Hierarchical approach to complexity with applications to dynamical systems

Abstract

A hierarchical approach to complexity of infinite stationary strings of symbols is introduced by investigating the scaling behavior of suitable quantities. The topological entropy, which estimates the growth rate of the number of admissible words, corresponds to the first-order indicator ${\mathit{C}}^{\mathrm{}\left(1\right)\mathrm{}}$. At the second level, a novel indicator ${\mathit{C}}^{\mathrm{}\left(2\right)\mathrm{}}$ is introduced which measures the growth rate of the number of irreducible forbidden words. Finally, a detailed analysis of 2D maps reveals that ${\mathit{C}}^{\mathrm{}\left(2\right)\mathrm{}}$ can be expressed in terms of the Lyapunov exponents.