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

We present an investigation of epsilon -entropy, h(epsilon), 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 epsilon -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(epsilon) similar to epsilon (-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 epsilon -entropy density of a three-dimensional velocity field is affected by the correlations induced by the sweeping of large scales. (C) 2000 Elsevier Science B.V. All rights reserved.