### Abstract

Let L_{A} = {f_{A, x} : x is a partition of [0, 1]} be a class of piecewise linear maps associated with a transition matrix A. In this paper, we prove that if f_{A, x} ∈ L_{A}, then the Liapunov exponent λ (x) of f_{A, x} is equal to a measure theoretic entropy h_{mA, x} of f_{A, x}, where m_{A, x} is a Markov measure associated with A and x. The Liapunov exponent and the entropy are computable by solving an eigenvalue problem and can be explicitly calculated when the transition matrix A is symmetric. Moreover, we also show that max_{x} λ (x) = max_{x} h_{mA, x} = log (λ_{1}), where λ_{1} is the maximal eigenvalue of A.

Original language | English |
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Pages (from-to) | 358-364 |

Number of pages | 7 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 338 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2008 Feb 1 |

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### Keywords

- Entropy
- Ergodic theory
- Liapunov exponents
- Piecewise linear map

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

**Piecewise linear maps, Liapunov exponents and entropy.** / Juang, Jonq; Shieh, Shih Feng.

Research output: Contribution to journal › Article

*Journal of Mathematical Analysis and Applications*, vol. 338, no. 1, pp. 358-364. https://doi.org/10.1016/j.jmaa.2007.05.035

}

TY - JOUR

T1 - Piecewise linear maps, Liapunov exponents and entropy

AU - Juang, Jonq

AU - Shieh, Shih Feng

PY - 2008/2/1

Y1 - 2008/2/1

N2 - Let LA = {fA, x : x is a partition of [0, 1]} be a class of piecewise linear maps associated with a transition matrix A. In this paper, we prove that if fA, x ∈ LA, then the Liapunov exponent λ (x) of fA, x is equal to a measure theoretic entropy hmA, x of fA, x, where mA, x is a Markov measure associated with A and x. The Liapunov exponent and the entropy are computable by solving an eigenvalue problem and can be explicitly calculated when the transition matrix A is symmetric. Moreover, we also show that maxx λ (x) = maxx hmA, x = log (λ1), where λ1 is the maximal eigenvalue of A.

AB - Let LA = {fA, x : x is a partition of [0, 1]} be a class of piecewise linear maps associated with a transition matrix A. In this paper, we prove that if fA, x ∈ LA, then the Liapunov exponent λ (x) of fA, x is equal to a measure theoretic entropy hmA, x of fA, x, where mA, x is a Markov measure associated with A and x. The Liapunov exponent and the entropy are computable by solving an eigenvalue problem and can be explicitly calculated when the transition matrix A is symmetric. Moreover, we also show that maxx λ (x) = maxx hmA, x = log (λ1), where λ1 is the maximal eigenvalue of A.

KW - Entropy

KW - Ergodic theory

KW - Liapunov exponents

KW - Piecewise linear map

UR - http://www.scopus.com/inward/record.url?scp=34548835002&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34548835002&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2007.05.035

DO - 10.1016/j.jmaa.2007.05.035

M3 - Article

AN - SCOPUS:34548835002

VL - 338

SP - 358

EP - 364

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 1

ER -