For how long time evolution of chaotic or random systems can be predicted

Abstract Traditionally, probability theory and the theory of strongly chaotic dynamical systems concern asymptotic behavior, where observation time tends to infinity. By contrast, predictions of finite-time evolution, despite their practical importance, have received considerably less attention. One approach to such predictions originates from a new question in the theory of open dynamical systems: how does escape depend on the position of a hole in phase space? This question leads naturally to comparing first-hitting probabilities of different regions. Here, we study the resulting finite-time prediction problem for fair-dice-like systems, whose strongly chaotic dynamics statistically resembles a stochastic process. Previous results showed that transport of orbits in phase space exhibits three distinct stages. In the first stage, elements of a refined Markov partition form a hierarchy of first-hitting probabilities: for trajectories initially distributed according to equilibrium, some regions are more likely to be visited first than others. The time interval over which this hierarchy persists thus defines a natural prediction horizon. A central question is how this horizon changes as the resolution of observation increases, that is, as refinements of the Markov partition are considered. Counterintuitively, refining the partition extends rather than shortens the prediction horizon. Previous results provided a linear lower bound on its growth under refinements of the partition. Using a different approach based on autocorrelation polynomials and generating functions for first-hitting probabilities, we prove that the prediction horizon actually grows exponentially.