are now widely used to characterize multiscale phenomena that occur in a variety of physical situations (Stanley & Ostrowski 1986, 1988; Pietronero & Tosatti 1986; Guttinger & Dangelmayr 1987; Feder 1988; Aharony & Feder 1989; Vicsek 1989; Family & Vicsek 1991). In its present form, the multi fractal approach is basically adapted to describe statistically the scaling properties of singular measures (Benzi et al. 1984; Yul et al. 1984; Halsey et al. 1986; Badii 1987; Collet et al. 1987; Feigenbaum 1987; Jensen et al. 1987; Paladin & Vulpiani 1987; Mandelbrot 1989a; Rand 1989). Notable examples of such measures include the invariant probability distribution on a strange attractor (Halsey et al. 1986; Collet et al. 1987; Rand 1989), the distribution of voltage drops across a random resistor network (Stanley & Ostrnwski 1986, 1988; Feder 1988; Bunde & Havlin 1991), the distribution of growth probabilities on the boundary of diffusion-limited aggregate ( Feder 1988; Meakin 1988; Vicsek 1989) and the spatial distribution of the dissipation field of fully developed turbulence (Mandelbrot 1974; Pal adin & Vulpiani 1987; Frisch & Orszag 1990; Meneveau & Sreeni va.san 1991).