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.

In the following we discuss something else.

1. Notes from Donnay’s paper (see also here). Let be a connected bounded domain on the plane, and be the billiard map induced on . Let be a smooth curve, and for . Assume is a 2-nd order deformation of around the base point , that is, ,

, but . Assume the perturbation is so localized such that are not effected. Let be the new billiard map, which preserves the orbit segment . But we always have that and are transverse at in .

Denote the three tangent vectors at . It suffices to show . Note that

–the backward focusing function of satisfies ;

–the forward focusing function of satisfies .

–the mirror formula says that .

Then we have (since we assumed ). Therefore .

For certainty let’s assume . Then we have , or equally, : the -beam focuses before the -beam.

Another way to see this is via computing the slopes. In a proper coordinate system, the slope of a vector satisfies the following . One can show that .

2. Notes from Donnay’s paper. Let be a smooth Riemannian surface, , and be a small neighborhood of , be a geodesic passing through with , for all . Consider a normal variation of geodesics with such that the Jacobi field for all . Let be the geodesic flow on , be two families of geodesics through .

Let be 2nd order perturbation of the metric inside along the geodesic such that for all (the existence of such a perturbation needs be verified). In particular is still a geodesic with respect to . However, and are now transverse at . To show this, we need to compare the solutions of Riccati equations (for )

- and for , with the initial condition ;
- and for , with the final condition .

Take the first one for example. Let and . Then ; . A solution is given by

. Since , one gets for all and hence . Similarly (via time-reversal) we have and

.

3. Let be a closed manifold, and be a nonsingular flow on . A cross-section is a co-dimension 1 closed submanifold transverse to the flow, such that every flow-line returns to after leaving it. More precisely, there exists a smooth function , and a diffeomorphism on such that . Such is called the first-return map of , and is called the suspension of .

The existence of such sections is rare. For example, if , then no such section could exist, since every closed -dimensional closed submanifold in is a boundary and separates .

A Birkhoff section is a cross section with boundary such that

- is invariant under the flow;
- is transverse to the flow;
- the angle between and tends to 0 at the same rate that .

4. Consider the complex projective plane . There is an injective complex plane , whose complement is a copy of , which a complex projective line of points at infinity. Consider a regular circle in : . We can complexify this equation and convert it into a homogeneous equation on : . The solution of this equation is called the complexification of a real circle.

Note that the two points and are the two common solutions (for any choice of ). These two are called the circular points at infinity, or isotropic points in .

5. Let be a homeomorphism, be a non-periodic point. Kodama and Matsumoto constructed a cocycle with . Such is given by which converges uniformly. More precisely, for each ,

let be a sufficiently small neighborhood of such that , are all disjoint. Let be a bump function with . Then we extend the domain of such that

- for : ;
- for : ;
- elsewhere.

A special feature of such a cocycle is that for all ; and for all .

Now let . Note that for any . In particular one can pick with , and get . Therefore

.