The exponential map is chaotic: An invitation to transcendental dynamics

TL;DR

This paper provides an elementary proof that the complex exponential map exhibits chaotic behavior as a dynamical system, confirming a long-standing conjecture and making the result accessible with minimal background.

Contribution

It offers a simple, conceptual proof of the exponential map's chaos, bridging a gap in understanding with minimal prerequisites.

Findings

Confirmed the chaos of the exponential map on the complex plane.

Provided an accessible proof suitable for undergraduate-level understanding.

Reinforced the conjecture made by Fatou and proved by Misiurewicz.

Abstract

We present an elementary and conceptual proof that the complex exponential map is "chaotic" when considered as a dynamical system on the complex plane. (This result was conjectured by Fatou in 1926 and first proved by Misiurewicz 55 years later.) The only background required is a first undergraduate course in complex analysis.