Further results on periods and period doubling for iterates of the trapezoid function

The trapezoidal function λfe(x), is defined for fixed e ∈ (0,1] and for λ ∈ [1,2] by λfe(x) = λ for |x − 1| < 1 − e and λfe(x) = λ(1 − |x − 1|)/(1 − e) for 1 ≥ |x − 1| ≥ 1 − e. For a fixed e, this is a one parameter family of endomorphisms of the interval [0,2]. The structure of the periods (or cycles) of these mappings is studied. In addition, the metric properties of the corresponding bifurcation diagrams are considered; in particular, the rate of convergence of a sequence of bifurcation points in the (x,λ) plane is studied. It is shown to be different from that found by Feigenbaum and others for mappings which are not flat at the top. The limiting case e = 1 is of special interest. For cycles not containing a point x ∈ [e,2 − e], the period quadruplicates instead of doubling as it does in the usual case