Berretti A., Celletti A., Chierchia L., Falcolini C. Natural Boundaries for Area Preserving Twist Maps (35K, LaTeX) ABSTRACT. We consider KAM invariant curves for generalizations of the standard map of the form $(x',y')=(x+y',y+\eps f(x))$, where $f(x)$ is an odd trigonometric polynomial. We study numerically their analytic properties by Pad\'e approximant method applied to the function which conjugates the dynamics to a rotation. In the complex $\eps$ plane, natural boundaries of different shapes are found. In the complex $\theta$ plane the analyticity region appears to be a strip bounded by a natural boundary, whose width tends linearly to $0$ as $\eps$ tends to the critical value.