Superconvergence of period doubling cascade in trapezoid maps Its Rigorous proof and superconvergence of 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, the period doubling cascade occurs and the symbolic sequence of periodic points is the Metropolis-Stein-Stein sequence R∗m and the convergence of the onset point am of the period 2 m solution to the accumulation point ac is exponentially fast. In the previous paper, we proved these results. In this paper, we give the detailed description of the proof on the results. Rigorously, we show that ǫm = ba−2 m c γ −ζm G∞(ac) (1 + hm), δm = γ m(acγ ) m (1 + lm), lim m→∞ hm = 0, lim m→∞ lm = 0, where ǫm ≡ ac − am, δm ≡ ǫm−ǫm+1 ǫm+1−ǫm+2 , b and γ are the smaller size of the trapezoid and the ratio of its two slopes, respectively. γ = 1 corresponds to the symmetric trapezoid. Further, we study the period doubling cascade starting from period p(≥ 3) solution. We show ǫm ∝ (γ ap,c) −2mγζm , δm ≃ γ m−1(ap,cγ ) m , where ap,c is the accumulation point of the onset of the period p× 2 m solution.