Inverse Function Theorems for Arc-analytic Homeomorphisms

TL;DR

This paper investigates the properties of blow-analytic and arc-analytic homeomorphisms, establishing conditions under which these maps are Lipschitz and their inverses retain blow-analyticity, using motivic integration techniques.

Contribution

It proves that for semialgebraic homeomorphisms, blow-analyticity and Lipschitz inverse imply Lipschitz continuity and blow-analytic inverse, extending understanding of these classes of maps.

Findings

Blow-analytic and arc-analytic equivalence for semi-algebraic maps.

Lipschitz inverse implies Lipschitz and blow-analytic inverse.

Use of motivic integration to prove properties of arc-analytic homeomorphisms.

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.