Automorphisms of Cⁿ : The Andersén-Lempert Theory

Gupta, Tanuj (2017) Automorphisms of Cⁿ : The Andersén-Lempert Theory. Masters thesis, Indian Institute of Science Education and Research Kolkata.

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Abstract

For each entire function f in C, (z₁,..., zn) → (z₁,..., zn-1, zn + f(z₁)) is an automorphism of Cⁿ. Unlike C, Aut Cⁿ (n ≥ 2) is quite big. A simple class of automorphisms of Cⁿ consists of shears (additive and multiplicative). In this thesis, we first discuss some properties of shears. The main result that we present says that any automorphism of Cⁿ can be approximated, uniformly on compact sets, by composition of shears. This result is due to Andersén and Lempert. Most of the results about shears are taken from the paper by Jean-Pierre Rosay and Walter Rudin: Holomorphic maps from Cⁿ to Cⁿ and the form of Andersén-Lempert theory presented here is taken from the book Stein Manifolds and Holomorphic Mappings: The Homotopy Principle in Complex Analysis by Franc Forstnerič.

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