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 ∘ h is a Lipschitz quotient mapping. This corrects errors in the construction given earlier in [7]. 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 ℝn→ℝⁿ, n ≥ 3.