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# Are there examples of fractal caustics?

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.

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Oliver Knill
Fri Jun 12 13:34:37 CDT 1998