Volume 105, Number February 1998 Yueh-Gin Gung and Dr Charles Y Hu Award for Distinguished Service 105 The Use of Tagged Partitions in Elementary Real Analysis 107 The Dynamics of a Family of One-Dimensional Maps 118 Hunter S Snevily Douglas B West The Bricklayer Problem and the Strong Cycle Lemma 131 Craig M Johnson A Computer Search for Free Actions on Surfaces 144 Division Algebras-Beyond the Quaternions 154 Graphical Discovery of a New Identity for Jacobi Polynomials 163 Philippe Revoy The Generalized Level of a Non Prime Finite Field Is Two 167 Wolfgang KOhn Zuzana KOhn Cutting High-Dimensional Cakes 168 Kempe Revisited 170 Linda R Sons Russell A Gordon Susan Bassein John C McConnell NOTES Brian Gerard Lawrence Roberts UNSOLVED PROBLEMS Joan Hutchinson Stan Wagon PROBLEMS AND SOLUTIONS REVIEWS Gerald L Alexanderson Jean Pedersen William Goldman 175 II Invertible • Polyhedron Models 186 Distributed by Snyder Engineering Topology and Geometry 192 By Glen E Bredon TELEGRAPHIC REVIEWS AN OFFICIAL PUBLICATION OFTHE MATHEMATICAL ASSOCIATION OF AMERICA 195 NOTICE TO AUTHORS The MONTHLY publishes articles, as well as notes and other features, about mathematics and the profession Its readers span a broad spectrum of mathematical Interests, and include professional mathematicians as well as students of mathematics at all collegiate levels Authors are invited to submit articles and hotes that bring interesting mathematical ideas to a wide audience of MONTHLY readers The MONTHLY'S readers expect a high standard of exposition; 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Alice's first postgraduate position was at Connecticut College followed by one at The lohns Hopkins Applied Physics Laboratory She then held positions at the University of Michigan, Douglass College, Swarthmore College, Drexel Institute of Technology, and the University of Connecticut before 1998] AWARD FOR DISTINGUISHED SERVICE TO ALICE TURNER SCHAFER 105 returning to Connecticut College where she advanced to full Professor Moving to Wellesley College (by now Richard was at M.I.T), she soon became department head and the Helen Day Gould Professor of Mathematics, retiring in 1980 Indefatigable, Professor Schafer continued teaching, at Simmon's College and in the management program at Radcliffe College Seminars Upon Richard's retirement from M.I.T., they moved to Arlington, VA, where Alice became Professor of Mathematics at Marymount University, retiring once again in 1996 While living in the Boston area, Professor Schafer joined with then-graduate student Linda Rothschild and Bhama Srinivasan to organize a group of women mathematicians and students who met every few weeks to discuss common problems and goals The group anticipated both the A WM and a similar organization in Europe At the Atlantic City mathematics meetings in 1971, Mary Gray led a women's caucus of the Mathematics Action Group in organizing the A WM Alice Schafer served as the second president and under her guidance the Association was incorporated, secured financial footing, and established an office at Wellesley College Professor Schaefer prepared A WM to become a full member of the Conference Board of the Mathematical Sciences, and she and Mary Gray attained international recognition for A WM through its sponsorship of programs at the International Congress of Mathematicians at Vancouver Essential to the high regard in which A WM is now held by men and women are the excellent mathematical invited talks at its sessions, a feature begun by Schafer Even after her presidency, Alice Schafer has continued for two decades to give dedicated service and guidance to A WM Her successors in the presidency rely on her wisdom and good counsel In recognition of Professor Schafer's contributions, A WM now awards an annual prize in her honor for excellence in mathematics by undergraduate women Throughout her career, Professor Schafer sought to eliminate barriers to women in mathematics and to promote human rights for all mathematicians She directed the Wellesley Mathematics Project (continued jointly with Wesleyan University) aimed at reducing fear of mathematics for women She helped to prepare lists of women who were eligible for grants and fellowships, including invited lectureships She chaired the AMS Committee on Postdoctoral Fellowships and the Committee on Human Rights, and served on Committee Wand the National Council for the American Association of University Professors She has chaired the mathematics section of the American Association for the Advancement of Science Professor Schafer has served on the CBMS Committee on Women in the Mathematical Sciences for six years and has worked for many years for the MAA Women and Mathematics Program Three times in recent years, through the People-to-People program, she led delegations to China-one connecting women research mathematicians, one concerning mathematics education, and one concerning women's issues in mathematics and science Professor Schafer is known for her love of people, her boundless energy, and her fierce determination for a just cause Her lifetime achievements and her pioneering efforts to secure opportunities for all mathematicians make her a most worthy recipient of the Yueh-Gin Gung and Dr Charles Y Hu award for Distinguished Service to Mathematics Northem Illinois University, DeKalb, IL 60115-2854 sons@math.niu.edu 106 AWARD FOR DISTINGUISHED SERVICE TO ALICE TURNER SCHAFER [February The Use of Tagged Partitions in Elementary Real Analysis Russell A Gordon The purpose of this paper is to present alternate proofs of several well known results in elementary real analysis An alternate proof of a theorem provides a new way of looking at the theorem and this fresh perspective is often enough to justify the new approach However, a new proof of an old result that is conceptually easier and points the way to generalizations of the result has obvious benefits This is the case, in my opinion, for several of the proofs presented in this paper The results to be considered here all depend on the Completeness Axiom; every nonempty bounded set of real numbers has a supremum Throughout this paper, the universe is the set of real numbers, denoted by R Several useful statements that are equivalent to the Completeness Axiom are given in the following list: Every Cauchy sequence converges Every bounded monotone sequence converges Every bounded sequence contains a convergent subsequence The intersection of a nested sequence of closed and bounded intervals is non empty One of these equivalent statements provides the theoretical basis for results such as the Intermediate Value Theorem, the Extreme Value Theorem, and the integrability of continuous functions All of the proofs in this paper use a consequence of the Completeness Axiom that involves tagged partitions of an interval The motivation for this concept can be found in the theory of the Riemann integral Although tagged partitions usually appear only in the context of Riemann sums, we will show that tagged partitions can be used successfully to prove results about differentiable functions and continuous functions as well In other words, the method of tagged partitions is quite versatile For the reader who chooses to skim this article as opposed to reading it fully, I would like to highlight the proofs of Theorems 3, 10, and 14 The proof of Theorem is a good illustration of this new approach while the proofs of Theorems 10 and 14 are simpler than the standard proofs found in current textbooks We begin with the definition of 8-fine tagged partitions This concept has its origins in the theory of the Henstock integral A thorough treatment of the Henstock integral can be found in [2] Definition A partition of an interval [a, b] is a finite collection of non-overlapping closed intervals whose union is [a, b] A tagged partition of [a, b] is a partition of [a, b] with one point, referred to as the tag, chosen from each interval comprising the partition A tagged partition of [a, b] will be denoted by {(C i , [Xi-I' xJ): :::; i :::; n}, where a =xo xJ): ~ i ~ n} is a 8-fine tagged partition of [a, b] and let So = {i: ci E G} Since the intervals in the partition are non-overlapping, 00 E E [Uk)' (Xi - Xi-I) ~ k~l iES G where [Uk) denotes the length of the interval I k In addition, if [Xi-I' xJ n =1= 0, then c i E H That is, any tagged interval that intersects H has a tag that belongs to H and the sum of the lengths of the intervals that not intersect H is governed by the sequence {Ik} Suppose that F: [a, b] ~ R is differentiable at each point of [a, b] and let E > O For each x E [a, b], there exists 8(x) > such that H IF(t) - F(x) - F'(x)(t - x)1 ~ Elt - xl for all t E [a, b] that satisfy It - xl < 8(x) If {(Ci,[X i - I ' Xi)): ~ i ~ n} is a 8-fine tagged partition of [a, b], then (omitting some algebraic details) li~ P'(cJ(xi - xi-d - (F(b) - F(a)) =Ii~ (F'(cJ(Xi -Xi-I) - I (F(xJ -F(Xi-d))1 n ~ E IF'(cJ(x i - Xi-I) - (F(xJ - F(xi-d)1 i~l n ~ E E(Xi - xi-d i~l =E(b-a) 108 mE USE OF TAGGED PARTITIONS [February In other words, every 8-fine tagged partition of [a, b] generates a Riemann sum of F' that is close to F(b) - F(a) This represents a proof that, in some sense, every derivative is integrable and this observation is the motivation for the development of the Henstock integral The interested reader should consult [1] for an elementary discussion of the generality of this integral When working with the Riemann integral, one normally thinks of the intervals as being chosen first (each interval with length less than a prescribed constant 8) then a tag is picked for each interval There is no question as to the existence of tagged partitions in this case The positive function essentially reverses this process The tags must be chosen first; then intervals of the "right size" are chosen for each tag For an arbitrary positive function 8, the existence of 8-fine tagged partitions is no longer obvious If the infimum of the set {8(x): x E [a, b]} is positive, then it is clear that 8-fine tagged partitions of [a, b] exist-this is essentially the constant case once again If the infimum is (as is the case in Examples and 2), then a proof of the existence of 8-fine tagged partitions is required This is the content of the following theorem Theorem If is a positive function defined on the interval [a, b], then there exists a 8-fine tagged partition of [a, b] Proof" Let E be the set of all points x E (a, b] for which there exists a 8-fine tagged partition of [a, x] The set E is not empty since it contains the interval (a, a + 8(a))-the one element set {(a, [a, x])} is a 8-fine tagged partition of [a, x] for each x E (a, a + 8(a)) Let z = sup E and note that z E [a, b] To complete the proof, it is sufficient to prove that z belongs to E and that z = b E or there is a point u E E such that z - 8(z) < !Jli be a 8-fine tagged partition of [a, u] and let !Jli =!Jli U {(z, [u, z])} Then !Jli is a 8-fine tagged partition of [a, z] and this shows that z E E Now suppose that z < b Let v be a point in [a, b] such that z < v < z + 8(z) and let !Jli =!Jli1 U {(z, [z, v])} Then !Jli2 is a 8-fine tagged partition of [a, v] and it follows that vEE, a contradiction to the fact that z is an upper bound of the set E We conclude that z = b • u Since z = sup E, either z E < z In the latter case, let This proof of the existence of 8-fine tagged partitions makes direct use of the Completeness Axiom One may also prove this result using the Nested Intervals Theorem (statement in the introduction); the details are left to the reader When requested to give a proof of this result, students often try a direct approach; the actual construction of a 8-fine tagged partition This is not difficult if the number of points where the function "goes to 0" is finite Such attempts by students offer good opportunities to discuss the full generality of functions and sets The similarities between the proof of the existence of 8-fine tagged partitions of [a, b] and the proof (at least one of the standard proofs) that the interval [a, b] is a compact set are evident This is no accident-the two statements are actually equivalent However, compact sets are a difficult concept for many students since the typical student finds open covers, finite subcovers, and manipulations with large collections of sets rather abstract A positive function seems easier to visualize and the end result, a tagged partition, is easy to grasp: start with a piece of string, cut it into pieces of various lengths, and mark a point on each piece In addition, the definition of a 8-fine tagged partition seems a little more motivated than the open cover definition of a compact set For the record, I am not 1998] THE USE OF TAGGED PARTITIONS 109 advocating the elimination of the concept of compact sets; I just feel that this concept should not appear early in a first course in real analysis Tagged partitions can be used to prove the standard results on continuous functions that involve the Completeness Axiom such as the Intermediate Value Theorem, the Extreme Value Theorem, and the uniform continuity theorem The usual proofs of these results use properties of st?quences and are not difficult The proofs using 8-fine tagged partitions are not any easier, but they illustrate another way to think about these theorems In this method of proof for the Intermediate Value Theorem, the existence of the positive function is a simple consequence of the definition of a continuous function However, unlike the proof using the Nested Intervals Theorem, the following proof does not yield a method for finding the point c Theorem Suppose that f: [a, b] ~ R is continuous on [a, b] If L is a number between fCa) and fCb), then there exists a point c E (a, b) such that fCc) = L Proof' Suppose that f(a) < L < f(b); the proof for f(b) < L < fCa) is similar Assume that fCx) =1= L for all x E [a, b] Since f is continuous at each point x of [a, b], if fCx) < L, there satisfy It - xl < if f(x) > L, there satisfy It - xl < exists 8(x) > such that fCt) < L for all t 8(x); exists 8(x) > such that f(t) > L for all t 8(x) E [a, b] that E [a, b] that This defines a positive function on [a,b] Let {(Ci,[xi_px;l): ~ i ~ n} be a 8-fine tagged partition of [a, b] Note that for each index i either f(x) < L for all x E [Xi-I' x;l or f(x) > L for all x E [Xi-I' xJ Since f(x o) = f(a) < L, we find that f(x) < L for all x E [x o, Xl] Since f(x l ) < L, we find that f(x) < L for all x E [Xl' x ] After a finite number of similar steps, we find that f(b) = f(x n ) < L, a contradiction Hence, there exists a point c E (a, b) such that fCc) = L • We next prove that a continuous function defined on [a, b] is bounded on [a, b] The proof of this result using subsequences is an indirect proof, but with 8-fine tagged partitions, a direct proof is possible Theorem Iff: [a, b] ~ R is continuous on [a, b], then f is bounded on [a, b] Proof' Since f is continuous on [a, b], for each X E [a, b] there exists a positive number 8(x) such that If(t) - f(x)1 < for all t E [a, b] that satisfy It - xl < 8Cx) This defines a positive function on [a, b] Let {Cc i , [Xi-I' xJ): ~ i ~ n} be a 8-fine tagged partition of [a, b] and let M = max{ If(c) I: ~ i ~ n} Given a point X E [a, b], there is an index j such that X E [x j _ P x) and thus If(x)1 ~ If(x) - f(cj)1 + If(cj)1 < + M This shows that the function f is bounded by + M • The proof of the preceding result reveals that the continuity hypothesis is not all that crucial The continuity of f at the point x is only used to obtain a local bound for the function f A function f is locally bounded at a point x if there exist positive numbers M and such that If(t) I ~ M for all t that satisfy It - xl < A slight modification in the proof of Theorem yields the following stronger result 110 THE USE OF TAGGED PARTITIONS [February Theorem If f: [a, b] ~ R is locally bounded at each point of [a, b], then f is I bounded on [a, b] Proof' Since f is locally bounded at each point of [a, b], for each x E [a, b] there exist positive numbers M(x) and 8(x) such that If(t) I ::;; M(x) for all t E [a, b] that satisfy It - xl < 8(x) This defines a positive function on [a, b] Let {(c i , [X i - i , Xi)): 1.::;; i ::;; n} be a 8-fine tagged partition of [a, b] and let M = max{M(c): ::;; i ::;; n} Given a point x E [a, b], there is an index j such that x E [x j _ i , x) and thus If(x) I ::;; M(c j ) ::;; M This shows that the function f is bounded by M • Corollary Iff: [a, b] ~ R has one-sided limits at each point of [a, b], then f is bounded on [a, b] Proof' It is a routine exercise to prove that a function with one-sided limits at a point is locally bounded at that point • The Extreme Value Theorem states that a continuous function defined on a closed interval [a, b] assumes its maximum and minimum values Once it has been established that such a function is bounded on [a, b] (Theorem 3), it is necessary to find points c, d E [a, b] such that f(d ::;; f(x) ::;; f(d)for all x E [a, b] One way to proceed is to let M = sup{f(x): x E [a, b]}, assume that f(x) < M for all x E [a, b], and define a continuous function g on [a, b] by g(x) = 1/(M - f(x)) The fact that g is then bounded on [a, b] leads to a contradiction Here is a proof that makes direct use of 8-fine tagged partitions Theorem Iff: [a, b] ~ R is continuous on [a, b], then there exist points c, d E [a, b] such that f(c) ::;; f(x) ::;; f(d) for all x E [a, b] Proof' We prove that there exists a point d E [a, b] such that f(x) ::;; f(d) for all x E [a, b]; the proof of the existence of a point c is quite similar (or one can consider the function -f) Let M = sup{f(x): x E [a, b]} and suppose that f(x) < M for all x E [a, b] Since f is continuous on [a, b], for each x E [a, b] there exist positive numbers 8(x) and a(x) such that f(t) < M - a(x) for all t E [a, b] that satisfy It - xl < 8(x) (For example, one could let a(x) = (M - f(x)) /2.) This defines a positive function on [a, b] Let {(c i , [X i - i , x;l): ::;; i ::;; n} be a 8-fine tagged partition of [a, b], let a = min{ a(c): ::;; i ::;; n}, and note that a is a positive number Fix x E [a, b] and choose an index j such that x E [xj-l> x j ] It follows that f(x) O Since D has measure zero, there exists a sequence Uk} of open intervals such that D ~ U k=l Ik and x E [a, b] such that 1998] THE USE OF TAGGED PARTITIONS 113 < elM Define a positive function on [a, b] as follows: if x$ D use the continuity of f at x to choose 8(x) > so that If(t) - f(x)1 < e/2 for all t E [a, b] that satisfy It - xl < 8(x); if xED choose 8(x) > so that (x - 8(x), x + 8(x» ~ Ik for some index k r.k~ll(Ik) Let {(C i , [X i - 1, xJ): ~ i ~ n} So = {i: Since the intervals be a 8-fine tagged partition of [a, b] and define $ Ci [X i - ' Xi] D} and SD = {i: Ci ED} are non-overlapping, we find that n E w(f, [X i - , x;l)( Xi - xi-d i~l ~ E e( Xi - Xi-l) ~ e(b - a) + 2M + E 2M( Xi - Xi- 1) E IUd k~l < e(b - a + 2) Hence, the function f is Riemann integrable on [a, b] • Finally, we consider the use of 8-fine tagged partitions to prove results in which the derivative is involved One of the simplest results of this type is the fact that a function with a positive derivative on an interval is increasing on that interval This result is usually proved in calculus textbooks as an easy application of the Mean Value Theorem Tracing the roots of the Mean Value Theorem leads to the Extreme Value Theorem, so it becomes apparent that the Completeness Axiom is needed in the proof of this monotonicity result Since there are several intermediate results prior to the Mean Value Theorem (in the usual scheme), it is easy to forget that the Completeness Axiom is relevant to this result There are several advantages to the proof given here-namely, the Completeness Axiom is more apparent, continuity is not used explicitly, and few preliminary results are needed We want to prove that a function that has a positive derivative at each point of an interval is increasing on that interval As a reminder that there is indeed something to prove here, consider the function F: [ -1, 1] -7 R defined by F(x) = {X/2 + X2 sin(1/x), 0, if x =1= 0; if x = o This function has a positive derivative at 0, but it is not increasing on any open interval that contains O This indicates that a proof of some sort is needed for the result under discussion The purpose of the mono tonicity result is to extract global information (F is increasing on an interval) from local information (F ' is positive at each point) Theorem 11 Suppose that F: [a, b] -7 R is differentiable at each point of [a, b] (appropriate one-sided limits are assumed at a and b) If F'(x) > for each x E [a, b], then F is increasing on [a, b] 114 THE USE OF TAGGED PARTITIONS [February Proof' For each x E [a, b] use the fact that F'(x) > to choose 8(x) > so that F(t) - F(x) ->0 t-x for all t E [a, b] that satisfy < It - xl < 8(x) This defines a positive function on [a, b] Suppose that a :::;; u < v :::;; b and let {(Ci' [Xi-I> xJ): :::;; i :::;; n} be a 8-fine tagged partition of [u, v] For each index i, we find that F(Xi-l) :::;; F(c) :::;; F(x) and at least one of these inequalities is strict It follows that n F(v) - F(u) = E (F(x;) - F(Xi-d) > i=1 which is equivalent to F(u) < F(v) Therefore, the function F is increasing on [a, b] • The statement of Theorem 11 is not quite as general as that found in most calculus books Using a simple continuity argument, one can use the preceding result to prove the following result Theorem 12 Suppose that F: [a, b] ~ R is continuous on [a, b] and differentiable at each point of (a, b) If F'(x) > for each x E (a, b), then F is increasing on [a, b] Variations of the argument found in the proof of Theorem 11 can be used to prove the following facts: a If F'(x) ~ for each x b If F'(X) = for each x E E [a, b], then F is nondecreasing on [a, b] [a, b], then F is constant on [a, b] However, it is also possible to use Theorem 11 to prove each of these facts The details are left for the interested reader In addition, the hypotheses of Theorem 11 can be weakened in several ways Some of these versions of the monotonicity result involve upper and lower derivates, but we will be content to prove a simpler version This version illustrates how 8-fine tagged partitions can deal with an exceptional set that is countable A property is said to hold nearly everywhere if the set of points where it fails to hold is countable Theorem 13 Suppose that F is continuous on [a, b] If F is differentiable nearly everywhere on [a, b] and if pi > nearly everywhere on [a, b], then F is nondecreasing on [a,b] Proof: Let D be the set of all points x E [a, b] such that either F I (x) does not exist or F'(x) :::;; and express D as a sequence {d k : k E Z+} Let E> O For each x E [a, b] \D; use the fact that F'(x) > to choose 8(x) > so that F(t) - F(x) ->0 t-x for all t E [a, b] that satisfy < It - xl < 8(x) If x = d k , use the continuity of F at x to choose 8(x) > so that IF(t) - F(x)1 < E/2 k for all t E [a, b] that satisfy It - xl < 8(x) This defines a positive function on [a, b] Suppose that a :::;; u < v :::;; b and let {(c i , [Xi-I> xJ): :::;; i :::;; n} be a 8-fine tagged partition of [u, v] By combining intervals if necessary, we may assume that each tag occurs only once 1998] THE USE OF TAGGED PARTITIONS 115 Let So = {i: Ci $ D} and SD = {i: ci ED} Note that F(x) - F(X i_ l ) > for each i E So and that F(x) - F(X i- l ) > -2(E/2 k ) for some unique k for each i E SD It follows that n F(v) - F(u) = E (F(x;) - F(X i- I )) i~l = E (F(x;) - F(Xi_d) + E (F(x;) - F(xi-d) = -2E Since E > was arbitrary, we find that F(v) nondecreasing on [a, b] ~ F(u) Therefore, the function F is • Our final result is the fact that an absolutely continuous singular function is constant This result, which is outside the realm of elementary real analysis, is important in Lebesgue integration theory In most current textbooks (see [3] for instance), the proof of this result uses the Vitali Covering Lemma This lemma is familiar to students at this level since it is used in the proof that monotone functions are differentiable almost everywhere Since the concepts involved in the Vitali Covering Lemma are difficult for many students, the following proof may be easier to understand Theorem 14 Suppose that F is absolutely continuous on [a, b] If F' = almost everywhere on [a, b], then F is constant on [a, b] Proof: Let E be the set of all points x E [a, b] for which either F'(x) does not exist or F'(x) =1= O By hypothesis, the set E has measure zero Let E > and choose a positive number YJ such that L:i~IIF(t) - F(s)1 < E whenever Us i, tJ: ~ i ~ n} is a finite collection of non-overlapping intervals in [a, b] that satisfy L:i~llti - sil < YJ Since E has measure zero, there exists a sequence Uk} of open intervals such that E ~ U~~IIk and L:~~II(Ik) < YJ Define a positive function l5 on [a, b] as follows: if x $ E use the fact that F'(x) = to choose l5(x) > so that IF(t) - F(x)1 ~ Elt - xl for all t E [a, b] that satisfy It - xl < l5(x); if x E E choose l5(x) > so that (x - l5(x), x + l5(x)) ~ Ik for some index k Let {(C i , [Xi-I' Xi]): ~ i ~ n} be a l5-fine tagged partition of [a, b] and define So = {i: c i $ E} and SE = {i: ci Note that IF(x) - F(Xi-I)1 ~ E(X i - Xi-I) for each i E iES E 116 (Xi - xi-d ~ E l(Ik) E < E} E So and that YJ k~l THE USE OF TAGGED PARTITIONS [February It follows that IF(b) -F(a)1 =liE(F(X;) -F(Xi_1))1 ~L IF(x;) - F(Xi_1)1 ~ L E(Xi -xi-d + + L IF(x;) - F(Xi_1)1 E iESo ~E(b-a+l) Since E > was arbitrary, we find that F(b) = F(a) Similarly, it can be shown that F(x) = F(a) for all x E (a, b) This completes the proof • My aim in this paper has been to demonstrate the versatility of 8-fine tagged partitions and their use outside the context of integration theory Although I not anticipate the use of 8-fine tagged partitions to transform the teaching of real analysis, I hope this discussion provides new insight into old results By introducing 8-fine tagged partitions early, the transition to integration theory and the abstract notion of a compact set can be made easier In addition, some of the ideas here would make good "research" questions for advanced undergraduates REFERENCES R G Bartle, Return to the Riemann Integral, Amer Math Monthly 103 (1996), 625-632 R A Gordon, The integrals of Lebesgue, Denjoy, Perron, and Henstock, Graduate Studies in Mathematics, Vol 4, American Mathematical Society, Providence, RI, 1994 H L Royden, Real analysis, 3rd ed., Macmillan, New York, 1988 RUSSELL A GORDON received a BA from Blackburn College, an MS from Colorado State University, and a PhD from the University of Illinois His dissertation, written under the influence of Jerry Uhl, mutated into a graduate textbook on nonabsolute integration He has also written a textbook on elementary real analysis, which explains his current interest/preoccupation with this subject Beyond academia, his three sons keep him busy, especially an active year old A weekly date with his wife Brenda is a welcome change of pace from a very busy life He also enjoys running, playing basketball, and hiking in the mountains Whitman College, Walla Walla, WA 99362 gordon@whitman.edu 1998] TIlE USE OF TAGGED PARTITIONS 117 The Dynamics of a Family of One-Dimensional Maps Susan Bassein The purpose of this paper is to classify the dynamics of the following two-parameter family of very simple maps from the unit interval to itself: for given < a < b :;:; 1, let [ be the map from [0, 1] to [0, 1] whose graph consists of two and straight line segments extending from (0, b) to (a, 1) to (1,0), as illustrated in Figure ° °:;:; y b ~ ~ ~ -x a Figure The graph of I(x) with a = 0.4 and b = 0.3 Algebraically, I - b [(x) = { b+ x ifO:;:;x such that rex) = x The smallest such n is called the prime period of x and the orbit of x is called an ° n-cycle Throughout this paper, I and J represent closed intervals larger than a single point The length of an interval I is deno~ed by III The images of a set Sunder repeated mapping by I are denoted I(S), 12 (S), and so on Because our maps are continuous, the images of I are also closed intervals A map I from an 'infinite subset S of [0, 1] to itself is chaotic if: (1) it is topologically transitive: if U and V are open intervals, each of which contains a point in S, there is an n > such that r(U Ii S) and V Ii S ° have a point in common and (2) periodic points are dense in S: if U is an open interval that contains a point of S, then it contains a periodic point in S Although the traditional definition of chaos [5, p 50] includes the criterion of sensitive dependence, [3] shows that sensitive dependence follows from (1) and (2) Further, if S is all of [0, 1], then [8] shows that even (2) is a consequence of (1) Intuitively, (1) means that even if the orbit of a particular point in S doesn't "wander" throughout S, the set of orbits of its neighbors in S does Section provides some insight into the significance of (2) We can trace the single application of the map to a point x by drawing line segments from (x, x) to (x,/(x» to (j(x),/(x», as illustrated in Figure A portion of the orbit of x can be drawn by repeating this process to move from (j(x), I(x» to (j(x), 12(X» to (j2(X), j2(x», and so on Note that the map has a fixed point at x = 1/(2 - a), where the graph crosses the line y = x, which means 1998] DYNAMICS OF A FAMILY OF ONE-DIMENSIONAL MAPS 119 y ~ _ _ _ _ _ _ _ _ _ _ _ _ _ _- L_ _ _ _ _ _ _ _ ~ X c Figure The fixed point c and a portion of the orbit of x = 0.15 that /(1/(2 - a» = 1/(2 - a) For notational convenience, we denote this fixed point by c Because the absolute value of the slope of the graph is greater than at the fixed point, the fixed point is repelling, which means that points near the fixed point are mapped further away from the fixed point In particular, if I c [a, 1], then 1/(1)1 > III because 1/(1)1/111 = Islopel = 1/(1 - a) > To classify the dynamics of the maps in the family, we interpret the a and b of each map as the coordinates of a point in a unit square in (a, b)-space and decompose that square into the regions shown in Figure Each region corresponds to a subfamily of maps that have similar dynamics and can be analyzed by means of the same general strategy b L- ~ a Figure Regions of different dynamics in (a, b)-space 120 DYNAMICS OF A FAMILY OF ONE-DIMENSIONAL MAPS [February NON-CHAOTIC DYNAMICS WITH AND WITHOUT AN ATTRACTING CYCLE The simplest region to analyze is the one defined by b > - a + a , which is shaded in the diagram on the left in Figure The significance of this inequality results from - a + a > a and f(l - a + a 2) = a, which, as shown by the dashed lines in the graph in the middle of Figure 4, implies that f([O, aD c [b,l] £; [1 - a + a 2, 1] so that f2([O, aD = [0, feb)] £; [0, a] y b y a a b "'_a l - -_ _ _ "' -' x " -'- x Figure The region with b > - a + a , a typical graph, and an attracted orbit Because f is linear on both [0, a] and [a, 1], f2 is linear on [0, a] It follows that if Ie [0, a], then If2(I)1 = (f2(0)/a) III < III In particular, there is a point x E [0, a] for which f2(X) = x and for every y E [0, a], y =F x we have If2(y) - f2(X) 1= (J2(0)/a)ly - xl < Iy - xl· This inequality implies that, as illustrated on the right on Figure 4, the 2-cycle {x, f(x)} is attracting Further, the basin of attraction is all of [0, 1] except the fixed point, which means the orbit of every point except c approaches the 2-cycle, because, as one can see from the graph, the orbit of every point except c eventually enters the interval [0, a] In particular, f is not chaotic on any set The condition b = - a + a defines the lower boundary of the region we are considering When that condition holds, the 2-cycle is no longer attracting because f2([0, aD = [0, a]; in fact, f2(X) = a - x for x E [0, a] and every point in [0, a] except the one in the 2-cycle has period THE REGIONS FOR WHICH b < c Before we perform detailed analyses on the regions in (a, b)-space for which b < c, which are shaded in the diagram on the left side of Figure 5, it is useful to determine some general consequences of y b a b ~~ _ ~ ~x c Figure The regions for which b < c and a typical graph 1998] DYNAMICS OF A FAMILY OF ONE-DIMENSIONAL MAPS 121 this condition on b In particular, in each of those regions there is some set upon which f acts chaotically The presence of chaos for these regions can be deduced from more general results about maps with non-degenerate homoclinic points [5, §1.16]; the simplified proof here is tailored to the simple form of the maps being considered The horizontal dashed line in the graph on the right hand side of Figure shows a geometric meaning of b < c that plays a central role in creating chaotic behavior: the fixed point is contained in the interior of f([O, aD The point d will be used in Proposition and Section Proposition Assume b < c If c E I, then there is an n such that r(I) = [0, 1] ° Proof' First we claim that there is an m ~ such that E fm(I), so that [c,l] C fmc!) If E I, then we are done If a E I, then E f(J) and we are done Otherwise, I C (a, 1) and If(I)1 = (1/(1 - a» III > III because the slope of the right hand side of the graph is 1/(1 - a) Therefore, the images of I expand exponentially until, for some k, either E fk(I) or a E fk(J) If a E fk(J), then E fk+l(J), which completes the proof of the claim It follows from [c, 1] E fm(I) that [0, c] cfm+l(J) and then [b, 1] C fm+Z(J) If b :s; a, then [a, 1] cfm+Z(J), so [0,1] cfm+3(I) and we are done If b > a, then write [b,l] = [b, c] U [c, 1] From [b, c] c [a, c], we obtain f([b, cD c [c, 1] and If([b, cDI = (1/(1 - a»I[b, c]l Then either a E P([b, cD or IfZ([b, cDI = (1/(1 - a»ZI[b, c]l Therefore, the images of [b, c] expand exponentially under repeated applications of fZ until they contain [a, c] Since [c,l] c P([c, ID, we • then have [a, 1] cfP(J) for some p Then [0, 1] CfP+l(!) Proposition If for every interval I c [0, 1], there is an n such that r(I) = [0, 1], then f is chaotic on [0,1] and there are no attracting periodic orbits Proof' The map f is topologically transitive because every I eventually maps to all of [0, 1] It is easy to show that periodic points are dense, without appealing to [8]: since for every I there is an n > such that r(I) = [0, 1] :::) I, the continuity of r implies that there is some x E I such that rex) = X We can also show sensitive dependence without appealing to [3]: since for every I there is an n > such that r(I) = [0,1], for every x E I there is ayE I such that Ir(x) - r(y)1 ~ t Finally, if there were an attracting periodic orbit, each of its points would be contained in an interval that was contracted toward the orbit points by repeated application of f, contrary to the hypothesis • ° ° We make use of the following proposition in Section to show that in certain cases there is chaos hiding in apparently orderly dynamics Its proof provides additional insight into the conditions defining chaos Proposition Assume b < C Define So = {x for which there is an n ° > such that rcx) = c} and let S = closure of So Then f is chaotic on S Proof' We have f(S) c S because f(So) c So and f is continuous We use Figure to trace a "backwards orbit" in So from c to show that So, and therefore S, is infinite Start with Co = c and construct cl , cz, satisfying fCc k+ l) = c k by reversing the process illustrated in Figure 2, as follows Let 122 DYNAMICS OF A FAMILY OF ONE-DIMENSIONAL MAPS [February