Masanori Kobayashi and Tzee-char Kuo

We consider the case of two real variables. A function g : R 2 ! R is blowanalytic if there exists a composition of simple blowings up, each centered at a point, = 1 n : X ! R 2 such that g is analytic ([7][4]). If a homeomorphism h : R 2 ! R 2 and its inverse, h 1 , both have blow-analytic components, we say h is blow-analytic. S. Koike([5]) was the rst to discover a blow-analytic homeomorphism of R 3 with itself which is not Lipschitz (see also [4]). L. Paunescu, in [9], has discovered one, also of R 3 , which does not preserve the multiplicity of analytic arcs. On the other hand, M. Suzuki([11]) and T. Fukui ([3]) have found some blowanalytic invariants which can be used to show, for example, that functions like x, x 2 y 3 , x 3 y 7 are not blow-analytically equivalent. A seemingly simple question, raised by Koike, is whether a blow-analytic homeomorphism h : R 2 ! R 2 can carry a line fx = 0g to a singular curve such as fx 2 = y 3 g or fx 3 = y 7 g. (These are as topological spaces; analytic structures are ignored. ) We shall answer the question in the a rmative. Let C be the germ of a singular curve in R 2 , which is unibranched in the sense that its complexi cation has only one branch ([12]).