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

TL;DR

This paper extends the theory of renormalization for critical circle maps to non-integer exponents, establishing hyperbolicity, universality, and rigidity results near odd integer exponents.

Contribution

It introduces a new renormalization operator for analytic circle maps with non-integer critical exponents and proves hyperbolicity and rigidity results near odd integer exponents.

Findings

Hyperbolicity of renormalization for maps with exponents close to odd integers.

Universality results for critical circle maps with non-integer exponents.

Rigidity in the $C^{1+eta}$ class for these maps.

Abstract

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.