Hyperbolicity of renormalization of critical circle maps
The paper is devoted to hyperbolicity of renormalization of critical circle maps and gives rigorous mathematical explanation of the universality phenomena. The renormalization theory of circle maps has developed alongside the unimodal theory. However, despite the similarity in the development of the two renormalization theories, the question of hyperbolicity of the horseshoe attractor for critical circle maps notably differs from that for unimodal maps. In the unimodal case, the proof of hyperbolicity includes two parts. The first (more simple) part is to endow the ambient space of the renormalization transformation with the structure of a complex-analytic manifold with respect to which renormalization is an analytic operator and to show that the stable sets of periodic points of this operator are codimension one submanifolds and that renormalization is a strict contraction in the stable direction. The second (much more difficult) part is to show the existence of an unstable direction. In contrast, in the case of critical circle maps it is well known how one may construct an unstable direction for normalization. The difficulty arises with the first part of the program: there is no obvious way to turn the renormalization into an analytic operator. To overcome the difficulty, a different renormalization construction (a renormalization of critical cylinder maps) is introduced. This construction is motivated by a so-called parabolic renormalization studied in \textit{M. Yampolsky} [Commun. Math. Phys. 218, 537--568 (2001; Zbl 0978.37033)]. By virtue of its construction, the cylinder renormalization is an analytic operator on a Banach manifold. It is shown that the stable sets of the periodic orbits of this operator are codimension one submanifolds and that every such orbit is hyperbolic. A natural functorial relation between the renormalization of cylinder maps and the usual one allows to transfer the results on the renormalization of critical circle maps. Some remaining open problems are discussed.
scientific article; zbMATH DE number 1986169