Uniqueness of unibranched curve in \(\mathbb{R}^2\) up to simple blowings up
A function \(g:\mathbb{R}^m \to\mathbb{R}\) is blow-analytic if there exists a composition \(\beta= \beta_1\circ \cdots \circ \beta_n: X\to\mathbb{R}^m\) of simple blowings up, each centered at a point, such that \(g\circ \beta: X\to\mathbb{R}\) is analytic. A homeomorphism \(h:\mathbb{R}^m \to\mathbb{R}^m\) is blow-analytic if both \(h\) and its inverse \(h^{-1}\) have blow-analytic components. Blow-analytic homeomorphisms do not behave so well as expected. For example, \textit{S. Koike} [J. Math. Soc. Japan 45, No. 2, 313-320 (1993; Zbl 0788.32024)] found a non Lipschitz blow-analytic homeomorphism of \(\mathbb{R}^3\). Also, L. Paunescu, has recently discovered a blow-analytic homeomorphism of \(\mathbb{R}^3\) which does not preserve the multiplicity of analytic arcs. The paper under review deals with blow-analytic homeomorphisms of \(\mathbb{R}^2\). The authors introduce some invariants to study such homeomorphisms some of them graph theoretic, and an intersection matrix, mod 2, for the components of the strict transform of the exceptional divisor of each \(\beta_i\). The article contains no proofs and we remark two interesting facts. First, the authors answer in the affirmative a question posed by Koike: there exist blow-analytic homeomorphisms \(h_1: \mathbb{R}^2 \to \mathbb{R}^2\) and \(h_2: \mathbb{R}^2 \to \mathbb{R}^2\) which map the line \(\{x=0\}\) to the singular curves \(\{x^2=y^3\}\) and \(\{x^3= y^7\}\), respectively. Secondly, they prove a result ``à la Paunescu'' for \(\mathbb{R}^2\): there exist a blow-analytic homeomorphism of \(\mathbb{R}^2\) which does not preserve the multiplicity of analytic arcs.
scientific article; zbMATH DE number 1143816