# Footnotes

[1] As pointed out
in Hsu * et al.* [1988], the calculation of the invariant set
is generally not very sensitive to the value of as long
as 0 < < 1. We use = 1/2

[2] Two algorithms that compute
the manifolds are documented in You * et al.* [1991], one that uses
a rigorous method and one that does not. Unfortunately, the rigorous
algorithm requires information about the Jacobian of **f**, and since explicitly
obtaining the Jacobian for the billiard system is very cumbersome,
we only use the nonrigorous algorithm. On the other hand, it was shown
in You * et al.* [1991] that pictures generated from the two algorithms
are generally indistinguishable.

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