Equivalence relations for two variable real analytic function germs

抄録

For two variable real analytic function germs we compare the blow-analytic equivalence in the sense of Kuo to other natural equivalence relations. Our main theorem states that C¹ equivalent germs are blow-analytically equivalent. This gives a negative answer to a conjecture of Kuo. In the proof we show that the Puiseux pairs of real Newton-Puiseux roots are preserved by the C¹ equivalence of function germs. The proof is achieved, being based on a combinatorial characterisation of blow-analytic equivalence, in terms of the real tree model.
We also give several examples of bi-Lipschitz equivalent germs that are not blow-analytically equivalent.