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# ST MATH 397

UMass

GPA 3.55

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This 2 page Class Notes was uploaded by Reuben Hudson DDS on Friday October 30, 2015. The Class Notes belongs to MATH 397 at University of Massachusetts taught by Staff in Fall. Since its upload, it has received 12 views. For similar materials see /class/232225/math-397-university-of-massachusetts in Mathematics (M) at University of Massachusetts.

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Date Created: 10/30/15

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 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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