Scaling behaviour of entropy estimates

Entropy estimation of information sources is highly non-trivial for symbol sequences with strong long-range correlations. The rabbit sequence, related to the symbolic dynamics of the nonlinear circle map at the critical point as well as the logistic map at the Feigenbaum point, is known to produce long memory tails. For both dynamical systems the scaling behaviour of the block entropy of order n has been shown to increase proportional to log n. In contrast to such probabilistic concepts, we investigate the scaling behaviour of certain nonprobabilistic entropy estimation schemes suggested by Lempel and Ziv (LZ) in the context of algorithmic complexity and data compression. These are applied in a sequential manner with the scaling variable being the length N of the sequence. We determine the scaling law for the LZ entropy estimate applied to the case of the critical circle map and the logistic map at the Feigenbaum point in a binary partition.