On the hyperbolicity of the period-doubling fixed point

Smania, Daniel

doi:10.1090/S0002-9947-05-03803-1

On the hyperbolicity of the period-doubling fixed point
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Trans. Amer. Math. Soc. 358 (2006), 1827-1846 Request permission

Abstract:

We give a new proof of the hyperbolicity of the fixed point for the period-doubling renormalization operator using the local dynamics near a semi-attractive fixed point (in a Banach space) and the theory of holomorphic motions. We also give a new proof of the exponential contraction of the Feigenbaum renormalization operator in the hybrid class of the period-doubling fixed point: our proof uses the non-existence of invariant line fields in the period-doubling tower (C. McMullen), the topological convergence (D. Sullivan), and a new infinitesimal argument.

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