### Abstract

Let us consider a family of maps Q_{a}(x) = ax(1 - x) from the unit interval [0, 1] to itself, where a ∈ [0, 4] is the parameter. We show that, for any β < 2, there exists a subset E ∋ 4 in [0, 4] with the properties (1) Leb([4 - ε, 4] - E) < ε^{β} for sufficiently small ε > 0, (2) Q_{a} admits an absolutely continuous BRS measure μ_{a} when a ∈ E, and (3) μ_{a} converges to the measure μ_{4} as a tends to 4 on the set E. Also we give some generalization of this results.

Original language | English |
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Pages (from-to) | 1-11 |

Number of pages | 11 |

Journal | Communications in Mathematical Physics |

Volume | 177 |

Issue number | 1 |

DOIs | |

Publication status | Published - May 3 1996 |

Externally published | Yes |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

**On continuity of Bowen-Ruelle-Sinai measures in families of one dimensional maps.** / Masato, Tsujii.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - On continuity of Bowen-Ruelle-Sinai measures in families of one dimensional maps

AU - Masato, Tsujii

PY - 1996/5/3

Y1 - 1996/5/3

N2 - Let us consider a family of maps Qa(x) = ax(1 - x) from the unit interval [0, 1] to itself, where a ∈ [0, 4] is the parameter. We show that, for any β < 2, there exists a subset E ∋ 4 in [0, 4] with the properties (1) Leb([4 - ε, 4] - E) < εβ for sufficiently small ε > 0, (2) Qa admits an absolutely continuous BRS measure μa when a ∈ E, and (3) μa converges to the measure μ4 as a tends to 4 on the set E. Also we give some generalization of this results.

AB - Let us consider a family of maps Qa(x) = ax(1 - x) from the unit interval [0, 1] to itself, where a ∈ [0, 4] is the parameter. We show that, for any β < 2, there exists a subset E ∋ 4 in [0, 4] with the properties (1) Leb([4 - ε, 4] - E) < εβ for sufficiently small ε > 0, (2) Qa admits an absolutely continuous BRS measure μa when a ∈ E, and (3) μa converges to the measure μ4 as a tends to 4 on the set E. Also we give some generalization of this results.

UR - http://www.scopus.com/inward/record.url?scp=0030525810&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0030525810&partnerID=8YFLogxK

U2 - 10.1007/BF02102427

DO - 10.1007/BF02102427

M3 - Article

AN - SCOPUS:0030525810

VL - 177

SP - 1

EP - 11

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 1

ER -