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

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Abstract

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.

Original languageEnglish
Pages (from-to)1-11
Number of pages11
JournalCommunications in Mathematical Physics
Volume177
Issue number1
DOIs
Publication statusPublished - May 3 1996
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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