### Abstract

The initial ensemble dependence of statistical laws in non-hyperbolic dynamical systems with infinite ergodicity are studied by use of the modified Bernoulli maps. We show that statistical laws crucially depend on the initial ensemble and that the time average for the Lyapunov exponent converges in distribution for the non-stationary regime. This is completely consistent with the Darling-Kac-Aaronson (DKA) limit theorem from the fact that the Lyapunov exponent is an L_{μ}
^{1}-class function. Next, we study the correlation function, which is not an L_{μ}
^{1}-class function. The most remarkable result is that the transformed correlation function also reveals uniform convergence in distribution in the same sense of the DKA limit theorem.

Original language | English |
---|---|

Pages (from-to) | 254-260 |

Number of pages | 7 |

Journal | Journal of the Korean Physical Society |

Volume | 50 |

Issue number | 1 I |

Publication status | Published - 2007 Jan |

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### Keywords

- Ergodic theory
- Non-stationary chaos

### ASJC Scopus subject areas

- Physics and Astronomy(all)

### Cite this

*Journal of the Korean Physical Society*,

*50*(1 I), 254-260.

**New aspects of the correlation functions in non-hyperbolic chaotic systems.** / Akimoto, Takuma; Aizawa, Yoji.

Research output: Contribution to journal › Article

*Journal of the Korean Physical Society*, vol. 50, no. 1 I, pp. 254-260.

}

TY - JOUR

T1 - New aspects of the correlation functions in non-hyperbolic chaotic systems

AU - Akimoto, Takuma

AU - Aizawa, Yoji

PY - 2007/1

Y1 - 2007/1

N2 - The initial ensemble dependence of statistical laws in non-hyperbolic dynamical systems with infinite ergodicity are studied by use of the modified Bernoulli maps. We show that statistical laws crucially depend on the initial ensemble and that the time average for the Lyapunov exponent converges in distribution for the non-stationary regime. This is completely consistent with the Darling-Kac-Aaronson (DKA) limit theorem from the fact that the Lyapunov exponent is an Lμ 1-class function. Next, we study the correlation function, which is not an Lμ 1-class function. The most remarkable result is that the transformed correlation function also reveals uniform convergence in distribution in the same sense of the DKA limit theorem.

AB - The initial ensemble dependence of statistical laws in non-hyperbolic dynamical systems with infinite ergodicity are studied by use of the modified Bernoulli maps. We show that statistical laws crucially depend on the initial ensemble and that the time average for the Lyapunov exponent converges in distribution for the non-stationary regime. This is completely consistent with the Darling-Kac-Aaronson (DKA) limit theorem from the fact that the Lyapunov exponent is an Lμ 1-class function. Next, we study the correlation function, which is not an Lμ 1-class function. The most remarkable result is that the transformed correlation function also reveals uniform convergence in distribution in the same sense of the DKA limit theorem.

KW - Ergodic theory

KW - Non-stationary chaos

UR - http://www.scopus.com/inward/record.url?scp=33846698870&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33846698870&partnerID=8YFLogxK

M3 - Article

VL - 50

SP - 254

EP - 260

JO - Journal of the Korean Physical Society

JF - Journal of the Korean Physical Society

SN - 0374-4884

IS - 1 I

ER -