Irreversible fractal-growth models like diffusion-limited aggregation (DLA) and the dielectric breakdown model (DBM) have confronted us with theoretical problems of a new type for which standard concepts like field theory and renormalization group do not seem to be suitable. The fixed-scale transformation (FST) is a theoretical scheme of a novel type that can deal with such problems in a reasonably systematic way. The main idea is to focus on the irreversible dynamics at a given scale and to compute accurately the nearest-neighbor correlations at this scale by suitable lattice path integrals. The next basic step is to identify the scale-invariant dynamics that refers to coarse-grained variables of arbitrary scale. The use of scaleinvariant growth rules allows us to generalize these correlations' to coarse-grained cells of any size and therefore to compute the fractal dimension. The basic point is to split the long-time limit (t -+ 00 ) for the dynamical process at a given scale that produces the asymptotically frozen structure, from the large-scale limit (r--+ 00 ) which defines the scale-invariant dynamics. In addition, by working at a fixed scale with respect to dynamical evolution, it is possible to include the fluctuations of boundary conditions and to reach a remarkable level of accuracy for a real-space method. This new framework is able to explain the self-orgauized critical nature and the origin of fractal structures in irreversible-fractal-growth models. It also provides a rather systematic procedure for the analytical calculation of the fractal dimension and other critical exponents. The FST method can be naturally extended to a variety of equilibrium and nonequilibrium models that generate fractal structures.

q-derivatives can be identified with the generators of fractal and multifractal sets with discrete dilatation symmetries. Besides providing a natural language in which to discuss homogeneous functions with oscillatory amplitudes, this also allows one to discuss cascade models with continuous scale changes. [S0031-9007 (97)03043-3] PACS numbers: 47.53.+n, 02.20.-a, 05.45.+b, 05.70.Jkq-analysis, q-special functions, and the quantum groups to which they are related is a very active and fast growing field of research, as testified by the enormous amount of literature that has recently appeared on the subject [1][2][3][4]. Yet the topic still has not received the attention it deserves from the statistical physics community. Our purpose here is to point out that q-derivatives are naturally suited for describing systems with discrete dilatation symmetries such as certain fractal and multifractal sets, the limit of q ! 1 corresponding to the case of continuous scale changes.We will introduce a slightly different notation than the usual [1][2][3][4]. In order to indicate the variable with respect to which a q-derivative is to be taken we use a subscript, and we use a superscript to indicate the dilatation factor q, in parentheses. Thus we define the q-derivative as

No abstract