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.
All Science Journal Classification (ASJC) codes
- Statistical and Nonlinear Physics
- Mathematical Physics