Motivation From Dynamics - A Brief Sketch

1 Introduction 1.1 Motivation from dynamics–a brief sketch This work is about the combinatorial aspects of rigidity phenomena in complex dynamics. It is motivated by discoveries of Douady-Hubbard [DH1], MilnorThurston [MT], and Sullivan made during the early 1980’s (see the preface by Hubbard in [Tan4] for a firsthand account). In the real quadratic family fa (x) = (x2 + a)/2,a ∈ R, it was proven [MT] that the entropy of fa as a function of a is continuous, monotone, and increasing as the real parameter varies from a = 5 to a = 8. A key ingredient of their proof is a complete combinatorial characterization and rigidity result for critically periodic maps fa , i.e. those for which the unique critical point at the origin is periodic. To any map fa in the family one associates a combinatorial invariant, called its kneading invariant. Such an invariant must be admissible in order to arise from a map fa . It was shown that every admissible kneading invariant actually arises from such a map fa , and that if two critically periodic maps have the same kneading invariant, then they are affine conjugate. In a process called microimplantation the dynamics of one map fa could be “glued” into that of another map fa0 where fa0 is critically periodic to obtain a new map fa∗a0 in this family. More precisely: a topological model for the new map is constructed, and its kneading invariant, which depends only on the topological data, is computed. The result turns out to be admissible, hence by the characterization theorem defines uniquely a new map fa0 ∗a . This construction interprets the cascade of period-doublings as the limit limn→∞ fan where an+1 = an ∗ a0 and a0 is chosen so that the critical point is periodic of period two. As an application, it is shown that there exists an uncountable family of maps with distinct kneading invariants but with the same entropy. Similar combinatorial rigidity phenomena wer