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Characteristic Exponents of Rational Functions

Volume 62 / 2014

Anna Zdunik Bulletin Polish Acad. Sci. Math. 62 (2014), 257-263 MSC: 30D99, 37F10. DOI: 10.4064/ba62-3-6


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


  • Anna ZdunikInstitute of Mathematics
    University of Warsaw
    Banacha 2
    02-097 Warszawa, Poland

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