``On voit courir après l'ombre Tant de fous qu'on n'en sait pas.''

(``One sees chasing after shadows more fools than one can count.'')

-- La Fontaine, *Fables*. (1668)

*
Finally, there is much further work that should be done both in general
high dimensional shadowing, and in the N-body shadowing in particular.
We point out some possible directions for further research.
*

There are two distinct questions addressed in this thesis. One
deals with shadowing of high-dimensional systems in general, and the
other deals with shadowing of large *N*-body systems in particular.
It is not clear that these questions are close enough to each
other for generalities to be drawn from conclusions for the *N*-body
problem. For example, the previous chapter noted that scaling the
problem by increasing the number of moving particles, while holding the
total number of particles constant, may not be the most realistic scaling
method. From the perspective of shadowing in general, it
seems that this method changes as little as possible in the system
while the dimensionality of the problem is increased, thus arriving at
a more ``pure'' result about how shadowing behaves as the
dimensionality increases. But astronomers may be more
interested in how the problem scales as the collisionality decreases
with increasing *N*; thus having *M* moving particles amongst 100*M*
fixed ones seems a more apt model for studying this question, because
the gravitational potential becomes smoother as the total number of
particles is increased (assuming the total mass is kept constant, so
each particle has mass 1/*N*).

As QT point out, these are two separate
questions, even for the *N*-body problem: even if, in general,
shadowing becomes more difficult as the number of dimensions increases,
the *N*-body problem becomes less chaotic (*i.e., * a smaller Lyapunov
exponent) as the potential becomes more smooth with increasing *N*.
The question of how these two processes interact to affect shadowing of
large *N*-body systems is still open. Perhaps, even if Conjecture 1 is
correct, the collisionality of large *N*-body systems decreases enough
with increasing *N* to offset the decreasing average shadow length with
increasing dimensionality.

In the first section of this chapter, I look at possible future work for
shadowing of *N*-body systems; in the second section, I look at future
work for shadowing in general.

Sun Dec 29 23:43:59 EST 1996

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