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
Famous quotes containing the words definition, iterated and/or functions:
“I’m beginning to think that the proper definition of “Man” is “an animal that writes letters.””
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)
“The customary cry,
‘Come buy, come buy,’
With its iterated jingle
Of sugar-bated words:”
—Christina Georgina Rossetti (1830–1894)
“If photography is allowed to stand in for art in some of its functions it will soon supplant or corrupt it completely thanks to the natural support it will find in the stupidity of the multitude. It must return to its real task, which is to be the servant of the sciences and the arts, but the very humble servant, like printing and shorthand which have neither created nor supplanted literature.”
—Charles Baudelaire (1821–1867)