### 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