Nielsen-Thurston reducibility and renormalization

Consider an orientation preserving homeomorphism $f$ of the 2-disk with an infinite set of nested periodic orbits $\{\mathcal {O}_n\}_{n\ge 1}$, such that, for all $m>1$, the restriction of $f$ to the complement of the first $m$ orbits, from $\mathcal {O}_1$ to $\mathcal {O}_m$, is $m-1$ 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.