The universality phenomena in smooth families of circle homeomorphisms with one critical point, the so-called critical circle maps, are analogous to Feigenbaum universality and are explained by hyperbolicity of the so-called cylinder renormalization operator. So far the theory is complete only in the case of critical circle maps with the critical point of order $3$ (or any other odd positive integer). In this talk I will extend the cylinder renormalization operator to the new functional space that includes critical circle maps with the critical point of an arbitrary order. Then applying perturbation argument, I will show that in the space of critical circle maps of bounded type and with the critical point of a fixed order close to 3, the periodic orbits of the cylinder renormalization operator are hyperbolic.

This is a joint work with Michael Yampolsky.