Equivalence Relations for Two Variable Real Analytic Function Germs

For two variable real analytic function germs we compare the blowanalytic equivalence in the sense of Kuo to the other natural equivalence relations. Our main theorem states that C 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 C 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. The natural equivalence relations we first think of are the C coordinate changes for r = 1, 2, · · · ,∞, ω, where C stands for real analytic. Let f , g : (R, 0) → (R, 0) be real analytic function germs. We say that f and g are C (right) equivalent if there is a local C diffeomorphism σ : (R, 0) → (R, 0) such that f = g ◦ σ. If σ is a local bi-Lipschitz homeomorphism, resp. a local homeomorphism, then we say that f and g are bi-Lipschitz equivalent, resp. C equivalent. By definition, we have the following implications: (0.1) C-eq. ⇐ bi-Lipschitz eq. ⇐ C-eq. ⇐ C-eq. ⇐ · · · ⇐ C-eq. ⇐ C-eq. By Artin’s Approximation Theorem [2], C equivalence implies C equivalence. But the other converse implications of (0.1) do not hold. Let f , g : (R, 0) → (R, 0) be polynomial functions defined by f(x, y) = (x + y), g(x, y) = (x + y) + x for r = 1, 2, · · · . N. Kuiper [14] and F. Takens [21] showed that f and g are C equivalent, but not C equivalent. In the family of germs Kt(x, y) = x 4 + txy + y, the phenomenon of continuous C moduli appears: for t1, t2 ∈ I, Kt1 and Kt2 are C equivalent if and only if t1 = t2, where I = (−∞,−6], [−6,−2] or [−2,∞), see example 0.5 below. On the other hand, T.-C. Kuo proved that this family is The second named author was partially supported by the JSPS Invitation Fellowship Program. ID No. S-07026 1991 Mathematics Subject Classification. Primary: 32S15. Secondary: 14B05, 57R45.