Nielsen-Thurston reducibility and renormalization

Consider an orientation preserving homeomorphism of the 2-disk with an infinite set of nested periodic orbits , such that, for all , the restriction of to the complement of the first orbits, from to , is times reducible in the sense of Nielsen and Thurston. We define combinatorial renormalization operators for such maps, and study the fixed points of these operators. We also recall the corresponding theory for endomorphisms of the interval, and give elements of comparison of the theories in one and two dimensions.