On blow-analytic equivalence(Singularities and o-minimal category)
Toshizumi Fukui · Institutional Repositories DataBase (IRDB) · 2007
This is a resume for the talk, with the title above, at 29 November 2007 at RIMS workshop.This is a joint work with Laurentiu Paunescu.Motivated by the classification problem of analytic frnction germs, T.-C.Kuo ([31]) introduced the notions of blow-anaJytic maps and blow-analytic equivalence.We start the article explaining this motivation to deflne blow-analytic cquivalence.He discovered a finite classification theorem for analytic function germs with iso- lated singularities and also shows some important triviality theorems.We are going to report several facts known now about the blow-analytic triviality arrd invariants.We then discuss Lipschitz property of blow-analytic maps an\'o show blow-analytic homeomorphism can be far from Lipschitz map.We also discuss exotic pathologies on a blow-analytic homeomorphisIn: this is illustrated by the exaxnples in \S 7. We then introduce a strengthened notion, called blow-analytic isomorphism, and discuss the behavior of their jacobians.In \S 8, we present a version of thc Inverse Mapping Thcorcni for blow-aIialytic isomorphisms. MotivationsThe notion of blow-anaJytic equivalence arises from attempts to classify analytic function germs.One is temptcd to use thc following $equi\tau^{ u}a1_{C^{\backslash }1}\downarrow cc\iota\cdot c^{\backslash }1atio\iota 1$ .Definition 1.1.Let $k=0,1,2,$ $\ldots,$ $\infty,$ $\omega$ .We say that two analytic functiou- germs $f,$ $g$ : $R^{n}.0arrowR,$ $0$ are $C^{k}$ -equivalent if there is a $C^{k}-diffeomorphism$ -germ $h:R^{n},$ $0arrow R^{n}.0$so that $f=g\circ h$ .However, the following example, due to H. Whitney, shows that the $C^{1}$ -equivalence is $ah\cdot eady$ too fine for the classification purpose.Example 1.2 ([41]).Consider the functions $/l$ .: $R^{2},0arrow R,$ $0<t<1$ , defincd by $f_{t}(x, y)=xy(y-x)(y-tx)$ .Then $f_{t}$ is C'-equivalent to $f_{t'}$ , if and only if $r=t'$ .As for tha ( $\sim$ ノ-equivalence. the functions $(;t:_{Y}l)$ ) $\vdasharrow?i^{2}+y^{2k+1}$ .$k\geq 1$ , for instance, are $C^{(}$ -equivalent to the regular function $(x\cdot, y)\mapsto y$ .Hence it seems hopeless to expect a decent classification theory.Now we consider the blowing-up $\pi$ : $\Lambda''arrow R^{2}$ : at $0$ .This map is illustrated by $th($ , following picture,