### Abstract

We consider a family of chaotic skew tent maps. The skew tent map is a two-parameter, piecewise-linear, weakly-unimodal, map of the interval F_{a,b}. We show that F_{a,b} is Markov for a dense set of parameters in the chaotic region, and we exactly find the probability density function (pdf), for any of these maps. It is well known, that when a sequence of transformations has a uniform limit F, and the corresponding sequence of invariant pdfs has a weak limit, then that invariant pdf must be F invariant. However, we show in the case of a family of skew tent maps that not only does a suitable sequence of convergent sequence exist, but they can be constructed entirely within the family of skew tent maps. Furthermore, such a sequence can be found amongst the set of Markov transformations, for which pdfs are easily and exactly calculated. We then apply these results to exactly integrate Lyapunov exponents.

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

Pages (from-to) | 365-376 |

Number of pages | 12 |

Journal | Chaos, Solitons and Fractals |

Volume | 12 |

Issue number | 2 |

DOIs | |

State | Published - 2 Jan 2001 |

### Fingerprint

### Cite this

*Chaos, Solitons and Fractals*,

*12*(2), 365-376. https://doi.org/10.1016/S0960-0779(99)00204-0

}

*Chaos, Solitons and Fractals*, vol. 12, no. 2, pp. 365-376. https://doi.org/10.1016/S0960-0779(99)00204-0

**Probability density functions of some skew tent maps.** / Billings, Lora; Bollt, E. M.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Probability density functions of some skew tent maps

AU - Billings, Lora

AU - Bollt, E. M.

PY - 2001/1/2

Y1 - 2001/1/2

N2 - We consider a family of chaotic skew tent maps. The skew tent map is a two-parameter, piecewise-linear, weakly-unimodal, map of the interval Fa,b. We show that Fa,b is Markov for a dense set of parameters in the chaotic region, and we exactly find the probability density function (pdf), for any of these maps. It is well known, that when a sequence of transformations has a uniform limit F, and the corresponding sequence of invariant pdfs has a weak limit, then that invariant pdf must be F invariant. However, we show in the case of a family of skew tent maps that not only does a suitable sequence of convergent sequence exist, but they can be constructed entirely within the family of skew tent maps. Furthermore, such a sequence can be found amongst the set of Markov transformations, for which pdfs are easily and exactly calculated. We then apply these results to exactly integrate Lyapunov exponents.

AB - We consider a family of chaotic skew tent maps. The skew tent map is a two-parameter, piecewise-linear, weakly-unimodal, map of the interval Fa,b. We show that Fa,b is Markov for a dense set of parameters in the chaotic region, and we exactly find the probability density function (pdf), for any of these maps. It is well known, that when a sequence of transformations has a uniform limit F, and the corresponding sequence of invariant pdfs has a weak limit, then that invariant pdf must be F invariant. However, we show in the case of a family of skew tent maps that not only does a suitable sequence of convergent sequence exist, but they can be constructed entirely within the family of skew tent maps. Furthermore, such a sequence can be found amongst the set of Markov transformations, for which pdfs are easily and exactly calculated. We then apply these results to exactly integrate Lyapunov exponents.

UR - http://www.scopus.com/inward/record.url?scp=0343462471&partnerID=8YFLogxK

U2 - 10.1016/S0960-0779(99)00204-0

DO - 10.1016/S0960-0779(99)00204-0

M3 - Article

AN - SCOPUS:0343462471

VL - 12

SP - 365

EP - 376

JO - Chaos, solitons and fractals

JF - Chaos, solitons and fractals

SN - 0960-0779

IS - 2

ER -