Lyapunov exponents and related concepts for entire functions

TL;DR

This paper investigates the growth behavior of the spherical derivative and related Lyapunov exponents for entire functions, especially on Julia sets, providing insights into their dynamical complexity.

Contribution

It introduces new results on the growth rates of spherical derivatives and Lyapunov exponents for entire functions within complex dynamics.

Findings

Growth rates of $ ext{sup}_{z ext{ in } U} (f^n)^ ext#(z)$ tend to infinity.

Integral of $(f^n)^ ext#(z)^2$ over $U$ diverges as $n$ increases.

Growth behavior of $(f^n)^ ext#(z)$ for $z$ in Julia set analyzed.

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 $z\in J(f)$.