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The aim of this expository paper is to introduce the well-behaved uniformizing averages, which are useful in resummation theory. These averages associate three essential, but often antithetic, properties: respecting convolution; preserving realness; reproducing lateral growth. These new objects are serviceable in real resummation and we sketch two typical applications: the unitary iteration of unitary diffeomorphisms and the real normalization of real, local, analytic, vector fields.

Using the techniques developed by Jean Ecalle for the study of nonlinear differential equations, we prove that the $q$-difference equation $$x{\sigma }_{q}y=y+b(y,x)$$ with $\left({\sigma }_{q}f\right)\left(x\right)=f\left(qx\right)$ ($q\>1$) and $b(0,0)={\partial }_{y}b(0,0)=0$ is analytically conjugated to one of the following equations : $$x{\sigma }_{q}y=y\text{ou}x{\sigma }_{q}y=y+x$$