Characteristic Exponents of Rational Functions

Anna Zdunik ¹

¹ Institute of Mathematics University of Warsaw Banacha 2 02-097 Warszawa, Poland

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 62 (2014) no. 3, pp. 257-263

Résumé

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 $z\mapsto z^{\pm d}$.

Détail

DOI : 10.4064/ba62-3-6

Keywords: consider characteristic exponents rational function hat mathbb hat mathbb degree exponent chi average log respect measure maximal entropy exponent chi defined maximal characteristic exponent periodic orbits prove chi chi only conformally conjugate mapsto