Hexi Ye : Rational Functions with Identical Measure of Maximal Entropy

Posted by Paul Reschke , part of the Graduate Student Colloquium.

At

March 5, 2012, 4:15 p.m.

In

SEO 636

Abstract

Let $f$ be a generic rational function with degree $d \geq 3$. We are going to show that for any rational function $g$ with $\mu_f=\mu_g$ (the same measure of maximal entropy), $f$ and $g$ have some common iterate--i.e., $f^n=g^m$. Moreover, we are going to give non-exceptional functions $f$ and $g$, with the same degree and measure, such that there is no $\sigma$ in $PSL_2(\mathbb{C})$ and no $n \geq 1$ with $f^n=\sigma \circ g^n$ (which cannot happen in the polynomial case). This talk is meant to be accessible for graduate students of all levels; no technical knowledge of complex dynamics is expected.