Geodesic
Period doubling, entropy, and renormalization
Fundamenta Mathematicae, Tome 155 (1998) no. 3, pp. 237-249
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We show that in any family of stunted sawtooth maps, the set of maps whose set of periods is the set of all powers of 2 has no interior point. Similar techniques then allow us to show that, under mild assumptions, smooth multimodal maps whose set of periods is the set of all powers of 2 are infinitely renormalizable with the diameters of all periodic intervals going to zero as the period goes to infinity. 
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Jun Hu; Charles Tresser. Period doubling, entropy, and renormalization. Fundamenta Mathematicae, Tome 155 (1998) no. 3, pp. 237-249. doi: 10.4064/fm-155-3-237-249

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