Period doubling for \(C^{2+\epsilon}\) mappings (Q1918062)

The paper deals with the problem of period-doubling cascades, developed by Feingenbaum and thoroughly studied by Lanford for analytic families, and gives a rigorous treatment of \(C^{2+\varepsilon}\) families, \(0<\varepsilon< 1\). As in the Feingenbaum-Lanford theory, the approach is based on the study of a nonlinear doubling operator \(T\) on a Hölder space \(E= C^{2+\varepsilon}(I)\) of \(C^{2+\varepsilon}\)-functions on \(I= [a,b]\). The existence of a fixed point \(g\) of \(T\) is established, the spectrum of the Gâteaux derivative \(DT(g)\) of \(T\) at \(g\) (\(T\) is no longer Fréchet differentiable on \(E\)) is studied by means of introduced localized norms, and a version of the theory of stable manifolds is used in order to deduce information on the behaviour of \(T\) near \(g\). The methods presented in the paper are not direct (i.e., they use analytic theory); however, the authors hope that their attitude may be applicable in other renormalization problems for which a direct approach might be not available.