ABSTRACT
Common to essentially all complex systems is the existence of a hierarchical organization, with transfers across the different levels from the smallest to the largest ones and vice-versa. Here, we propose that hierarchies are generically formed dynamically due to a spontaneous breakdown of continuous scale invariance into discrete scale invariance. We present a short introduction to 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. Proposed illustrations are diffusion-limited-aggregation clusters, rupture in heterogeneous systems, earthquakes, animals (a generalization of percolation) among several other systems. We refer to several papers and a recent review for further information, especially on the use for predictions.
