Rigidity, universality,and hyperbolicity of renormalization for critical circle maps with non-integer exponents

We construct a renormalization operator which acts on analytic 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 cylinder renormalization of analytic critical circle maps. 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 $C^{1+\alpha }$-rigidity for such maps.