Discrete-scale invariance and complex dimensions

We discuss the concept of discrete-scale invariance and how it leads to complex critical exponents (or dimensions), i.e. to the log-periodic corrections to scaling. After their initial suggestion as formal solutions of renormalization group equations in the 1970s, complex exponents have been studied in the 1980s in relation to various problems of physics embedded in hierarchical systems. Only recently has it been realized that discrete-scale invariance and its associated complex exponents may appear "spontaneously" in Euclidean systems, i.e. without the need for a pre-existing hierarchy. Examples are diffusion-limited-aggregation clusters, rupture in heterogeneous systems, earthquakes, animals (a generalization of percolation) among many other systems. We review the known mechanisms for the spontaneous generation of discrete-scale invariance and provide an extensive list of situations where complex exponents have been found. This is done in order to provide a basis for a better fundamental understanding of discrete-scale in invariance. The main motivation to study discrete-scale invariance and its signatures is that it provides new insights in the underlying mechanisms of scale invariance. It may also be very interesting for prediction purposes. (C) 1998 Elsevier Science B.V. All rights reserved.