Characteristic Exponents of Rational Functions

We consider two characteristic exponents of a rational function $f:\hat{\mathbb{C}}\to\hat{\mathbb{C}}$ of degree $d\ge 2$. The exponent $\chi_a(f)$ is the average of $\log \|f'\|$ with respect to the measure of maximal entropy. The exponent $\chi_m(f)$ can be defined as the maximal characteristic exponent over all periodic orbits of $f$. We prove that $\chi_a(f)=\chi_m(f)$ if and only if $f(z)$ is conformally conjugate to $z\mapsto z^{\pm d}$.