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Rigidity, universality, and hyperbolicity of renormalization for critical circle maps with non-integer exponents
Published online by Cambridge University Press: 25 September 2018
Abstract
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.
MSC classification
- Type
- Original Article
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- Copyright
- © Cambridge University Press, 2018
References
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