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4 Conclusions
As we have seen, on a suitable set of configurations the global transition map
F
f
induced by a local rule of a 1D CA coincides with the shift map. In the case
of elementary rules, we have presented the behavior of such kind of subshifts
with respect to some components of chaoticity such as (positive) transitivity,
topological mixing and strong transitivity. The set of all ECA transitive rules
turns out to be the set theoretic union of all mixing and all strong transitive
rules.
In the more general context of DTDS, transitivity plays a central role in order
to establish topological chaoticity. We have renamed the notion of transitivity
given in [7] as positive transitivity and we have introduced the new notion of
full transitivity. The relations among these notions and other properties of a
DTDS can be summarized in the following results: the existence of a dense orbit
implies full transitivity and moreover, when the phases space is perfect, the
positive transitivity. Full transitivity (and thus also positive transitivity) implies
indecomposability. If the phases space has infinite cardinality, full transitivity
and regularity together imply sensitivity to initial condition. Finally we have
shown an example of a homeomorphic and full transitive DTDS on a compact
space which is not positively transitive.
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