Definitions and Notation From Discrete Dynamics
Let S be a set and let F : S → S be a map from S to itself. The iterate of F with itself n times is denoted
A point P ∈ S is periodic if F(n)(P) = P for some n > 1.
The point is preperiodic if F(k)(P) is periodic for some k ≥ 1.
The (forward) orbit of P is the set
Thus P is preperiodic if and only if its orbit OF(P) is finite.
Read more about this topic: Arithmetic Dynamics
Famous quotes containing the words definitions, discrete and/or dynamics:
“Lord Byron is an exceedingly interesting person, and as such is it not to be regretted that he is a slave to the vilest and most vulgar prejudices, and as mad as the winds?
There have been many definitions of beauty in art. What is it? Beauty is what the untrained eyes consider abominable.”
—Edmond De Goncourt (1822–1896)
“One can describe a landscape in many different words and sentences, but one would not normally cut up a picture of a landscape and rearrange it in different patterns in order to describe it in different ways. Because a photograph is not composed of discrete units strung out in a linear row of meaningful pieces, we do not understand it by looking at one element after another in a set sequence. The photograph is understood in one act of seeing; it is perceived in a gestalt.”
—Joshua Meyrowitz, U.S. educator, media critic. “The Blurring of Public and Private Behaviors,” No Sense of Place: The Impact of Electronic Media on Social Behavior, Oxford University Press (1985)
“Anytime we react to behavior in our children that we dislike in ourselves, we need to proceed with extreme caution. The dynamics of everyday family life also have a way of repeating themselves.”
—Cathy Rindner Tempelsman (20th century)