The Links Between Smale’s Mean Value Conjecture and Complex Dynamics

Abstract

In this paper we study the links between Smale’s Mean Value Conjecture (SMVC) and the convergence of critical points under iteration. We begin by introducing SMVC and discussing what progress has been made towards proving this conjecture. From there we give a brief introduction to a few concepts in Complex Dynamics and Complex Analysis including conjugacy, conjugations by 1/z, and orbits. We then show that conjugacy conserves SMVC. At this point, we go through the quadratic case to show the patterns that we are looking for as well as to show a basic structure of how this problem is being looked at. From there we look at the cubic case, going through SMVC for the cubic polynomial. We then introduce the Petal Theorem and show how it is used for the cubic case specifically. We conclude with the connections between the SMVC in the cubic case and the convergence of the same critical points explained and discuss future work.