Smale’s Mean Value Conjecture and Complex Dynamics

Abstract

A version of the Smale Mean Value conjecture asserts that for a non-linear complex polynomial of the form $f(z)=z+a_{2}z^{2}+\dots +a_{d}z^{d}$ , there exists a critical point c of f for which $|f(c)/c|\leq 1$ . It is natural to conjecture further that in addition, one may assume c converges to the origin under iteration of f. We verify this more precise conjecture in the case of quadratics and cubics.