Billiard map and rigid rotation

Can a billiard map be locally conjugated to a rigid rotation? We prove that the answer to this question is positive in the category of formal series. We also present numerical evidence that for “good” rotation angles the answer is also positive in an analytic category. •We study local dynamics of the billiard near a periodic orbit of period 2.•We show that dynamics can be formally conjugated to a rigid rotation.•We give numerical evidence that the effect is not only formal: the series converge.