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Quasiregular mappings of polynomial type in ℝ²

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 

 
 
 
 R
 
 
 2
 
 
 \mathbb {R}^{2}
 

 which are a composition of a quadratic polynomial and an affine stretch.