Rigidity of bi-Lipschitz equivalence of weighted homogeneous   function-germs in the plane
  The main goal of this work is to show that if two weighted homogeneous (but not homogeneous) function-germs $(\C^2,0)\to(\C,0)$ are bi-Lipschitz equivalent, in the sense that these function-germs can be included in a strongly bi-Lipschitz trivial family of weighted homogeneous function-germs, then they are analytically equivalent. 
Alexandre Fernandes and Maria Ruas
math.AG