弧-解析同胚的反函数定理
Inverse Function Theorems for Arc-analytic Homeomorphisms
摘要：
我们称一个局部同胚$f: (R^n,0)\to(R^n,0)$ blow-analytic如果它在与一个具有光滑无处稠密中心的有限个blow-up复合后成为实解析的。如果$f$的图是半代数的，那么根据Bierstone和Milman的定理，$f$是吹解析的当且仅当它是弧解析的：参数化的实解析弧的$f$的图像也是实解析弧。对于半代数同胚$f$，证明了如果$f$是blow-analytic的，且$f$的逆是Lipschitz，则$f$是Lipschitz的，且$f$的逆是blow-analytic的.证明是由motivic集成参数，使用添加剂的弧空间上的不变量。
英文摘要：
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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