The Lempel-Ziv Complexity of Non-Stationary Chaos in Infinite Ergodic Cases

Abstract

The large deviation properties of the Lempel-Ziv complexity are studied using a one-dimensional non-hyperbolic chaos map called the “modified Bernoulli map”, where the transition between stationary and non-stationary chaos is clearly observed. The upper limit of the Lempel-Ziv complexity in the non-stationary regime is theoretically evaluated, and the relationship between the algorithmic complexity and the Lempel-Ziv complexity is discussed. Non-stationary processes are universal phenomena in non-hyperbolic systems, and they are usually characterized by an infinite ergodic measure and intrinsic long time tails, such as 1/ƒν spectral fluctuations. It is shown that the Lempel-Ziv complexity obeys universal scaling laws and that the Lempel-Ziv complexity has the L1-function property, which guarantees the Darling-Kac-Aaronson theorem for an infinite ergodic system. The most striking result is that the maximum diversity appears at the transition point from stationary chaos to non-stationary chaos where the exact 1/ƒ spectral process is generated.