Exit-times and ϵ-entropy for dynamical systems, stochastic processes, and turbulence

We present an investigation of ϵ-entropy, h(ϵ), in dynamical systems, stochastic processes and turbulence. This tool allows for a suitable characterization of dynamical behaviours arising in systems with many different scales of motion. Particular emphasis is put on a recently proposed approach to the calculation of the ϵ-entropy based on the exit-time statistics. The advantages of this method are demonstrated in examples of deterministic diffusive maps, intermittent maps, stochastic self- and multi-affine signals and experimental turbulent data. Concerning turbulence, the multifractal formalism applied to the exit-time statistics allows us to predict that h(ϵ)∼ϵ−3 for velocity–time measurement. This power law is independent of the presence of intermittency and has been confirmed by the experimental data analysis. Moreover, we show that the ϵ-entropy density of a three-dimensional velocity field is affected by the correlations induced by the sweeping of large scales.