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2010•Conformal Geometry and Dynamics of the American Mathematical SocietyOpen access

Quasiregular mappings of polynomial type in ℝ²

Alastair Fletcher, Dan F. M. Goodman

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Abstract

Complex dynamics deals with the iteration of holomorphic functions. As is well known, the first functions to be studied which gave non-trivial dynamics were quadratic polynomials, which produced beautiful computer generated pictures of Julia sets and the Mandelbrot set. In the same spirit, this article aims to study the dynamics of the simplest non-trivial quasiregular mappings. These are mappings in R2 \mathbb {R}^{2} which are a composition of a quadratic polynomial and an affine stretch.

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What this paper is about

Complex dynamics deals with the iteration of holomorphic functions. As is well known, the first functions to be studied which gave non-trivial dynamics were quadratic polynomials, which produced beautiful computer generated pictures of Julia sets and the Mandelbrot set. In the same spirit, this article aims to study the dynamics of the simplest non-trivial quasiregular mappings. These are mappings in R2 \mathbb {R}^{2} which are a composition of a quadratic polynomial and an affine stretch.

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Available abstract

Complex dynamics deals with the iteration of holomorphic functions. As is well known, the first functions to be studied which gave non-trivial dynamics were quadratic polynomials, which produced beautiful computer generated pictures of Julia sets and the Mandelbrot set. In the same spirit, this article aims to study the dynamics of the simplest non-trivial quasiregular mappings. These are mappings in R2 \mathbb {R}^{2} which are a composition of a quadratic polynomial and an affine stretch.

Key concepts: Julia set, Holomorphic function, Mandelbrot set, Quadratic equation, Type (biology), Polynomial, Algorithm, Annotation

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