lp Billiard
Here are some orbits of billiards. This billiard has appeared in the
Final Exam (PDF).
The first one is the circular
billiard, p=2, which is an integrable situation.
The second one is the case p=4, where we suspect the Kolmogorov
Sinai entropy to be positive. Nobody knows how to prove this.
In 1995, in a SURF undergraduate research project,
Monwhea Jeng worked with me on the problem. We proved
here is the article [PDF] the following result
|
Theorem (Jeng-Knill): except for p=1, p=infinity or p=2, the topological entropy
of the lp billiard is positive.
|
This result sits on theorems of
John Mather (who proved a theorem
on billiards with flat points = zero curvature), Andrea Hubacher (who proved a theorem on billiards
with unbounded curvature) and
Sigurd Angenent
who proved that the non-existence of some invariant curves implies positive topological entropy.
We also computed the entropy numerically.
Here is a newly made phase space picture showing a few orbits. The x-coordinate is the
place on the table (here the angular polar coordinate), the y-coordinate shows the cos
of the impact angle. We see an obvious dihedral symmetry, which is the 4 fold symmetry of
the table as well as the time reversal symmetry, where the orbits are run backwards. The big
islands in the middle are the shortest 2-periodic connections which are physical paths on
the x-axes or y axes. We see a large blue ``chaotic sea" as well as ``elliptic islands of stability".
The reason for the frustratingly hard problem to compute the ``Kolmogorov-Sinai entropy"
(mathematically of course, not just
measuring as done in the above mentioned paper. Measuring is what physisists do. Mathematicians prove
things!) is the presence of these islands. These islands are actually
dense in the phase space. The key problem is the non-commutativity of the matrix product, when
computing the Jacobean dT
n of the n-th iterate, which is a 2x2 matrix. The multiplicative
ergodic theorem of Oseledec deals with a non-commutative situation so that we can not just use an
ergodic theorem to compute the exponential growth rate of ||dT
n|| as in one dimensions
where one can write this as ||dT(T
n-1x)|| .... ||dT(x)||.
When I was young, I threw essentially every mathematics I could find at this problem
("Anosov-Woitokowski cone methods", "transport approaches", "subharmonicity",
"random Schroedinger operators" (Thouless), "isospectral Toda deformations" (of the associated
Jacobi operators), "multi-linear algebra",
"determinants" or "cocycle deformations", everything failed). It appears to need a new idea.
It might even be that the entropy is zero and that the measurements do fool us
(which I personally do not believe to be the case). I think the l
4 table has positive
Kolmogorov-Sinai entropy.
