Return times of polynomials as meta-Fibonacci numbers

Abstract

We consider generalized closest return times of a complex polynomial of degree at least two. Most previous studies on this subject have focused on the properties of polynomials with particular return times, especially the Fibonacci numbers. We study the general form of these closest return times. The main result of this paper is that these closest return times are meta-Fibonacci numbers. This result applies to the return times of a principal nest of a polynomial. Furthermore, we show that an analogous result holds in a tree with dynamics that is associated with a polynomial.