The entropy of entire transcendental functions

TL;DR

This paper demonstrates that the topological entropy of any entire transcendental function is infinite, using Bowen's definition, Ahlfors five islands theorem, and polynomial-like mappings, with entropy concentrated on the Julia set for certain meromorphic functions.

Contribution

It establishes the infinity of topological entropy for entire transcendental functions and links entropy concentration to the Julia set for meromorphic functions without wandering domains.

Findings

Topological entropy of entire transcendental functions is infinite.

Entropy is concentrated on the Julia set for meromorphic functions with no wandering domains.

Uses Bowen's definition, Ahlfors five islands theorem, and polynomial-like mappings.

Abstract

We use Bowen's definition of topological entropy and Ahlfors five islands theorem, as well as the theory of polynomial-like mappings, to show that the topological entropy of any entire transcendental function is infinity. In addition the entropy is concentrated on the Julia set for each meromorphic function which has no wandering domains.