Exit-Times and {\Large $ϵ$}-Entropy for Dynamical Systems, Stochastic Processes, and Turbulence

We present a comprehensive investigation of $ϵ$-entropy, $h(ϵ)$, in dynamical systems, stochastic processes and turbulence. Particular emphasis is devoted 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-affine and multi-affine signals and experimental turbulent data. Concerning turbulence, the multifractal formalism applied to the exit time statistics allows us to predict that $h(ϵ)\sim {ϵ}^{-3}$ 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 3-dimensional velocity field is affected by the correlations induced by the sweeping of large scales.