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

Masato Tsujii

doi:10.1007/BF02102427

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

Masato Tsujii

Research output: Contribution to journal › Article › peer-review

26 Citations (Scopus)

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 μ₄ as a tends to 4 on the set E. Also we give some generalization of this results.

Original language	English
Pages (from-to)	1-11
Number of pages	11
Journal	Communications in Mathematical Physics
Volume	177
Issue number	1
DOIs	https://doi.org/10.1007/BF02102427
Publication status	Published - May 3 1996
Externally published	Yes

All Science Journal Classification (ASJC) codes

Statistical and Nonlinear Physics
Mathematical Physics

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10.1007/BF02102427

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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.

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

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