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Are there examples of fractal caustics?

We have seen that the coordinate functions of caustics tex2html_wrap_inline880 can be "rough". Can the image of tex2html_wrap_inline880 become "fractal"? More precisely one can ask: do there exist convex tables T such that the image of tex2html_wrap_inline880 has Hausdorff dimension tex2html_wrap_inline1052 for some tex2html_wrap_inline1466 ? 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 tex2html_wrap_inline1468 of tex2html_wrap_inline1030 to tex2html_wrap_inline880 . Let tex2html_wrap_inline1474 be the table defined by a function tex2html_wrap_inline1168 . If tex2html_wrap_inline1030 is a Weierstrass function then the graph G of tex2html_wrap_inline1030 has noninteger box-counting dimension [9]. The transformation tex2html_wrap_inline1486 : tex2html_wrap_inline1488 maps G onto the caustic tex2html_wrap_inline880 . If it were invertible near tex2html_wrap_inline1494 , the image of tex2html_wrap_inline880 would inherit the fractal properties of G. Let's see: the Jacobian of tex2html_wrap_inline894 is

displaymath1502

Because tex2html_wrap_inline1504 , one has tex2html_wrap_inline1506 and unfortunately, tex2html_wrap_inline1508 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.

tex2html_wrap1530 tex2html_wrap1532
tex2html_wrap1534 tex2html_wrap1536
When plotting caustics using larger and larger frequencies in tex2html_wrap_inline1030 , 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.

tex2html_wrap1538 tex2html_wrap1540
tex2html_wrap1542 tex2html_wrap1544
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