On a family of mild functions

Abstract

We prove that the function $P_{\alpha }(x)=\exp (1-x^{-\alpha })$ with $\alpha >0$, is $1/\alpha$-mild. We apply this result to obtain a uniform $1/\alpha$-mild parametrization of the family of curves $\{xy={\epsilon }^2\mid (x,y)\in (0,1{)}^2\}$ for $\epsilon \in (0,1)$, which does not have a uniform $0$-mild parametrization by work of Yomdin. More generally we can parametrize families of power-subanalytic curves. This improves a result of Benjamini and Novikov that gives a $2$-mild parametrization.

References in corpus (1)

• Smooth parameterizations of power-subanalytic sets and compositions of Gevrey functions

Cited by in corpus (1)

• Mild parametrizations of power-subanalytic sets