Inverse Function Theorems for Arc-analytic Homeomorphisms

Toshizumi Fukui; Krzysztof Kurdyka; Adam Parusiński

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arXiv · 1003.0826

Inverse Function Theorems for Arc-analytic Homeomorphisms

Toshizumi Fukui·Krzysztof Kurdyka·Adam Parusiński

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Abstract

We call a local homeomorphism $f: (R^n,0)\to(R^n,0)$ blow-analytic if it becomes real analytic after composing with a finite number blowings-up with smooth nowhere dense centers. If the graph of $f$ is semi-algebraic then, by a theorem of Bierstone and Milman, $f$ is blow-analytic if and only if it is arc-analytic: the image by $f$ of a parametrized real analytic arc is again a real analytic arc. For a semialgebraic homeomorphism $f$ we show that if $f$ is blow-analytic and the inverse of $f$ is Lipschitz, then $f$ is Lipschitz and the inverse of $f$ is blow-analytic. The proof is by a motivic integration argument, using additive invariants on the spaces of arcs.

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BibTeX RIS

Toshizumi Fukui, Krzysztof Kurdyka, Adam Parusiński. 2010-03-03. Inverse Function Theorems for Arc-analytic Homeomorphisms. https://arxiv.org/abs/1003.0826

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