Non-sequential Recursive Pair Substitutions and Numerical Entropy Estimates in Symbolic Dynamical Systems

We numerically test the method of non-sequential recursive pair substitutions to estimate the entropy of an ergodic source. We compare its performance with other classical methods to estimate the entropy (empirical frequencies, return times, and Lyapunov exponent). We have considered as a benchmark for the methods several systems with different statistical properties: renewal processes, dynamical systems provided and not provided with a Markov partition, and slow or fast decay of correlations. Most experiments are supported by rigorous mathematical results, which are explained in the paper.