Igors Gorbovickis: Renormalization for critical circle maps with non-integer exponents

An analytic critical circle map with critical exponent $\alpha$ is a
homeomorphism of the circle $\mathbb T=\mathbb R\slash\mathbb Z$ that
is analytic everywhere in $\mathbb T$ except possibly at one point at
which in some local chart it can be represented as $x\mapsto
\psi(x|x|^{\alpha-1})$, for some analytic diffeomorphism $\psi$. We
construct a renormalization operator which acts on analytic critical
circle maps whose critical exponent $\alpha$ is not necessarily an odd
integer $2n+1$, $n\in\mathbb N$. When $\alpha=2n+1$, our definition
generalizes the cylinder renormalization operator previously
constructed by Yampolsky. In the case when $\alpha$ is close to an odd
integer, we prove hyperbolicity of renormalization for maps of bounded
type. We use it to prove universality and $C^{1+\beta}$-rigidity for
such maps. The universality phenomenon here is analogous to the
Feigenbaum-Coullet-Tresser universality in one parameter families of
unimodal maps. This is a joint work with Michael Yampolsky.