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

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Abstract

Let us consider a family of mapsQ a (x)=ax(1−x) from the unit interval [0,1] to itself, wherea∈[0,4] is the parameter. We show that, for any β<2, there exists a subsetE∋4 in [0,4] with the properties

  1. (1)

    Leb([4−ɛ,4]−E) < ɛβ for sufficiently small ɛ>0,

  2. (2)

    Q a admits an absolutely continuous BRS measure µa whena∈E, and

  3. (3)

    µa converges to the measure µ4 asa tends to 4 on the setE. Also we give some generalization of this results.

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Authors and Affiliations

  1. Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Ohokayama, Meguro, 152, Tokyo, Japan

    Masato Tsujii

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  1. Masato Tsujii
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Communicated by J.-P. Eckmann

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Tsujii, M. On continuity of Bowen-Ruelle-Sinai measures in families of one dimensional maps. Commun.Math. Phys. 177, 1–11 (1996). https://doi.org/10.1007/BF02102427

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  • DOI: https://doi.org/10.1007/BF02102427

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