Quadratic Convergence in Period Doubling to Chaos for Trapezoid Maps

The trapezoid map ge x. is defined for fixed eg 0, 1. by ge x.sxre for xgw0, ex, ge x.s1 for xg e, 2ye., and ge x.s 2yx.re for xgw2ye, 2x. For a given e and the associated one-parameter family lge x.: 1-l -24, letting ln e. be the smallest value of l )1 for which a fixed xg e, 2ye., say xc, is a periodic point of period 2n, Beyer and Stein conjectured in 1982 that, for any e-1, the parameter sequence ln e.4`1 is quadratically convergent. In this paper the conjecture is proved. Further, the quadratic convergence is generalized to nonisosceles trapezoid maps