The Periodic Points of Renormalization

Abstract

It will be shown that the renormalization operator, acting on the space of smooth unimodal maps with critical exponent a > 1, has periodic points of any combinatorial type. A central question in the theory of dynamical systems is whether small scale geometrical properties of dynamical systems are determined by the combinatorial properties of the system. Indeed, such universality of small scale geometry was discovered by Coullet-Tresser and Feigenbaum. They studied infinitely renormalizable unimodal maps of the period doubling type and observed that the geometry of the invariant Cantor set of such maps converges when looking at smaller and smaller scales. Furthermore they observed that the limiting geometry was universal, in the sense that the small scale geometry of these Cantor sets depends only on the local behavior of the map around the critical point. This local behavior is specified by the critical exponent. To explain the universality of geometry, they defined the period doubling renormalization operator on a suitable space of unimodal maps. This operator acts like a microscope: the image under the renormalization operator is a unimodal map describing the geometry and dynamics on a smaller scale. The universality of geometry was understood by conjecturing that the renormalization operator has a unique hyperbolic fixed point. In particular, the infinitely renormalizable unimodal maps form the stable manifold of the fixed point of the renormalization operator. The first step in proving these conjectures is showing the existence of a renormalization fixed point.