Empirical One-Step Conditional Entropy in Infinite Ergodic Systems: Vanishing Entropy Rate, Sparse-Transition Scaling, and Mittag-Leffler Fluctuations

Abstract

Empirical entropy rates are widely used to quantify unpredictability from symbolic or time-series data, yet their interpretation is subtle in weakly chaotic dynamics, where ordinary Lyapunov exponents vanish and invariant measures are infinite. We address this issue by studying the empirical one-step conditional entropy for the fixed finite partitions considered below in one-dimensional intermittent maps with infinite invariant measures. For the modified Bernoulli map and the Boole transformation in the infinite-measure weak-chaos regime, we prove that this per-step empirical entropy converges to zero. Thus, the usual entropy-rate normalization becomes asymptotically blind to subexponential instability. The finite-time information sum, however, remains informative. Rare transitions between long laminar phases occur on the return-sequence scale, and their empirical self-information contributes an additional logarithmic factor. Under the stated regularity and moment assumptions, this mechanism yields a two-term estimate for the ensemble mean decay, supported by numerical simulations. Although the raw entropy rate vanishes, self-normalized fluctuations remain nontrivial and are numerically consistent with normalized Mittag-Leffler laws. A comparison with generalized Lyapunov sums shows that the corresponding information sum is not a Krengel entropy estimator, but a computable, partition-dependent finite-time measure of sparse symbolic transitions. These results clarify what empirical Markov entropy can, and cannot, measure in infinite-measure weak chaos.