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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