The Exponential Map Is Chaotic: An Invitation to Transcendental Dynamics

The Exponential Map Is Chaotic: An Invitation to Transcendental Dynamics

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 was conjectured by Fatou in 1926 and first proved by Misiurewicz 55 years later.) The only background required is a first undergraduate course i...

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Veröffentlicht in:The American mathematical monthly 2015-12, Vol.122 (10), p.919-940
Hauptverfasser: Shen, Zhaiming, Rempe-Gillen, Lasse
Format: Artikel
Sprache:eng
Schlagworte:
Chaos theory
Complex analysis
Curricula
Distance functions
Dynamical systems
Exponential functions
Geometric planes
Hyperbolic geometry
Infinity
Mathematical constants
Mathematical theorems
Mathematics
Topological theorems
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Beschreibung
Zusammenfassung: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 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.
ISSN:0002-9890
1930-0972
DOI:10.4169/amer.math.monthly.122.10.919