### 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/f^{ν} spectral fluctuations. It is shown that the Lempel-Ziv complexity obeys universal scaling laws and that the Lempel-Ziv complexity has the L^{1}-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/ f spectral process is generated.

Original language | English |
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Pages (from-to) | 503-515 |

Number of pages | 13 |

Journal | Progress of Theoretical Physics |

Volume | 116 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2006 Sep |

### ASJC Scopus subject areas

- Physics and Astronomy(all)

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## Cite this

*Progress of Theoretical Physics*,

*116*(3), 503-515. https://doi.org/10.1143/PTP.116.503