Franks’s Lemma is a major tool in the study of differentiable dynamical systems. It says that along a simple orbit segment , the perturbation of can be realized via a perturbation of the map (which preserves the orbit segment). Moreover, such a perturbation is localized in a neighborhood of , and it can be made arbitrarily -close to .
There have been various generalizations of Franks’ Lemma. Some constraints have been noticed when generalizing to geodesic flows and billiard dynamics, since one can’t perturb the dynamics directly, but have to make geometric deformations. See D. Visscher’s thesis for more details.
Let be a strictly convex domain, be the orbit along the/a diameter of . Clearly is 2-period. Let be the radius of curvatures at , respectively. Then
, where stands for the diameter of .
Note that the two entries on the diagonal are always the same. Therefore any linearization with different entries on the diagonal can’t be realized as the tangent map along a periodic billiard orbit of period 2. In other words, even through there are three parameters that one can change: the distance , the radii of curvature at both ends , the effects lie in a 2D-subspace of the 3D .
Visscher was able to prove that generically, for each periodic orbit of period at least 3, every small perturbation of is actually realizable by deforming the boundary of billiard table. For more details, see Visscher’s paper:
A Franks’ lemma for convex planar billiards.