
Full Citation 
Material Information 

Title: 
A Counterexample to the bounded orbit conjecture 

Alternate Title: 
bounded orbit conjecture 

Physical Description: 
vii, 21 leaves : ill. ; 28 cm. 

Language: 
English 

Creator: 
Boyles, Stephanie Marion, 1954 

Copyright Date: 
1980 
Subjects 

Subject: 
Homeomorphisms ( lcsh ) Mathematics thesis Ph. D Dissertations, Academic  Mathematics  UF 

Genre: 
bibliography ( marcgt ) nonfiction ( marcgt ) 
Notes 

Thesis: 
ThesisUniversity of Florida. 

Bibliography: 
Bibliography: leaves 1920. 

General Note: 
Typescript. 

General Note: 
Vita. 

Statement of Responsibility: 
by Stephanie M. Boyles. 
Record Information 

Bibliographic ID: 
UF00099375 

Volume ID: 
VID00001 

Source Institution: 
University of Florida 

Holding Location: 
University of Florida 

Rights Management: 
All rights reserved by the source institution and holding location. 

Resource Identifier: 
alephbibnum  000098313 oclc  06719188 notis  AAL3758 

Downloads 

Full Text 
A COUNTEREXAMPLE TO THE BOUNDED ORBIT CONJECTURE
By
Stephanie M. Boyles
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE
UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
Copyright 1980
by
Stephanie M. Boyles
To My MotheA and FatheA
ACKNOWLEDGEMENTS
I would like to express my appreciation to several people who
have contributed in various ways toward the completion of this
dissertation.
To my mentor, Professor Gerhard X. Ritter, I extend my deepest
thanks. By suggesting that I solve such an old Mathematical problem,
his confidence in my ability has helped me both professionally and
personally. His friendship, support, and constant willingness to
listen and advise will never be forgotten. I am especially indepted
to him and Professor Serge Zarantonello for sharing with me the benefit
of their experiences when I was selecting a career in mathematics.
To my fellow graduate students in mathematics I offer a toast.
Your attitude of c3mradery and festive natures have made these last
four years most enjoyable. I also owe many thanks to the mathematics
professors at the University of Florida for their love of mathematics
and their interest in the students.
Most of all I wish to thank my parents, Marion and Charles Boyles,
for their unending encouragement, faith, and love. Were it not for
them and their great respect for education, I would never have reached
this goal. I owe them more than I can ever repay and dedicate this
work to them.
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS ................................................. iv
ABSTRACT ....................................................... vi
Chapter
I. PRELIMINARIES ........................................... 1
1. Notation and definitions ............................ 1
2. History of the Bounded Orbit Conjecture ............. 3
II. A FIXED POINT FREE HOMEOMORPHISM WITH BOUNDED ORBITS .... 6
1. Definition of h ..................................... 6
2. h is fixed point free ............................... 10
3. Every point has bounded orbit ....................... 10
4. h is a homeomorphism ................................ 11
III. CONCLUDING REMARKS ...................................... 13
1. Consequences ........................................ 13
2. Related questions ................................... 13
BIBLIOGRAPHY ....................................... ............. 19
BIOGRAPHICAL SKETCH ....................................... ..... 21
Abstract of Dissertation presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
A COUNTEREXAMPLE TO THE BOUNDED ORBIT CONJECTURE
By
Stephanie M. Boyles
June 1980
Chairman: Gerhard X. Ritter
Major Department: Mathematics
A long outstanding problem in the topology of Euclidean spaces
is the Bounded Orbit Conjecture, which states that every homeomorphism
of the plane onto itself, with the property that the orbit of every
point is bounded, must have a fixed point. It is well known that the
conjecture is true for orientation preserving homeomorphisms. We
provide a counterexample to the conjecture by constructing a fixed
point free orientation reversing homeomorphism which satisfies the
hypothesis of the conjecture.
Chapter I contains a summary of definitions and notation used in
the text, as well as a historical perspective of the Bounded Orbit
Conjecture. In Chapter II, we present the counterexample to the
Bounded Orbit Conjecture. This homeomorphism can be outlined as
follows. On the complement of the strip Ixl < 1, h is a reflection
across the yaxis. In the strip itself, h is defined on the images of
an arc A = {(x,y) : Ixl < 1 and y = 0} and extended in a piecewise
fashion to the remainder of the strip. With this construction, every
point on A has a bounded orbit even though the orbit of A itself is
unbounded. Consequences of the counterexample in Chapter II are
revealed in the first part of Chapter III; in the final section of
Chapter III, we furnish two problems in planar fixed point theory
which remain unsolved.
CHAPTER I
PRELIMINARIES
The first section of this chapter introduces notation and
definitions used in the sequel. For a more detailed approach the
reader is referred to (11). In the second section we provide a
historical perspective of the Bounded Orbit Conjecture. This
summary begins with Brouwer's original Translation Theorem and
mentions results obtained by other researchers since that time.
1. Notation and definitions. The letters p, q, v are used to denote
points in E2. Line segments and open arcs are designated by S and A,
respectively. The letter t always stands for a real number between 0
and 1; and i, j, k, m, n take only integer values. We let x and y
2
correspond to the x and y coordinates of a point in E2
A homeomorphism of the plane is a bijective continuous transforma
tion of 2dimensional Euclidean space E2. Homeomorphisms of the plane
fall into two categories, orientation preserving and orientation rever
sing. The intuitive idea is simple. A homeomorphism of the plane
preserves orientation if the image of every clockwise oriented simple
closed curve is again clockwise oriented; it reverses orientation if the
image of every clockwise oriented simple closed curve is counterclockwise
oriented. Note that both a translation and a rotation are orientation
preserving, while a reflection is an orientation reversing homeomorphism.
The rigorous definition of this concept is given in terms of homology
theory as follows. Let h be a homeomorphism from En onto itself, and
let h' denote the unique extension of h to Sn = Enu{}, defined by
h'() = and h' restricted to En equals h. The homeomorphism h' is
said to be orientation preserving or orientation reversing according
as the induced isomorphism h',: H (Sn) , Hn(Sn) is or is not the
identity. The map h is orientation preserving if h' is orientation
preserving; it is orientation reversing if h' is orientation reversing.
We note that the composition of an odd (even) number of orientation
preserving homeomorphisms is an orientation reversing (preserving)
homeomorphism.
The orbit of a point p is defined as the set of all hk (p), where k
ranges over the integers. Note that under a rotation, as well as a
reflection, the orbit of every point is bounded. Whereas, if h is a
translation, the orbit of every point is unbounded.
A set of n points in Ek is said to be in general position if no m+2
of them lie in an mplane. Let v, v2, ..., vn be n points in general
position in Ek. Denote the closed convex hull of {vl, v2, ..., vn
(i.e. the smallest convex set containing {vl, v2' ..." Vn) by
. By a linear map we shall mean a map
h :
defined by
zn tn.v. n tih(v.) where En t. < 1.
i=l i=l i=l
2. History of the Bounded Orbit Conjecture. In 1912, L. E. J. Brouwer
proved his famous translation theorem (9) which states that if h is an
orientation preserving homeomorphism of E2 onto itself having no fixed
points, then h is a translation. By a translation, Brouwer meant that
for each point p in E2, hn(p)  as n ; that is, the orbit of
every point is unbounded. Thus, if h is an orientation preserving
homeomorphism of E2 onto itself such that the orbit of some point is
bounded, then h must have a fixed point. Obviously, Brouwer's theorem
cannot be true for orientation reversing homeomorphisms. For instance,
let h be a reflection across the yaxis followed by the upward shift
f(x,y) = (x ylxl+l) in the strip determined by Ixl < 1 (figure 1).
Then every point (x,y) with IxI 1 has a bounded orbit, even though h
is fixed point free.
The question arose as to whether or not any homeomorphism of E onto
itself with the property that the orbit of every point is bounded must
have a fixed point. This eventually became known as the bounded orbit
problem, and a considerable amount of research relating to this problem
has accumulated in the literature (1), (2), (3), (4), (6), (7), (8), (10),
(12), (13), (15), (16), (17), and (18). Since the Bounded Orbit Conjecture
is true for orientation preserving homeomorphisms (1), any counterexample
to the conjecture must be orinetation reversing. However, if a homeo
morphism h is orientation reversing with bounded orbits, then h2 is
orientation preserving with bounded orbits and, according to Brouwer's
theorem (9), must have a fixed point. Thus, one considers the question:
does every homeomorphism, whose square has a fixed point, leave some
point fixed itself? In 1974, Gordon Johnson (12) answered this question
in the negative by giving an example of a fixed point free homeomorphism
of E2 onto itself with the property that each of its iterates has a fixed
point. We note that Johnson's example is not a counterexample to the
Bounded Orbit Conjecture since the orbits of points are not bounded.
In a 1975 paper Brechner and Mauldin (7) showed that if there exists
a fixed point free homeomorphism h of E2 onto itself such that the orbits
of bounded sets are bounded, then there is a compact continuum M in E2
which does not separate the plane and which is invariant unde: h. The
next year Bell (3) announced that every homeomorphism of the plane onto
itself leaving a nonseparating continuum M invariant has a fixed point in
M. Thus, any counterexample to the Bounded Orbit Conjecture must be orien
tation reversing and must contain a compact set whose orbit is unbounded.
Initially, the concept of a compact set whose orbit is unbounded, with
every point on that compact set having bounded orbit, seems contradictory.
However, by modifying an example in (14), Brechner and Mauldin (7) used
the standard stereographic projection in order to obtain a homeomorphism
of the plane in which the orbit of an arc is unbounded, even though the
orbit of every point on that arc is bounded. We note that the example
mentioned above is not fixed point free.
The objective of this dissertation is to give a counterexample to
the Bounded Orbit Conjecture; that is, to construct a fixed point free
orientation reversing homeomorphism of E2 onto itself with the property
that the orbit of every point is bounded. The counterexample constructed
in Chapter II has the property that any arc connecting a point p with its
image h(p) has unbounded orbit. To see that every counterexample to the
Bounded Orbit Conjecture must exhibit this property, let f be a
5
homeomorphism of the plane onto itself, and suppose that A is an arc
connecting a point p with its image f(p). If the orbit of A is bounded,
then the closure of the orbit of A, together with the bounded components
contained in this closure, is a nonseparating continuum which is
invariant under f. Thus, by Bell's theorem (3), if the orbit of A
is bounded, then f must leave some point fixed and cannot be a counter
example to the Bounded Orbit Conjecture.
CHAPTER II
A FIXED POINT FREE HOMEOMORPHISM WITH BOUNDED ORBITS
In this chapter we present the counterexample to the Bounded
Orbit Conjecture. The definition of the homeomorphism h, which will
serve as the counterexample, is given in section one. Section two
explains why the homeomorphism is fixed point free. In section three
we prove that the orbit of every point is bounded. We conclude this
chapter by showing that h is indeed a homeomorphism.
1. Definition of h. Let B denote the open strip {(x,y) : xl < 1}.
If p is not in B, define h(p) = h(x,y) = (x,y), where e indicates
definition. Thus, on the complement of B the map is a reflection
across the yaxis.
To describe the homeomorphism on B, we first define h on a
countable set of points which will be the vertices of convex polygons
in B. For all m > 0 and k > 1, let
+m,O = ( m/(m+l) 0 ), and
V+m,k = ( m/(m+) k 1/(m+i) ).
i=l
For all j and k > 0, define
h(v ) k v
j,k (1)k+lj,k+l
Denote by Sjk. Extend h linearly on S ,k by
Deoe
defining
h(Sj,k) = h() , for j, k 0.
Let A = {(x,y) : Ixi < 1 and y = 0}. Thus, hk(A) = U j=_ Sj,k
(figure 2). Note that the positive orbit of A forms an unbounded
sequence of arcs contained in the half of B having nonnegative ycoord
inate, and that the intersection of any two of these arcs is empty.
One may view the homeomorphism h on h (A), k > 0, as the following
composition: h = v'r's, where s is a shift of every segment Sj,k either
"left" one segment if k is odd (i.e. for all j, Sj,k Sjl,k ), or
"right" one segment if k is even (i.e. for all j, Sj,k Sj+,,k ), r is
a reflection through the yaxis, and v is an upward projection to hk+l(A).
To describe h on the regions between hk (A) and hk(A), k > 0,
consider the following two sequences of points on each vertical segment
:
1/n
v v + ( 
j,k j,k1 j,k
(n1)/n
v E v + (v
j,k j,k1 j
1/n (nl)/n
Note that {vj,k n>3 and {vj,k
and vj,k, respectively. Let
v
j,k1
v
,k j
)/n, where n > 3, and
)(nl)/n, where n > 2.
,k1
n>2 are sequences converging to vj,k_
t t t
Sj,k = and
t
Akt Uj= Sj,k'
where t = 1/n for n > 3, or t = (nl)/n for n > 2. Thus, for every
positive integer k {Ak,i/n}n>3 and {Ak,(nl)/n}n>2 are disjoint
sequences of open arcs between hk (A) and hk(A) limiting to hkl(A)
and hk(A), respectively. For k > 1, and for all j, define
1/n 1/(n2)
h(v ) v
j,k (1)kj,k+1
1/3 2/3
h(v ) v and
j,k j,k+l
(nl)/n (n+l)/(n+
h(v ) = v k+l
j,k (1) j
, for n > 4,
+2)
i,k+1
and k > 1, extend h linearly to a
S(nl)/n n > 2, and S/n n >
j,k j,k
, for n > 2.
homeomorphism on the
3, by defining
h(S(nl)/n) = h()
j,k jl,k j,k
and
jl,k j,k
h(S /n) = h(
j,k j1,k j,k
j1,k j,k
We define h on the region between Ak,(nl)/n and Ak,n/(n+l)
where n > 2, by mapping each vertical segment joining Ak,(nl)/n
to Ak,l/n. Refer to figure 3 for k odd.
For all j
segments
We map the region between Ak,1/4 and Ak,1/2 to the region between
Ak+1,1/2 and Ak+1,3/4 as follows: for all j, k > 1, and n = 3,4, we
define h as a linear map so that
1/(n1) 1/(n1) 1/n
h()
j1,k j,k j+(1+(l)k)/2,k
1/(n1) 1/(n1) 1/n
= and
jl,k j,k j+(l+(l)k)/2,k
1/n 1/n 1/(n1)
h()
j1,k j,k j(l+(l) )/2,k
1/n 1/n 1/(n1)
= .
j1,k j,k j(l+(l)k)/2,k
For odd k refer to figure 4. Observe that, for n > 2,
h(Ak,(nl)/n) = Ak+l,(n+l)/(n+2) Note also that the image under h
of a vertical segment joining Ak,1/2 to hk(A) is a vertical segment
joining Ak+1,3/4 to hk+l(A). Similarly, a vertical segment joining
hkl(A) to Ak,1/4 is mapped under h to a vertical segment joining hk(A)
to Ak+1,1/2. On the region between Ak,1/4 and Ak,l/2 h may also
be viewed as the composition of three maps: a shift s, a reflection r
through the yaxis, and an upward projection a to the region between
Ak+1,1/2 and Ak+1,3/4. Note that when k is odd Ak,1/4 shifts "right"
and Ak,1/2 shifts "left," the reverse occurs when k is even; s is the
identity on Ak,1/3 whether k is odd or even (figure 4).
For a point p in B on or below the reflection of h(A) through the
xaxis, define h(p) = h (p). We extend h linearly on the region
between h1(A) and A by sending every vertical segment joining
h1(A) to A to the segment .
2. h is fixed point free. Clearly no point in the complement of B
remains fixed under h. For every k, h is fixed point free on hk(A)
since the intersection of hk(A) and hk+l(A) is empty. If p is a point
between hk (A) and hk(A), then h(p) is between hk(A) and hk+l(A); thus,
p f h(p).
3. Every point has bounded orbit. If p is in the complement of B, then
the orbit of p is the two point set p and h(p).
Observe that
hn(vk,0) = hn( k/(k+l) 0 )
2n+k
= ( (l)n(n+k)/(n+k+l) ,
i=n+k+l
1/i ), for n,k > 1, and
hn(vk,) = hn( k/(k+1) 0 )
2nk
S( (l)n(nk)/(nk+l) ,
i=nk+l
1/i ), for k > 1, n > k.
n
Since ( i=l 1/i log n)  y as n + ~, where y is the Euler constant,
one can show that, for fixed k,
2n+k
lim E
n~ i=n+k+l
2nk
1/i = lim E
n* i=nk+l
1/i = log 2.
Hence, for every j, the positive orbit of vj,0 is bounded. If p is
in , then hn(p) is in implies
that the positive orbit of every point on A is bounded. Observe
that if p is on hk(A), then hk(p) is on A; thus, the positive orbit
of every point on hk(A) is bounded for every k > 0.
To check points between hk1(A) and hk(A), for k > 1, we first
consider a point p in the region above Ak,1/2 and below h (A). Let
k1
q be the point on hk(A) which is directly above p (i.e. q is the
intersection of hk(A) with the vertical line containing p). Since
q has bounded positive orbit and hn(p) is directly below hn(q), for all
n > 0, the positive orbit of p must also be bounded. Next suppose
p is between Ak,l/(n+l) and Ak,l/n, where n > 2. By construction
there is a positive integer m such that hm(p) is between Ak+m,1/2 and
hkm(A). Thus, the above argument shows that p has a bounded positive
orbit. In fact, since a point between hk(A) and hk+l(A), k > 0, is
between A and h(A) after k applications of h, all points in B have
bounded positive orbits.
One can see that the negative orbits of points in B are bounded
by recalling that h'(p) = h(p).
4. h is a homeomorphism. It follows from the construction that h is
bijective, and that h is continuous at every point (x,y) with Ixl f 1.
To show that h is continuous on the lines x = +1 we first consider a
point (l,y), where y > 0. Since h is a reflection on the left side
of the line x = 1, it suffices to show that if (xm,Ym) + (l,y),
then h(xm,ym) + (l,y), where {(xm ,y)} is contained in B.
12
Observe that B is partitioned into convex po
lines x = n/(n+l) and the arcs hk(A), wh
Thus, for m sufficiently large (xm, m) is co
bounded by the lines x = n /(n +1), x = (
k 1 k
h (A) and h m(A), where n > 4. Hence,
m 
n +k +1
m
lygons bounded by the
ere n and k are integers.
ntained in a quadrilateral
nm1)/nm, and the arcs
n +k +1
I/i iY : Y
m
1=n
m
1/i.
Let h(xm'ym) = (xm ',m'). By viewing the image of the quadrilateral
one can see that
n +k +3 nm+km+l
(nm2)/(n 1)
i=n +2 i=n 1
m m
Since xm 1, nm must increase without bound; thus, x 1. To
see that ym' y observe that
nm+km+1 nm+km+1 n +k +1 n +k +3
m 1/i m 1/i
i=nm +1 i=n m 1i=nm i=n+2
The differences of the sums on each side of the bound decrease to 0
as m m; hence, ymYm' 0 implies that ym' y. The arguments
for the remaining points of form (l,y) are analogous to the argument
given above.
CHAPTER III
CONCLUDING REMARKS
1. Consequences. By modifying an example of Bing's (5), Brechner
and Mauldin (7) gave an example of a fixed point free orientation
preserving homeomorphism of E3 onto itself with the property that
the orbit of every point is bounded. Examples of fixed point free
orientation preserving and orientation reversing homeomorphisms of
E3 onto itself, with the property that the orbit of every point is
bounded, follow as easy corollaries from the example we constructed
in Chapter II. The homeomorphism f defined by f(x,y,z) = (h(x,y),z),
where h is the homeomorphism constructed above, verifies the corollary
for the orientation reversing case. For the orientation preserving
case simply define g(x,y,z) = (h(x,y),z). Observe that the first
homeonorphism can be viewed as a reflection across the yzplane,
followed by an orientation preserving map; while the second homeo
morphism is the composition of a reflection across the yzplane, an
orientation preserving homeomorphism, and a reflection across the xyplane.
2. Related questions. Research in the area of planar fixed point theory
continues to be quite active. We state two unsolved problems which are
closely related to the counterexample presented in Chapter II. Let h
be an orientation reversing homeomorphism of E2 onto itself having the
property that the orbit of every point is bounded. Define h to be
bounded at p if there exists an open set U containing p such that the
orbit of U is bounded. Let D be the set of all points p such that h
2
is bounded at p. Observe that D is open in E2. To see that D is dense
in E2, let F = E2D, and suppose that F contains some open set U. Define
F {p E F : the orbit of p is contained in B },
where Bn = {p E2 : Ip < n}. Since each Fn is closed and F = Un=, F ,
the Baire Category Theorem implies that some F contains an open set W.
Therefore, since h is bounded at every point in W, we obtain the
contradiction that W is in D as well as F.
In the example we give,D has infinitely many components. The
question as to whether D can have finitely many components without h
having a fixed point remains unsolved.1 Even when D has just two
components, the answer to this question is not known.
Another question of a similar nature can be phrased as follows:
if h is a continuous function on E2 with the property that the positive
orbit of every bounded set is bounded, then does h have a fixed point?
As mentioned in Section 2 of Chapter I, when h is a homeomorphism,
this question can be answered in the affirmative.
IThis problem was communicated to me by Professor R. Dan Mauldin.
* h2n(p)
xax1 S
XaXis
FIGURE 1
A fixed point free homeomorphism with some bounded orbits
Fyaxis
h(q1)
/I
h2nnl (p)
I
I
S\1
FIGURE 2
A counterexample to the Bounded Orbit Conjecture
FIGURE 3 k_ k
Map of h on region between h (A) and h(A)
Map of h cn region
1/4
j/2 1 k A k 1,1
between and /3
51/4
FIGURE 4
between Ak,1/4 and Ak,1/2
BIBLIOGRAPHY
1. Stephen Andrea, On homeomorphisms of the plane which have no
fixed points, Abh. Math. Sem. Univ. Hamburg. 30 (1967), 6174.
2. The plane is not compactly generated by a free
mapping, TAMS, 151 (1970), 481498. MR 42 #2445.
3. Harold Bell, A fixed point theorem for plane homeomorphisms,
BAMS, 82 (1976), 778780.
4. A fixed point theorem for, plane homeomorphisms,
Fund. Math., 100TT978), 119128.
5. R. H. Bing, The elusive fixed point property, Amer. Math.
Monthly, 76 (1969), 119132. MR 38 #5201.
6. S. Boyles, A Counterexample to the Bounded Orbit Conjecture,
TAMS (to appear).
7. B. L. Brechner and R. D. Mauldin, Homeomorphisms of the plane,
Pac. J. of Math., 59 (1975), 375381. MR 52 #9199.
8. EC+ Homeomorphisms of Euclidean spaces, Top. Proc.,
1 (1976), 335343.
9. L. E. J. Brouwer, Beweis des ebenen Translationesatzes, Math.
Ann., 72 (1912), 3954.
10. M. L. Cartwright and J. E. Littlewood, Some fixed point theorems,
Ann. of Math., 54 (2) (1951), 137.
11. J. G. Hocking and G. S. Young, Topology, Addisonlesley Pub. Co.,
Reading, Mass., 1961.
12. Gordon Johnson, An example in fixed point theory, PAMS, 44 (1974),
511514.
13. B. v. Kerekjarto, On a geometric theory of continuous groups,
Ann. Math., 59 (1925), 105117.
14. W. K. Mason, Fixed points of pointwise almost periodic homeo
morphisms on the 2sphere, preprint.
15. Deane Montgomery, Pointwise periodic homeomorphisms, Amer. J.
Math., 59 (1937), 118120.
20
16. E. v. Sperner, Uber die fixpunktfreien Abbilfungen der Ebene,
Abh. Math. Sem. Hamburg, 10 (1934), 147.
17. H. Terasaka, On quasi translations in En, Proc. Japan Acad.,
30 (1954), 8084.
18. S. M. Ulam, Problems in Modern Mathematics, Wiley and Sons,
Inc., New York, 1960. MR 22 #10884.
BIOGRAPHICAL SKETCH
Stephanie Marion Boyles was born in Atlanta, Georgia, on
December 29, 1954. In 1974, she obtained her Bachelor of Science
degree in mathematics from the University of Georgia. Two years
later she received her Master of Science degree in applied
mathematics from Michigan State University. She then entered the
University of Florida and in 1980, attained the degree of Doctor
of Philosophy in mathematics. She is a member of the American
Mathematical Society, the Mathematical Association of America, and
Phi Kappa Phi.
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is
fully adequate in scope and quality as a dissertation for the degree
of Doctor of Philosophy.
erpard X. Ritter, Chairman
7Associate Professor of Mathematics
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is
fully adequate in scope and quality as a dissertation for the degree
of Doctor of Philosophy.
Elroy J. Bolduc
Professor o Subject Specialization
Teacher Education
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is
fully adequate in scope and quality as a dissertation for the degree
of Doctor of Philosophy.
J mes Edgar Keesling
/ professor of Mathematics
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is
fully adequate in scope and quality as a dissertation for the degree
of Doctor of Philosophy,
Patrick J seph McKenna
Assistant Professor of Mathematics
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is
fully adequate in scope and quality as a dissertation for the degree
of Doctor of Philosophy.
Sergio E. Z anton lo
Assistant rfessor of Mathematics
This dissertation was submitted to the Graduate Faculty of the Depart
ment of Mathematics in the College of Liberal Arts and Sciences and to
the Graduate Council and was accepted as partial fulfillment of the
requirements for the degree of Doctor of Philosophy.
June, 1980
Dean, Graduate School

