GEOMETRIC SINGULARITY THEORY BANACH CENTER PUBLICATIONS, VOLUME 65 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 2004 AN INVERSE MAPPING THEOREM FOR ARC-ANALYTIC HOMEOMORPHISMS TOSHIZUMI FUKUI Department of Mathematics, Faculty of Science, Saitama University 255 Shimo-Okubo, Urawa 338-8570, Japan E-mail: tfukui@rimath.saitama-u.ac.jp KRZYSZTOF KURDYKA Laboratoire de Math´ ematiques, Universit´ e de Savoie Campus Scientifique, 73 376 Le Bourget-du-Lac Cedex, France E-mail: kurdyka@univ-savoie.fr LAURENTIU PAUNESCU School of Mathematics and Statistics University of Sydney, NSW 2006, Australia E-mail: laurent@maths.usyd.edu.au Abstract. We show that a subanalytic map-germ (R n , 0) → (R n , 0) which is arc-analytic and bi-Lipschitz has an arc-analytic inverse. Let U be an open subset of R n or more generally a smooth real analytic variety. Following [8] we say that a map f : U → R k is arc-analytic if f ◦ α is analytic for any analytic arc α :(−ε, ε) → U . In general these functions are very far from being analytic, in particular there are arc-analytic functions which are not subanalytic [9], not continuous [2], with a non-discrete singular set [10]. Hence it is natural to consider only arc-analytic maps with subanalytic graphs. In the late seventies T.-C. Kuo introduced the notion of blow-analytic mappings, i.e., mappings which become analytic after a com- position with appropriate proper bi-meromorphic maps (e.g. composition of blowings-up with smooth centers). He suggested that the equivalence relation for germs of analytic functions, defined by blow-analytic homeomorphisms, should give “the canonical” strat- 2000 Mathematics Subject Classification : Primary 32B20; Secondary 14Pxx. Key words and phrases : arc-analytic, Lipschitz, sub-analytic, homeomorphism. The authors thank JSPS for making possible their collaboration. The paper is in final form and no version of it will be published elsewhere. [49]