The transition from deterministic chaos to astochastic

Deterministic dynamical systems of Langevin type recently introduced by Beck & Roepsdorf Physica 145 A (1987) 1] generate Langevin dynamics in a suitable limit. We discuss the transition from chaos to randomness in these systems under the point of view of the attractor dimension and dynamical en-tropy and show that the limit of a stochastic process is reached via a shift of the deterministic properties towards the innnitesimal length scales. 1 Dynamical systems of Langevin type In a series of articles Beck and collaborators 1, 2, 3, 4] studied the emergence of randomness from deterministic chaotic dynamics. They investigated deterministic chaotic maps called maps of Langevin type, because in an appropriate scaling limit the dynamics of these maps converges to stochastic dynamics governed by a Langevin equation. The most comprehensively studied system of this type has the form x n+1 = T(x n) y n+1 = y n + x n (1) with = e ? and = p. It is related to the solution of the ODE _ Y = ?Y + L (t) with L (t) = p X n (t ? nn)x n (2) by Y (t) = e ?(t?nn) y n nn t < (n + 1) : (3) Observing the process Y (t) at a constant sampling rate t in the limit ! 0 the observed process converges to a Langevin process, if the dynamics T of the x variable 1