Topological equivalence of polynomials and quasi-isometric mappings of the plane (Q1921782)

Stoilow's theorem implies that a quasiregular map \(f\) of the plane can be represented in the form \(f=\varphi\circ w\), where \(\varphi\) is analytic and \(w\) is a quasiconformal homeomorphism. The author studies a similar decomposition for quasi-isometries of the plane and proves that (1) if \(f\) is a quasi-isometry then \(f=\varphi\circ w\), where the function \(\varphi\) is a polynomial and \(w\) is a homeomorphism and that (2) if \(f\) is a polynomial then \(f=\varphi\circ w\), where \(\varphi\) is a quasi-isometry and \(w\) is a homeomorphism.