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Of course, non-quadratic irrationals have continued fraction expansions,
too. By Lagrange's theorem, these continued fractions are aperiodic,
and hence the orbit of these initial points under the Gauss map is
aperiodic. Note that most numbers in are thus aperiodic. We
examine some beautiful examples, beginning with one due to Euler:
- e (the base of the natural logarithms) has an aperiodic continued
fraction expansion .
The elements of the orbit of this initial point are always of the form
, ,
or ,
which tend to 1, 0, and , respectively. Thus the
-limit set
of this orbit is the set , which, unlike the -
limit sets of continuous maps, is not invariant
under the Gauss
map since so G applied to this set simply gives
0. In other words, we have an
asymptotically periodic orbit which is
not asymptotic to a real orbit of the map. This cannot happen for a
discrete dynamical system with a continuous map.
- Stark [26]. If x is the positive root of , then
which has very large entries placed irregularly throughout. This
intermittency is a typical feature of a chaotic system [10].
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- Lambert, 1770--- cf [20]. The continued
fraction for is not known, in the sense that no pattern has
been identified. It begins
and some 17,000,000 elements of this continued fraction have been computed
by Gosper (see [4]). There are many open questions
about this continued fraction---for example, it is not known if the
elements of the continued fraction are bounded.
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Next: Lyapunov Exponents
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Previous: Periodic and Fixed