Definition For Iterated Functions
Let be a metric space, and let be a continuous function. The -limit set of, denoted by, is the set of cluster points of the forward orbit of the iterated function . Hence, if and only if there is a strictly increasing sequence of natural numbers such that as . Another way to express this is
where denotes the closure of set . The closure is here needed, since we have not assumed that the underlying metric space of interest to be a complete metric space. The points in the limit set are non-wandering (but may not be recurrent points). This may also be formulated as the outer limit (limsup) of a sequence of sets, such that
If is a homeomorphism (that is, a bicontinuous bijection), then the -limit set is defined in a similar fashion, but for the backward orbit; i.e. .
Both sets are -invariant, and if is compact, they are compact and nonempty.
Read more about this topic: Limit Set
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