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Rigidity, universality, and hyperbolicity of renormalization for critical circle maps with non-integer exponents
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 40 / Issue 5 / May 2020
- Published online by Cambridge University Press:
- 25 September 2018, pp. 1282-1334
- Print publication:
- May 2020
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- Article
- Export citation
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We construct a renormalization operator which acts on analytic circle maps whose critical exponent
$\unicode[STIX]{x1D6FC}$ is not necessarily an odd integer 
$2n+1$, 
$n\in \mathbb{N}$. When 
$\unicode[STIX]{x1D6FC}=2n+1$, our definition generalizes cylinder renormalization of analytic critical circle maps by Yampolsky [Hyperbolicity of renormalization of critical circle maps. Publ. Math. Inst. Hautes Études Sci.96 (2002), 1–41]. In the case when 
$\unicode[STIX]{x1D6FC}$ is close to an odd integer, we prove hyperbolicity of renormalization for maps of bounded type. We use it to prove universality and 
$C^{1+\unicode[STIX]{x1D6FC}}$-rigidity for such maps.
