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 y-axis. 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 2-dimensional 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 counter-clockwise

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 m-plane. 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 y-axis followed by the upward shift

f(x,y) = (-x -y-lxl+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 non-separating 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 non-separating 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 y-axis.

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+l-j,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 non-negative y-coord-

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 y-axis, 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,k-1 j,k

(n-1)/n
v E v + (v
j,k j,k-1 j

1/n (n-l)/n
Note that {vj,k n>3 and {vj,k

and vj,k, respectively. Let

v
j,k-1

v
,k j

)/n, where n > 3, and

)(n-l)/n, where n > 2.
,k-1

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 = (n-l)/n for n > 2. Thus, for every
positive integer k {Ak,i/n}n>3 and {Ak,(n-l)/n}n>2 are disjoint

sequences of open arcs between hk- (A) and hk(A) limiting to hk-l(A)
and hk(A), respectively. For k > 1, and for all j, define

1/n 1/(n-2)
h(v ) v
j,k (-1)k-j,k+1
1/3 2/3
h(v ) v and
j,k -j,k+l
(n-l)/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(n-l)/n n > 2, and S/n n >
j,k j,k

, for n > 2.

homeomorphism on the
3, by defining

h(S(n-l)/n) = h()
j,k j-l,k j,k

and
j-l,k j,k

h(S /n) = h( j,k j-1,k j,k

j-1,k j,k

We define h on the region between Ak,(n-l)/n and Ak,n/(n+l)
where n > 2, by mapping each vertical segment joining Ak,(n-l)/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/(n-1) 1/(n-1) 1/n
h()
j-1,k j,k j+(-1+(-l)k)/2,k
1/(n-1) 1/(n-1) 1/n
= and
j-l,k j,k j+(-l+(-l)k)/2,k
1/n 1/n 1/(n-1)
h()
j-1,k j,k j-(l+(-l) )/2,k
1/n 1/n 1/(n-1)
= .
j-1,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
hk-l(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 y-axis, 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
x-axis, define h(p) = -h- (-p). We extend h linearly on the region

between h-1(A) and A by sending every vertical segment joining

h-1(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 )

2n-k
S( (-l)n(n-k)/(n-k+l) ,
i=n-k+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

2n-k
1/i = lim E
n-* i=n-k+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 h-k(p) is on A; thus, the positive orbit
of every point on hk(A) is bounded for every k > 0.

To check points between hk-1(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 h-k(A) and h-k+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

nm-1)/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
(nm-2)/(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, ym-Ym' 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 yz-plane,

followed by an orientation preserving map; while the second homeo-

morphism is the composition of a reflection across the yz-plane, an

orientation preserving homeomorphism, and a reflection across the xy-plane.

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 = E2-D, 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)

x-ax1 S
X-aXis

FIGURE 1
A fixed point free homeomorphism with some bounded orbits

Fy-axis
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), 61-74.

2. The plane is not compactly generated by a free
mapping, TAMS, 151 (1970), 481-498. MR 42 #2445.

3. Harold Bell, A fixed point theorem for plane homeomorphisms,
BAMS, 82 (1976), 778-780.

4. A fixed point theorem for, plane homeomorphisms,
Fund. Math., 100TT978), 119-128.

5. R. H. Bing, The elusive fixed point property, Amer. Math.
Monthly, 76 (1969), 119-132. 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), 375-381. MR 52 #9199.

8. EC+ Homeomorphisms of Euclidean spaces, Top. Proc.,
1 (1976), 335-343.

9. L. E. J. Brouwer, Beweis des ebenen Translationesatzes, Math.
Ann., 72 (1912), 39-54.

10. M. L. Cartwright and J. E. Littlewood, Some fixed point theorems,
Ann. of Math., 54 (2) (1951), 1-37.

11. J. G. Hocking and G. S. Young, Topology, Addison-lesley Pub. Co.,
Reading, Mass., 1961.

12. Gordon Johnson, An example in fixed point theory, PAMS, 44 (1974),
511-514.

13. B. v. Kerekjarto, On a geometric theory of continuous groups,
Ann. Math., 59 (1925), 105-117.

14. W. K. Mason, Fixed points of pointwise almost periodic homeo-
morphisms on the 2-sphere, preprint.

15. Deane Montgomery, Pointwise periodic homeomorphisms, Amer. J.
Math., 59 (1937), 118-120.

-20-

16. E. v. Sperner, Uber die fixpunktfreien Abbilfungen der Ebene,
Abh. Math. Sem. Hamburg, 10 (1934), 1-47.

17. H. Terasaka, On quasi translations in En, Proc. Japan Acad.,
30 (1954), 80-84.

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