  In the symmetric and the asymmetric trapezoid maps, as a slope of the
trapezoid is increased, the period doubling cascade occurs and the symbolic
sequence of periodic points is the Metropolis-Stein-Stein sequence and the
convergence of the onset point of the period 2^m solution to the accumulation
point is exponentially fast. We reported these results previously. In this
paper, we give the detailed description of the proof on the results. Further,
we study the period doubling cascade starting from period p solution and show
the superconvergence of the period doubling cascade.
