Period doubling, entropy, and renormalization
Volume 155 / 1998
Fundamenta Mathematicae 155 (1998), 237-249 DOI: 10.4064/fm-155-3-237-249
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.