Abstract: The problem of the existence of an analytic normal form near an equilibrium point
of an area-preserving map and analyticity of the associated coordinate change is a classical
problem in dynamical systems going back to PoincarĂŠ and Siegel. One important class of
examples of area-preserving maps consists of the collision maps for planar billiards.
Recently, Treschev discovered a formal bi-axially symmetric billiard with locally linearizable
dynamics and conjectured its convergence. Since then, a Gevrey regularity for such a billiard
was proven by Q. Wang and K. Zhang, but the original problem about analyticity still remains
open. We extend the class of billiards by relaxing the symmetry condition and allowing
conjugacies to non-linear analytic integrable normal forms. To keep the formal solution unique,
odd table derivatives and the normal form are treated as parameters of the problem. We show
that for the new problem, the series of the billiard table diverges for general parameters by
proving the optimality of Gevrey bounds. The general parameter set is prevalent (in a certain
sense has full measure) and it contains an open set. In order to prove that on an open set
Taylor series of the table diverges we define a Taylor recurrence operator and prove that it
has a cone property. All solutions in that cone are only Gevrey regular and not analytic.