Period doubling in maps with a maximum of order (Q1118225)

We consider the scaling behaviour in period-doubling systems, exemplified by the one-dimensional map \(x_{n+1}=1-\lambda | x_ n|^ z\), which has a maximum of order z \((z>1)\). The Feigenbaum scaling factors \(\alpha\) and \(\delta\) are studied as functions of z, and more generally the scaling functions 1/\(\sigma\) and f(a). In particular, using the universal functions g(x) and h(x) we establish the inequality \(\delta <\alpha^ z\), which implies that \(\delta\) remains finite (\(\lesssim 30)\) in the limit \(z\to \infty\).