Class Note for MATH 397 at UMass
Class Note for MATH 397 at UMass
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This 2 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Massachusetts taught by a professor in Fall. Since its upload, it has received 26 views.
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Date Created: 02/06/15
Math 3970 10101 De nitions about attracting and repelling7corrected Throughout let 1 A 7 A be a function from a subset A of R into itself For each nonnegative integer n denote by f the nth iterate of 1 so that also 1 A 7 A Thus 1 0 is the identity function of A the rst iterate f1 f the second iterate f2 f o 1 etc Then for a point z E A the set f m n O 1 23 is the orbit of z under 1 De nition 1 An z E A is called a xed point of 1 when fz m If x is a xed point of 1 then f m z for every n O 1 2 3 and so the orbit of z under 1 is just the onepoint set De nition 2 Let p be a xed point of 1 Say that p attracts a point z E A and z is attracted to p when 1mm 1 7 p The basin of attraction of p is the set of all points z E A that are attracted to p The xed point p as well as its orbit is said to attract and to be an attractor when its basin of attraction includes A O p 7 619 l 6 for some 6 gt 0 In other words p is an attractor when all points of A that are suf ciently close to p are attracted to p De nition 3 Corrected Let p be a xed point of 1 Then p as well as its orbit is said to repel and to be a repellor when for some 6 gt O for each x E A O p7 6p 6 with z 31 p there is at least one power 71 such that f z p 7 619 6 In other words p repels when for some 6 gt O the orbit of each point z E A O p7 6p 6 other than ofp itself does not remain in p 7 619 l 6 De nition 4 A point p E A is said to be a periodic point7and its orbit is said to be a periodic orbit7if there is some integer k 2 2 for which fkp p In this case the least such k is called the prime period of p According to the preceding de nition a xed point is not considered to be periodic Some authors do so consider it In any case you could regard a xed point as a sort of degenerate case of a periodic point Suppose p is a periodic point of f with period k The also fk1p fp fk2p f2p etc Thus the entire orbit of p reduces to just the nite set 19 fp f2p fk 1p consisting of exactly k distinct points If p is a periodic point of f with period k then p is a xed point of the kth iterate fk A 7 A In this case we may consider the new De nition 5 Let p be a periodic point of f with period k Consider instead of f the function fk A e A Say that p attracts or repels when p attracts or repels respectively for fk In this situation also call the periodic orbit ofp under 1 a periodic attractor or periodic repellor respectively
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