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:
“Its a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was mine.”
—Jane Adams (20th century)
“The customary cry,
Come buy, come buy,
With its iterated jingle
Of sugar-bated words:”
—Christina Georgina Rossetti (18301894)
“Nobody is so constituted as to be able to live everywhere and anywhere; and he who has great duties to perform, which lay claim to all his strength, has, in this respect, a very limited choice. The influence of climate upon the bodily functions ... extends so far, that a blunder in the choice of locality and climate is able not only to alienate a man from his actual duty, but also to withhold it from him altogether, so that he never even comes face to face with it.”
—Friedrich Nietzsche (18441900)