On the structural decomposition of planar Lipschitz quotient mappings

We show that for each fixed non-constant complex polynomial $P$ of the plane there exists a homeomorphism $h$ such that $P\circ h$ is a Lipschitz quotient mapping. This corrects errors in the construction given earlier by Johnson et. al. [Michigan Math. J. $\mathbf{\text{47}}$ (2000), 15-31]. Further we introduce a stronger notion of pointwise co-Lipschitzness and characterise its equivalence to the standard pointwise definition whilst also highlighting its relevance to a long-standing conjecture concerning Lipschitz quotient mappings ${\mathbb{R}}^n\to {\mathbb{R}}^n,n\ge 3$.