The Rosencrantz Coin: Predictability and Structure in Non-Ergodic Dynamics-From Recurrence Times to Temporal Horizons

We examine the Rosencrantz coin that can "stick" in states for extended periods. Non-ergodic dynamics is highlighted by logarithmically growing block lengths in sequences. Traditional entropy decomposition into predictable and unpredictable components fails due to the absence of stationary distributions. Instead, sequence structure is characterized by block probabilities and Stirling numbers of the second kind, peaking at block size n/logn. For large n, combinatorial growth dominates probability decay, creating a deterministic-like structure. This approach shifts the focus from predicting states to predicting temporal horizons, providing insights into systems beyond traditional equilibrium frameworks.