bs{p}-Adic Confluence of $bs{q}$-Difference Equations

We develop the theory of $p$-adic confluence of $q$-difference equations. The main result is the surprising fact that, in the $p$-adic framework, a function is solution of a differential equation if and only if it is solution of a $q$-difference equation. This fact implies an equivalence, called ``Confluence, between the category of differential equations and those of $q$-difference equations. We obtain this result by introducing a category of ``sheaves on the disk $mathrm{D}^-(1,1)$, whose stalk at 1 is a differential equation, the stalk at $q$ is a $q$-difference equation if $q$ is not a root of unity $xi$, and the stalk at a root of unity is a mixed object, formed by a differential equation and an action of $sigma_xi$.