Lyapunov exponents and related concepts for entire functions

Abstract

Let $f$ be an entire function and denote by $f^\#$ be the spherical derivative of $f$ and by $f^n$ the $n$-th iterate of $f$. For an open set $U$ intersecting the Julia set $J(f)$, we consider how fast $\sup_{z\in U} (f^n)^\#(z)$ and $\int_U (f^n)^\#(z)^2 dx\:dy$ tend to $\infty$. We also study the growth rate of the sequence $(f^n)^\#(z)$ for $z\in J(f)$.Comment: 20 page

On the multipliers of repelling periodic points of entire functions

“…We shall need the additional information that given a sequence (w k ) satisfying |f (w k )| ≤ 1 for all k, the sequence (z k ) can be chosen such that |z k | and |w k | are of the same order of magnitude. This could be deduced from Pommerenke's method, but for completeness we include the following lemma whose proof is based on a different method which was also used in [18,11,14].…”