9. Victor Ivrii conjecture. Let be a strictly convex domain, be the billiard map on the phase space . Let be the Lebesgue measure of , and be the Lebesgue measure on .

**Conjecture 1.** for all with boundaries.

*Remark.* This is about a general domain , not a generic domain.

**Definition.** A point is said to be an *absolute looping point,* if . Let be the set of absolute looping points.

**Conjecture 2.** for all .

*Question:* When ?

8. In Boltzmann gas model, the identical round molecules are confined by a box. Sinai has replaced the box by periodic boundary conditions so that the molecules move on a flat torus.

On circular and elliptic billiard tables, for all , the -periodic orbits forms a continuous family and hence all the trajectories have the same length.

An invariant noncontractible topological annulus, , whose interior contains no invariant circles, is a Birkhoff instability region. The dynamics in an instability region

has positive topological entropy. Hence Birkhoff conjecture implies

that any non-elliptical billiard has positive topological entropy.

How to construct a strictly convex -smooth billiard table with metric positive entropy? b) How to construct a convex -smooth billiard table with positive metric entropy?

Recall Bunimovich stadium is not , and not strictly convex.

A periodic orbit of period corresponds to an (oriented) closed polygon with sides, inscribed in , and satisfying the condition on the angles it makes with the boundary. Birkhoff called these the harmonic polygons.

Then the maximal circumference of 2-orbit yields the diameter of . The minimax circumference of 2-orbit corresponds

to the width of .

7. Sinai’s Fundamental Theorem of Dispersing Billiards. Suppose the orbit of never hit the singular set . Then for any , any there exists an open neighborhood such that

- for any unstable curve : ;
- for any stable curve : ;

In particular every points can be connected to invariant manifolds of larger size by -paths.

Hopf argument. Consider the billiard map on a dispersing billiard table. A point is said to be *Birkhoff-regular* if the backward and forward Birkhoff averages exist at for every continuous function . Denote by the set of such points. Then by *Birkhoff ergodic theorem*.

A point is said to be **good**, if , the stable and unstable manifolds exist and are -saturated. Denote by the set of such points. Then by -regularity and Pesin’s *absolute continuity* of these invariant foliations.

Property. For any given unstable manifold , exists for -a.e. , and those points ‘whose are not -saturated’ consist of -null subset of (by Pesin’s * transverse absolute continuity* of stable foliation).

In particular if is -saturated, then it is also -saturated. Therefore and are -saturated for each . So if we start with , , and build inductively , the accessibility class of cornering in .

- and are -saturated for every .
- and lies in one ergodic component of .

Local ergodicty: For every , there exists an open neighborhood (mod ).

Proof. There exists and a rectangle surrounding of stable size and unstable size . Fix for some large. Although not every stable/unstable manifolds in can fully cross it, there are -paths initializing from each point , cornering in and fully crossing , say . Given two points , the two leaves and , being -saturated, admit a nontrivial stable holonomy . In particular and : there exists such that . So the pair can be connected by a -paths cornering in , so is the pair .

Let and consider the following Newton system: , while . So the derivative . In particular the constant kinetic surface is a global attractor of the system.

6. A smooth curve is *(relatively) focusing*, if every incoming parallel wave-front will become focusing after the first reflection on .

A focusing curve Γ is *absolutely focusing*, if every ray leave after finitely many reflections, and every incoming parallel wave-front become focusing after the last reflection in the series of consecutive reflections on .

The above were introduced by Bunimovich in 1988 (under a different name: admissible). The local condition is studied by Donnay in 1991, where he prove the hyperbolicity of a class of billiards. Bunnimovich (1992) proved that weaker global condition implies the stronger local property.

Consider the ellipse . Without loss of generality we assume .

The right half is absolutely focusing if and only if .

The larger half is never absolutely focusing, since it traps rays of positive volume.

We parametrize the ellipse by for .

Donnay (1991). The arc is absolutely focusing if the endpoints connection does not intersect the focal connection .

Proof. For such arc, only the winding orbits can hit it twice and parallel wave-fronts focus between each collision.

In particular the upper ellipse is absolutely focusing. Again the larger half is not.

Dynamics on regular islands is characterized by a balance between focusing and defocusing, while defocusing dominates focusing on chaotic sea. Dispersing is a special case of defocusing, in the sense that the focusing time is negative.

Conjecture (Bunnimovich 1991): for every hyperbolic billiard, each focusing component of the boundary is absolutely focusing.

Bunnimovich (2003) and with Grigo (2010) proved that if a focusing arc is not absolutely focusing, then for any large , can be included in some billiard table whose minimal free path leaving is larger than , and has some elliptic periodic points on . So the violation of the absolutely focusing cannot by compensated for by making the free path arbitrarily large.

and absolutely focusing is a necessary condition to construct universally chaotic billiard tables.

Note that if , then the incoming parallel wave-front focus during the -flight and leave as a parallel wave-front. Every arc of a round circle is absolutely focusing, so is the short enough focusing piece, since the curvature are almost constant and . Similarly any focusing boundary components with vanishing curvature are non-absolutely focusing, since . Bunimovich and Grigo introduced the following

A focusing curve is called *non-absolutely focusing of minimal length*, if every of its closed sub-arcs is absolutely focusing, but the curve itself is not absolutely focusing.

Let be a simply connected billiard table, whose boundary consists finitely many piece-wisely smooth components. The *defocusing mechanism* applies to absolutely focusing components.

Although compositions of twist maps, and iterates of a twist map, are in general no longer twist maps, Donnay observed that, then the iterates of the billiard map will still be twist maps (when restricting to absolutely focusing boundary components).

5. Reuleaux triangle: take an equilateral triangle with vertices A, B, C. Draw the arc BC on the circle centered at A, the arc CA on the circle centered at B, and the arc AB on the circle centered at C. The resulting figure is of constant width.

More generally, we start with a triangle ABC (say is the longest edge) and put a wood stick over with length . We fix the stick at and rotate it countclockwise until it reach the point . Then we fix the stick at and start a new rotation to hit . Finally we fix the stick at and start a new rotation to hit . The resulting domain is of prescribed width . Moreover, it is easy to see that the boundary consists of six arcs of different radii and is of length . Similar construction can be carried out over any convex polygon with an odd number of sides.

Click or go to here for the following gif:

The evolute of a curve is the locus of all its centers of curvature. Equivalently, an evolute is the envelope of the normals to a curve.

A vertex of a smooth curve is a critical point of the curvature. Equivalently, a vertex is a point at which the osculating circle has the third order tangency with the curve.

This is typically a local maximum or minimum of curvature, and is the definition used by some author (For example, a round circle has constant curvature, then every point would be a vertex). According to the four-vertex theorem, every closed curve must have at least four vertices.

At a vertex of , the evolute has a stationary point, generically, a cusp. A generic cusp is semi-cubic: in appropriate local coordinates, it is given by the equation .

4. Let be the ellipse given by , or , where and . Taking derivatives, we get , where is a vector tangent to . Suppose a particle at moving inward the interior of the ellipse, say its direction . After hitting again elastically, we record the new position and direction .

Theorem 4.4 in **Geometry and Billiards** (link).

(a) Suppose the trajectory between one collision does not intersect , so is the trajectory between next collision. Moreover they are tangent to a confocal ellipse.

(b). The quantity is preserved. That is, the .

**Proof.** (a). We reflect along the trajectory, say , and connect and intersect with the trajectory at . Then stays the same and specifies the longer axis of the confocal ellipse.

(b). Note that , since is symmetric. So since is parallel to . Therefore .

Note that , and is parallel to , we see , or equivalently, . Putting together, we get that . **QED**

Once again let () be an ellipse parametrized by , the inner unit velocity parametrized by the angle with the tangent direction, . Another integral of the billiard map is , where is the eccentricity of . What is the relation with ?

Let be a point on the ellipse, be the out-normal direction at , be an inward unit vector. Then the orbit along is tangent to a caustic , where .

—————————–

Let preserving the diagonal . So and . Substitute : . Solve : and hence and is either parabolic (if ) or hyperbolic (generic case).

(Not that meaningful if all of them are not convex….)

3. Let’s consider the inscribed square in the unit circle and squeeze the two side arcs. Consider the periodic 2 orbit. We have increase (from to ), and decrease from to . In particular varies

In particular the orbit starts as a parabolic one, turns into an elliptic one and then parabolic again. Moreover there are elliptic island surrounding that orbit developing and then shrinking.

The case . In this case . Although it is a rotation, a small perturbation will make it hyperbolic. So it is reasonable to put it in the parabolic category.

Suppose we put a twist at the collision: where . Then the time-reversibility is equivalent to say that . This is equivalent to say that the graph is symmetric with respect to the line : the symmetric point of with respect to the line is .

Let and be a tangent vector. There is a smooth curve such that and . For each , consider the ray on the table and the intersection point with the ray of (may be at infinite). The limiting point is the **forward focus point** of the tangent vector and the distance is the **forward focus distance** of . In particular we let . Similarly we define the **backward focus distance** , where be the time-reverse on

Let be the slope of , and .

Donnay: If , then .

2. Note that is a generating function of : if . It is easy to see . So we might say that is the generating function for a more proper coordinate with . But sometime we won’t distinguish them.

Moreover and . So we have the total differential . Taking exterior differential, we get . In particular is invariant.

Given , consider the normal line based on and let be the length of . Let the -axis be the tangent direction at . Then locally around and around the other endpoint. Then . Suppose is a critical point of . Then . Two possible choices:

– either , then is orthogonal to at the other endpoint and hence a periodic 2 orbit;

– or (the radius of curvature). In general we can pre-exclude this case.

Let’s consider an optical wave-front. For small times, such a wave front is diffeomorphic to a sphere. After a reflection, there will be (in general) points where the front ceases to be an immersed surface, for example points where different parts of the front self-intersect. The set of all points where the immersion fails is called the caustic.

Caustics also can also be observed as the optical wave-front pass through the homogeneous or non-flat medium, since these spheres become more and more deformed as time increase until the front ceases to be an immersed surface. For example after the light bundle passes a glass of water.

1. Let be a convex domain with boundary, be the (open) phase space and be the billiard map on . A point is said to be *glancing* if its orbit gets arbitrarily close to the upper and lower boundaries of . It is clear that the existence of an invariant curve implies no glancing point. The converse is also true:

Birkhoff: Assume no glancing point/orbit of , then there exists an invariant curve.

Sketch of Proof. For any , there exists such that points from never visit . So the union is an open subset of , whose boundary should be an invariant curve.

Further analysis shows that such a curve must be the graph of a Lipshitz continuous function . In fact Birkhoff proved that, for any invariant open set homeomorphic to , its boundary is a Lipschitz graph.

Mather: If the curvature of vanishes at some point, then for every , there exists a trajectory visiting and . For example the table enclosed by .

Lazutkin introduced a special coordinate near the boundary: , where is a normalizing constant. Under this new coordinate, the map near the boundary can be represented by , as the perturbation of .

Lazutkin: Consider a smooth billiard table with strictly positive curvature. Then there are invariant curves of positive volume near the boundary . In other words, there are caustics of positive volume near the boundary .

Here smoothness is determined by KAM theorem. Douady showed that is sufficient.

Mather: If there is a point of zero curvature on the boundary a convex table, then there is no caustics passing that point.

Proof. Suppose and pick a point . According to the formula , we see . So the backward and forward focusing lie on the two sides of the boundary.

Andrea Hubacher: a discontinuity in the curvature of does not allow caustics near the boundary. For example, tables obtained by the string construction around a triangle.

Knill noted that Hubacher obtained this result when she was an undergraduate student at ETH.

Proof. Suppose is the jumping point with radii and there are caustics approaching the boundary of phase space, say . On each caustics we can find a trajectory orthogonal to the normal line of at , say . Then he proved that , where .

Quantitative version: estimate the size of the region free of caustics near the boundary. In particular for a typical table by the string construction of a polygon, the free of caustics region is the annulus between table and the polygon.

Now consider a trajectory starting at , say , following the same caustics. According to Mather, these two trajectories can’t cross. To avoid backward crossing, we need . To avoid forward crossing, we need . Recall that . So we can’t avoid both crossings: the trajectories of , will cross and can’t be on some caustics. So the caustics, (if exist) will detour when coming near the jumping point and accumulate to a bump over , which violates the twisting+area-preserving property.

Open questions:

1. Are periodic orbits dense in the annulus for a smooth Birkhoff billiard? Are the set of periodic orbits not dense?

2. Does there exist a smooth convex billiards with positive Lyapunov exponents on a set of positive measure?

3. Is there a Birkhoff billiard with a caustic which is a fractal: a set with Hausdorff dimension between 1 and 2?

For example, the Bunimovich stadium are hyperbolic and ergodic whose underline table is .

Birkhoff-Poritski conjecture: Let be a convex billiard with boundary. If the billiard dynamics on is integrable, then is an ellipse.

This is a bundle of many problems, because it depends on the notion of integrability.

A special case has been treated by Misha Bialy, Convex billiards and a theorem by E. Hopf, *Math. Z*, 124 (1): 147–154, 1993:

Theorem: If is continuously foliated by invariant curves which are not null-homotopic, then is a round disk.

Recently a conditional version has been proved by Kaloshin and Sorrentino: On conjugacy of convex billiards

Theorem. If an integrable billiard is conjugate to an ellipse (resp. a circle) in a neighborhood of the boundary, then it is an ellipse (resp. a circle).

Guillemin problem: if the dynamics of two Birkhoff billiards are topologically conjugate, then their tables are similar.

Consider the trajectory along the minor axis of the ellipse (). Then

1) is parabolic (not Lyapunov stable),

2) is elliptic for ,

3) is parabolic (but Lyapunov stable),

4) goes back to elliptic for .

Now let’s choose as the parameter . We can paste two different half-ellipses: say the upper/lower with . Then that orbit will be hyperbolic if , which is true for all small and close to 1. Interesting domain is around the point . In particular hyperbolicity holds for all with .