CHAOTIC SYSTEMS - COUNTING THE NUMBER OF PERIODS

Characterization of chaotic motion may proceed both at an averaged ''macroscopic'' level, using such notions as Lyapunov exponents, dimensions and entropies, and at a ''microscopic'' level. In the latter case, the number of periodic orbits, being a topological invariant, plays an important role. For various one-dimensional mappings, the counting problem itself has many interesting facets and may be solved more or less completely in different ways. Recent progress in this counting problem is summarized with the hope that the explicit results obtained may be useful for classification of higher-dimensional dissipative chaotic systems.