Geometry, Topology and Dynamics Seminar
Hexi Ye
UIC
Rational functions with identical measure of maximal entropy
Abstract: We discuss when two rational functions $f$ and $g$ can have the same
measure of maximal
entropy. The polynomial case was completed by (Beardon, Levin,
Baker-Eremenko,$\cdots$,
1980s-90s), and we address the rational case following Levin-Prytycki
(1997). We
show: for general f of degree $d \geq 3$, if $\mu_f = \mu_g$, then $f$
and $g$ share an iterate
($f^n = g^m$ for some $n$ and $m$); further more for generic $f$ with
degree $d\geq 3$, $\mu_f = \mu_g$ implies that $g=f^n$ for some $n\geq 1$.
For generic $f\in Rat_2$, $\mu_f = \mu_g$ implies that $g= f^n$ or
$\sigma_f\circ f^n$ for some $n\geq 1$, where $\sigma_f$ permutes
two points in each fiber of $f$.
And we construct examples of $f$ and $g$ with $\mu_f = \mu_g$
such that $f^n \neq \sigma\circ g^m$ for any $\sigma \in PSL(2,\mathbb{C})$ and
$m,n\geq 1$.
Monday October 8, 2012 at 3:00 PM in SEO 636