Limiting Behaviour
Upon iteration, one may find that there are sets that shrink and converge towards a single point. In such a case, the point that is converged to is known as an attractive fixed point. Conversely, iteration may give the appearance of points diverging away from a single point; this would be the case for an unstable fixed point. When the points of the orbit converge to one or more limits, the set of accumulation points of the orbit is known as the limit set or the ω-limit set.
The ideas of attraction and repulsion generalize similarly; one may categorize iterates into stable sets and unstable sets, according to the behaviour of small neighborhoods under iteration.
Other limiting behaviours are possible; for example, wandering points are points that move away, and never come back even close to where they started.
Read more about this topic: Iterated Function
Famous quotes containing the words limiting and/or behaviour:
“Do we honestly believe that hopeless kids growing up under the harsh new rules will turn out to be chaste, studious, responsible adults? On the contrary, by limiting welfare, job training, education and nutritious food, won’t we plant the seeds for another bumper crop of out-of-wedlock moms, deadbeat dads and worse?”
—Richard B. Stolley (20th century)
“The methodological advice to interpret in a way that optimizes agreement should not be conceived as resting on a charitable assumption about human intelligence that might turn out to be false. If we cannot find a way to interpret the utterances and other behaviour of a creature as revealing a set of beliefs largely consistent and true by our standards, we have no reason to count that creature as rational, as having beliefs, or as saying anything.”
—Donald Davidson (b. 1917)