Quadratic Convergence in Period Doubling to Chaos for Trapezoid Maps

Abstract

The trapezoid mapge(x) is defined for fixede∈(0,1) byge(x)=x/eforx∈[0,e],ge(x)=1 forx∈(e,2−e), andge(x)=(2−x)/eforx∈[2−e,2]. For a giveneand the associated one-parameter family {λge(x): 1<λ<2}, letting λn(e) be the smallest value of λ>1 for which a fixedx∈(e,2−e), sayxc, is a periodic point of period 2n, Beyer and Stein conjectured in 1982 that, for anye<1, the parameter sequence {λn(e)}∞1is quadratically convergent. In this paper the conjecture is proved. Further, the quadratic convergence is generalized to nonisosceles trapezoid maps.