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We have seen that the coordinate functions of
caustics can be "rough". Can the image of become "fractal"?
More precisely one can ask: do there exist convex tables T such that the
image of has Hausdorff dimension
for some ? Corresponding
questions for graphs of real functions can be difficult and often one is
only able to estimate the box-counting dimension (see the case of the
Weierstrass function in [9]).
We could try to find the distortion of
the map from the graph
of to .
Let be the table
defined by a function . If is
a Weierstrass function then the graph G of has noninteger box-counting
dimension [9].
The transformation :
maps G onto the caustic . If it were
invertible near ,
the image of would inherit the fractal
properties of G. Let's see: the Jacobian of is
Because , one has
and unfortunately,
vanishes exactly on G. While this approach failed
to give caustics with fractional box counting dimension, the calculation
shows why: the caustic of a curve of constant width is the place,
where the wave front formed
by the ray of light starting orthogonally to the table of constant width
fails to be an immersed submanifold.
See a
Movie (30 K Gif movie) of the Legendre collapse collapse in
Figure 11a.
When plotting caustics using larger and
larger frequencies in , the curves usually get more and more cusps
(see Figure 12a). From the pictures we can
expect that the box-counting dimension or
even the Hausdorff dimension can be bigger than 1 for some caustics but
we do not know whether this is the case.
Next: Caustics and billiards in
Up: On nonconvex caustics of
Previous: Nowhere differentiable caustics on
Oliver Knill
Fri Jun 12 13:34:37 CDT 1998