Universal properties of maps of the circle with \(\epsilon\)-singularities
The authors study universal properties of circle diffeomorphisms or analytic circle homeomorphisms which have a single cubic critical point in terms of the renormalization transformation introduced by the second author et al. [Phys. Rev. Lett. 49, 132-135 (1982)]. Using some of the ideas of \textit{P. Collet}, \textit{J.-P. Eckmann} and \textit{O. E. Lanford III} [Commun. Math. Phys. 76, 211-254 (1980; Zbl 0455.58024)] they study the renormalization transformation on analytic functions of \(x| x|^{\epsilon}\) for \(\epsilon \geq 0\). The universal properties are consequences of the dynamics of the renormalization transformation near a certain fixed point. These properties are shown to hold when \(\epsilon =0\) and are obtained for \(\epsilon\) small by perturbation methods. For the homeomorphisms with cubic critical points one would like the results to hold for \(\epsilon =2\). Although this case is not covered by the results of this paper, the results do support the validity of the numerical experiments performed by the second author et al. in the paper cited above.
scientific article; zbMATH DE number 3868290
scientific article