Superconvergence of Period-Doubling Cascade in Trapezoid Maps Its Rigorous Proof and Superconvergence of the Period-Doubling Cascade Starting from a Period p Solution

In the symmetric and the asymmetric trapezoid maps, as a slope a of the trapezoid is increased, a period-doubling cascade occurs. The symbolic sequence of periodic points in this cascade is the Metropolis-Stein-Stein sequence R∗m, and the convergence of the onset points am of the period 2 m solutions to the accumulation point ac is exponentially fast. In a previous paper, we proved these results. In this paper, we give a detailed description of proofs regarding the results. Further, we study the period-doubling cascade starting from the period p ( ≥ 3) solution and show that the convergence of the onset points of the period p × 2m solutions is also exponentially fast. It is well known that in a class of one-dimensional maps, as a parameter is changed, there occurs a cascade of period-doubling bifurcations, and there exist several universal properties. 1) One of the universal quantities is the so-called Feigen-baum constant δ, which depends only on the exponent z characterizing the behavio