Many complex systems are characterized by a hierarchical organization, with transfers across the different levels from the smallest to the largest ones and vice-versa. We propose that hierarchies are generically formed dynamically due to a spontaneous breakdown of continuous scale invariance into discrete scale invariance. We present an 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. We discuss the different mechanisms and show a series of examples where discrete scale invariance has been found (irreversible growth and rupture phenomena, non-local connectivity problems, quenched heterogeneous systems, etc.).
D. Sornette,Discrete scale invariance and complex dimensions, Physics Reports 297, 239-270 (1998). http://xxx.lanl.gov/abs/cond-mat/9707012
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