Discrete-time dynamics, step-skew products, and pipe-flows

Abstract

Dynamical processes can be classified in various ways as deterministic or stochastic, and continuous or discrete time. All these types can be studied by the path-spaces they generate, and stationary measures on that path-space. Such measures are called the law of the dynamics. This article presents how a general ergodic dynamical system may be approximated in terms of their law, by a simple and restricted family of deterministic continuous-time skew-product systems. In these systems, a deterministic, mixing flow intermittently drives a deterministic flow through a topological space created by gluing cylinders. The resulting orbits mimic the law of the original dynamics. This comparison is made possible by introducing a secondary intermediary approximation of the ergodic dynamics. This third system is a step-skew dynamical system, in which a finite state Markov process drives a dynamics on topological disk. Each of these three representations have their advantages. It is proved that the distribution induced on the space of paths by these three dynamics can be made arbitrarily close to each other. This analysis reconfirms the old principle that it is impossible to decide whether a general timeseries is generated by a deterministic or stochastic process, and is of continuous or discrete time.