dynamics of one-dimensional maps: symbols, … · sequences, through the method of symbolic...

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DYNAMICS OF ONE-DIMENSIONAL MAPS: SYMBOLS,

UNIQUENESS, AND DIMENSION

DISSERTATION

Presented to the Graduate Council of the

North Texas State University in Partial

Fulfillment of the Requirements

For the Degree of

DOCTOR OF PHILOSOPHY

by

Karen M. Brucks, B.A., M.A.

Denton, Texas

May, 1988

Brucks, Karen M., Dynamics of One-Dimensional Maps:

Symbols. Onimienass. M DimeHSiSJQ- Doctor of Philosophy

(Mathematics), May, 1988, 106 pp., 14 illustrations,

bibliography, 46 titles.

This dissertation is a study of the dynamics of

one-dimensional unimodal maps and is mainly concerned with

those maps which are trapezoidal. The trapezoidal function,

f , is defined for e€(0,l/2) by f (x)=x/e for x€[0,e], e e

f (x)=l for x€(e,1-e), and f (x)=(1-x)/e for xe[l-e,l]. We e g

study the symbolic dynamics of the kneading sequences and

relate them to the analytic dynamics of these maps.

Chapter one is an overview of the present theory of

Metropolis, Stein, and Stein (MSS). In Chapter two a formula

is given that counts the number of MSS sequences of length n.

Next, the number of distinct primitive colorings of n beads

with two colors, as counted by Gilbert and Riordan, is shown

to equal the number of MSS sequences of length n. An

algorithm is given that produces a bisection between these

two quantities for each n. Lastly, the number of negative

2

orbits of size n for the function f(z)=z -2, as counted by

P.J. Myrberg, is shown to equal the number of MSS sequences

of length n. For an MSS sequence P, let H (P) be the unique common

oo extension of the harmonics of P. In Chapter three it is

proved that there is exactly one J(P)€[0,1] such that the

itinerary of \{P) under the map is H ^ P ) .

In Chapter four it is shown that only period doubling or

period halving bifurcations can occur for the family

Jl€[0,1 ]. Results concerning how the size of a stable orbit

changes as bifurcations of the family If occur are given.

Let /l€[0,l] be such that 1/2 is a periodic point of Jfg.

In this case 1/2 is superstable. Chapter five investigates

the boundary of the basin of attraction of this stable orbit.

An algorithm is given that yields a graph directed

construction such that the object constructed is the basin

boundary. From this we analyze the Hausdorff dimension and

measure in that dimension of the boundary. The dimension is

related to the simple /^-numbers, as defined by Parry.

ACKNOWLEDGMENT

I thank W.A. Beyer and P.R. Stein of Los Alamos National

Laboratory.

iii

TABLE OF CONTENTS

LIST OP ILLUSTRATIONS v

Chapter

page I. INTRODUCTION 1

Synopsis MSS Theory

II. MSS SEQUENCES, PERIODIC POINTS OF f(z) = z2-2, AND COLORINGS OF NECKLACES 14

MSSn

Colorings of Necklaces 2

Periodic Points of f(z) = z -2

III. UNIQUENESS FOR H (P) 32

Section One Section Two

IV. BIFURCATIONS 54

Introduction Bifurcations

Uniqueness and Bifurcations

V. HAUSDORFF DIMENSION OF BASIN BOUNDARIES . . . . 75

Introduction Graph Directed Constructions for

Basin Boundaries Examples

VI. QUESTIONS 99

BIBLIOGRAPHY 103

IV

LIST OF ILLUSTRATIONS

Figure

page 1. Graph of the map f 1

2. Formation of the sequence RLRC 7

3. Tree giving rise to aperiodic sequences 8

4. Geometric view of J and $ 42 00 00

- 1 - 1 5. Graphs of F^ R and F^ ^ 45

6a. Regular period doubling 54

6b. Reverse Period Halving 54

7a. Graph of hp(x) f when P is even 65

7b. Graph of hp(x) , when P is odd 65

8. Graph of hp(x) 71

9. Graph of r{x) 72

10a. Bifurcation diagram 74

10b. Bifurcation diagram 75

11. Graph of F r 1 and F^ 1 80

12. Directed graph for RC 93

13. Directed graph for RLRC 95

14. Directed graph for RLC 97

CHAPTER I

INTRODUCTION

This dissertation is a study of the dynamics of

one-dimensional unimodal maps, and is mainly concerned with

maps referred to as trapezoidal. For e € (0,1/2) define the

map f as follows.

(l/e)x,

1,

{1/e){1-x),

if 0 < x < e

if e < x < 1-e

if 1-e < x < 1.

The graph of f is shown in Figure one. e

1 ••

V.

i-e 1

Fig. 1 Graph of the map f

The method of Symbolic Dynamics is the main tool used

throughout this thesis. The belief that certain properties

of trapezoidal maps and techniques developed in the study of

trapezoidal maps may apply to a broader class of functions is

a principal reason for this study. Section one of Chapter

one is a synopsis of the results presented in Chapters two

through six. Section two of Chapter one is an introduction

to and partial overview of the theory of Metropolis, Stein,

and Stein (MSS) and shift maximal sequences. These

sequences, through the method of symbolic dynamics, enable

one to study the dynamics of unimodal maps.

Synopsis

Section one of Chapter 2 gives a formula that counts the

number of MSS sequences of length n. This formula depends on

work of L.-C. Sun and 6. Helmberg [43]. The results of this

section are used throughout Chapter two.

Section two considers the various colorings of a necklace

consisting of n beads, where each bead can be either red or

black. Informally, two colorings will be considered to be

the same if it is possible to get from one to the other by

moving the clasp, interchanging colors, or both. A coloring

is considered primitive if it has no proper subpattern. (For

example, the coloring red-black-red-black on four beads is

not primitive, since it has the proper periodic subpattern

red-black.) Gilbert and Riordan [17] give a formula that

counts the number of such primitive colorings. Metropolis,

Stein, and Stein [30] noted that for all n < 15, Gilbert

and Riordan's formula also gives the number of MSS sequences

of length n. In Section two it is shown that these two

quantities agree for all n. This is done by an algorithm

that for each n produces a bijection.

2

Lastly in Chapter two the periodic points of f{z) = z -2

are considered. Let f° be the identity map, and for a

positive integer n define fn inductively by f1 = f and fn =

f(fn 1). If, for some n > 0, fn(w) = w, then w is called a

periodic point of f and the minimum { n > 0 | fn(w) = w } is

called the period of w. For any z, the orbit of z is { fn(z)

n > 0 }. If x is a periodic point of f with period m, then

(fm)'(x) is constant on the orbit of x; thus the orbit of x

is said to be negative (positive) if (fm)'(x) < 0 (>0). For

a given positive integer n, Myrberg [35] attempts to count

the values of p such that zero is a periodic point of h(z) = 2

z -p of period n. In his analysis he describes sequences

that would later be called MSS sequences. Based on a

hypothesis that he is unable to prove, he finds the number of

such p to be the number of distinct negative orbits of order, 2

or size, n using the function f(z) = z -2. The number of

such p for n < 16, calculated by Myrberg, is equal to the

number of MSS sequences of length n. Let N denote the set of

positive integers. In Chapter two it is shown that the

number of MSS sequences of length n, for all n € N, is the

same as the number of distinct negative orbits of order n 2

using the function f(z) = z -2. Two proofs are given. Chapter three addresses the question of uniqueness for

the trapezoid family defined above. Fix e € (0,1/2). It is

known [2,27] that for each MSS sequence P there is exactly

one value of Jp € [0,1] so that the itinerary of under the

^Pf

map -Jpfg' denoted I e U ) » is P. In Chapter three for

certain (namely, the infinite harmonics) aperiodic shift

N

maximal sequences A € { R,L } , there is shown to be exactly

one J € [0,1] such that the itinerary of 4 under i f is A. A A A Q

Also, it is pointed out that if g is a unimodal map ,then to

prove

= { 4 € [0,1] I I^(J) is an MSS sequence or

is infinite and periodic }

is dense in [0,1], it would suffice to show that for any

N given aperiodic A € {R,L> there exists exactly one €

Kg

[0,1] with I (4) = A. We believe that is dense in e

[0,1]. It is generally believed, but still unproven, that 9 3

is dense in [0,1] where s(x) = 4x(l-x) [18,9 pp. 31;69].

Possibly some of the techniques developed in Chapter three

will prove useful in showing that ? is dense in [0,1]. s

Results of Jakobsen [21] show that [0,1] \ y has positive s

Lebesgue measure. It is not known whether [0,1] \ has e

positive Lebesgue measure or not.

Chapter four discusses the bifurcation diagram for both

the trapezoid family and the family is(x) = 4ix(l-x), X €

[0,1]. It is shown that only period doubling or period

halving bifurcations can occur for the family -lf_, and noted

that the same is known to hold for the family -Is. Also, some

questions and known results about the type of bifurcations

that can occur for certain one-parameter families are

discussed. Results concerning how the size of a stable orbit

changes as bifurcations of the family if occur are given.

Questions are raised concerning the bifurcation diagrams of

both families If_ and is. e

In Chapter five an algorithm is given that, for a given

MSS sequence P and e € (0,1/2), defines a graph directed

construction [29] such that the object constructed is

precisely the boundary of the basin of attraction for the

stable orbit of the map J_,f . Results of Mauldin and P e

Williams [29] are then used to calculate the Hausdorff

dimension of the boundary of the basin of attraction for the

stable orbit when P is of the form R, RLR, RL, or RLL. The

appearance of the simple ^-numbers, as discussed by W. Parry

[37] is seen. For convenience we will refer to these numbers

as the simple r-numbers. The case P « RLn, n>0, is also

discussed.

Chapter six lists unanswered questions that have arisen

in this study.

MSS Theory

In 1973 Metropolis, Stein, and Stein (MSS) [30] developed

a universal theory for a certain class of maps of [0,1] into

itself. A map f: [0,1] -» [0,1] is said to be a unimodal map

if f is continuous, f(0)= f(l) = 0, f(l/2) = 1 , f is

nondecreasing on [0,1/2], and nonincreasing on [1/2, 1]. One

can form a one-parameter family of maps from a given unimodal

map f by setting

fJI (x) - j f (x ) , i e [o , i j .

MSS formed finite sequences of R's and L's by fixing a

unimodal map f and considering the iterates of the maps if at

1/2 for i € [0,1]. More precisely, if for some 4 € [0,1] the

point 1/2 is a periodic point of if of period n, then MSS

formed a sequence b^bj. . *^-1 setting

R, if (if)i(l/2) > 1/2

1 ' L, if (Xf)i(l/2) < 1/2.

For convenience I will set b R = C, since (Jf)n(l/2) » 1/2.

Finite sequences of R's ans L's obtained in this manner are

called MSS sequences. MSS note that the MSS sequences appear

in a certain order and they give an explicit algorithm

independent of f for constructing these sequences in order.

The class of maps discussed by MSS was not precisely the

class of unimodal maps on [0,1]; however, the unimodal maps

discussed in this thesis are in the class of maps discussed

by MSS. For each positive integer n the set of all possible

MSS sequences of length n is denoted by MSSn< In general |A

will denote the cardinality of a set A. For example, RLLRC

is an MSS sequence of length five. In Figure two one sees

how the MSS sequence RLRC might arise.

Fig. 2 Formation of the sequence RLRC

I next wish to discuss shift maximal sequences and relate

these to MSS sequences. Collet and Eckmann [9] define shift

maximal sequences. Briefly, a sequence w of symbols L,R,C is

said to be admissible if w is an infinite sequence of L's and

R's or if w is a finite (or empty) sequence of L's and R's

followed by a C. Such sequences will be referred to as

words. The parity-lexicographical order is put on the set of

admissible words. This order is defined as follows. Set L <

C < R. Let w = {w.} and v = {v.} be two distinct admissible 1 i

words. Let k be the first index where they differ. If they

differ in the first position, i.e., k « 1, then w < v iff w.

< v1. Assume k > 1. If wi-*,wjc_i = vi*"*vk-i **as a n e v e n

number of R's, i.e., has even parity, then w < v iff w^ < vfe.

If there are an odd number of R's, then w < v iff v^ < w^. A

word is shift maximal if it is greater than or equal to all

of its right shifts. For example, RLLRC is shift maximal,

where as LLLC is not. The cardinality of the set of shift

maximal sequences is less than or equal to 0 • 0 n e c a n s e e

that the cardinality is £ , by considering the tree given in

Figure three. Note that below the root all left branches are

an R and all right branches are an LRRR.

aLRKLR

LRRR

L (UP*

Fig. 3 Tree giving rise to aperiodic sequences

Repeated cancatenation along any branch will result in a

N shift maximal sequence in {R,L} . For example, the sequence

RLRRLR LRRR LRRR R LRRR R R R R R ...

is shift maximal.

To relate MSS sequences to shift maximal sequences some

notation is needed. Let f be a map of [0,1] into itself.

Then the itinerary of a point x € [0,1] is a finite or

infinite sequence

I'fx) - (I 1 > 1 > o

of R's , L's or C's, where 1^ = R if f*(x) > 1/2, 1^ = L if

fi(x) < 1/2, and IA « C if f1(x) = 1/2. The sequence stops

after the first C. Thus, if f is unimodal, X € [0,1], and

1/2 is a periodic point for the map /If, then I^(J) is an MSS

sequence. The next lemma and theorem, which are taken from

Beyer, Mauldin, and Stein (BMS) [2], show that every MSS

sequence is shift maximal. This fact will be used throughout

this thesis.

LEMMA 1.1. Let f:[0,l] -* [0,1] be unimodal and X € (0,1).

If I J f (x) < I^*(y), in the parity-lexicographical order, and

x,y € [0,1], then x < y.

Proof. Assume the implication is false. For each u,v in

[0,1] with

I^f(u) < r*f(v)

and

v < u,

j| f \ f

let d(u,v) be the first integer where I (u) and I (v)

differ. Choose x,y € [0,1] such that i * d(x,y) is smallest.

CASE 1. Suppose i = 0. Then IQ(x) = L or C and IQ(y) = H or

10

I (x) = L and I (y) =» C or R. In either case x < y. O o

CASE 2. Suppose i > 0 and

I0(x) = iQ(y) - L.

Then

I(f(x)) < I(f(y))/

since the parity is not changed by dropping the initial L.

Now

f(x) < f(y),

since d(x,y) is minimal. Thus x < y, since x,y < 1/2 and f

is nondecreasing on [0,1/2].

CASE 3. Suppose i > 0 and

I Q ( X ) - IQ(y) - R.

The argument is similar to case two.

THEOREM 1.2. Let f: [0,13 -* [0,1] be unimodal. For any €

(0,1), is shift maximal. In particular, an MSS

sequence is shift maximal.

Proof. For all j > 0 and x € [0,1],

(/If (x) < -1 .

Suppose I^f(JI) is not shift maximal. Then there exists some

j > 0 such that

I ^ f ( / I ) < I X f ( ( J f J - ' U ) ) .

Then, by Lemma 1.1,

X <

which contradicts Lemma 1.1. This completes the proof of

Theorem 1.2.

11

At least three questions Immediately come to mind.

QUESTION 1. Given a unimodal map f, for which shift maximal

sequences P is it true that there exists some X € [0,1] so

that l X t {X ) - P?

QUESTION 2. What are necessary and sufficient conditions to

impose on a map f so that for every shift maximal sequence P

there exists some X € [0,1] such that I^(J) = P?

QUESTION 3. A map f, or the one-parameter family Xf, is said

to exhibit uniqueness if for each MSS sequence P there exists

at most one value of X such that I^(J) = P. What are

necessary and sufficient conditions to impose on a map f to

guarantee that f exhibits uniqueness?

Question 1 has been answered for various maps f [2,9,11],

some of which will presently be discussed. Questions two and

three are open. BMS [2] conjecture that concave smooth

unimodal functions on [0,1] exhibit uniqueness.

For the tent map

2x, if 0 < x < 1/2 t(x) =

2(1-x), if 1/2 < x < 1,

DQP [11] give necessary and sufficient conditions for an MSS

12

sequence P to have an associated value of J € [0,1] such that

1 4(J) = P. The set

{ P | P is an MSS sequence such that there does not

exist some i € [0,1] with I^*(J) = P >,

is shown to be infinite. Thus, there exist maps f such that

the set of shift maximal sequences that do not appear is

nonempty. (A shift maximal sequence P is said to appear, for

a given f, if there exists some ^ € [0,1] such that =

P.) In contrast to this fact BMS [2] prove the following

theorem.

THEOREM 1.3. Let f be a unimodal, Lipschitz continuous,

round-top concave function. For each shift maximal sequence

P there is a value of \ so that = P. In particular,

each MSS sequence occurs.

A map f is said to be round-top concave if f is concave and

if there exists an a 6 (0,1/2) so that f'(x) ( the derivative

of f ) exists and is continuous in (a,l-a) and f'(l/2) = 0.

The functions s(x) = 4x(l-x) and fg, e € (0,1/2), are

examples of maps that satisfy the hypotheses of Theorem 1.3.

Thus, the appearance of a pattern is not invariant under

conjugacy since Ulam and Von Neumann [44] showed that the

tent map t(x) is topologically conjugate to s(x).

13

REMARK 1.4. Note that the MSS sequences are precisely the

finite shift maximal sequences. This follows from Theorems

1.3 and 1.2.

Unpublished results by Milnor and Thurston, and Douady

and Hubbard claim to have established that s(x) = 4x(l-x)

exhibits uniqueness. BMS [2] have established uniqueness for

a class of maps that include the maps f with 0 < e < (3

sqrt(17) - ll)/4 = 0.3423..., but that does not include the

map f(x) « 4x(l-x). Metropolis and Louck [27] establish

uniqueness for all f^, e € (0,1/2). BMS also give an example

of a map that does not exhibit uniqueness It might be noted

that this example has found physical application [8].

CHAPTER II

2

MSS SEQUENCES, PERIODIC POINTS OF f(z) • z -2,

AND COLORINGS OF NECKLACES

Section one of Chapter two gives a formula that counts

the number of MSS sequences of a given length. In section

two an algorithm is given that yields a bisection between the

MSS sequences of length n and the distinct primitive

colorings on n beads, as discussed in Chapter one. Lastly,

in section three it is shown that the number of negative

orbits of size n for the function f(z) = z -2 is equal to the

number of MSS sequences of length n.

|MSSn|

L.-C. Sun and G. Helmberg [43] expand the set of

admissible words, given by Collet and Eckmann, to include all

finite sequences of R's and L's. They also extend the

parity-lexicographic order as follows. If w and v are

admissible words such that there is a k ) 1 with w =

W1" " *wkwk+l' ' * a n d V 58 wl'"*wk' t h e n w > v P a r i tY o f

v is odd, otherwise v > w. A finite sequence of R's and L's,

w is called shift maximal in the extended

parity-lexicographic order if w is greater than or equal to

all of its right shifts. For simplicity w will simply be

14

15

called shift maximal. A word of length n is said to be

primitive provided its smallest subperiod is also of length

n. Notice that if w = w^.-w is primitive, then for each j,

2 ^ j 5s n, w # w ^ . . .

Only those results of Sun Lichiang and G. Helmberg that

are used to establish their formula for jMSSn| are listed. I

provide my own proofs for all except Theorem 2.2 in order to

avoid their notation. Again note that these results will be

used throughout Chapter two.

LEMMA 2.1. If w = w ...wn € {R,L}n is shift maximal, then

w„...w ,C € MSS and if b,...b .C is in MSS then both 1 n-i n i II j. **

b ...b „L and b, ...b ,R are shift maximal. 1 n-1 1 n-l

Proof. Lemma 3.1 follows from elementary observations.

THEOREM 2.2. Let w = wi**-wn ^ {R»L}

n. Then the following

are equivalent.

(1) w is shift maximal.

2

(2) w00 is shift maximal and either w is primitive or w » v

(cancatenation of v with v), where v has odd parity and

v is primitive.

Proof. We will first show that (1) implies (2). Observe

that vf° is shift maximal iff for alii, 1 < i < n,

wi+l' ' ' V l ' ' wi - W1 •' wn-l wn-i+l' ' - V

16

Let i be such that 1 ^ i < n. First, w shift maximal implies

that w. ... w ^ w....w .. We need only consider, i+l n - 1 n-i

C ) wi+j- ' w n " "i- • wn-i'

Assume that (*) holds. Now, since w is shift maximal, the

parity of w]/* , wn_i *s 0 <^ an(* wn-i+l'**Wn - Thus,

w i + r - - W * - w i - W1' ' *wn-iwn-i+l" * *Wn * W'

and therefore w® is shift maximal.

Next, suppose w is not primitive. Express w as

w = wi * ' " " W1''* Wk'

where n/k > 2. The parity of w 1••*wk i s odd, since w is

shift maximal. Thus n/k = 2, since otherwise w would be less

than its right shift w ..•wfcw1...w^. Hence, (1) implies (2).

Assume that (2) holds and let i be such that 1 ^ i < n.

It will be shown that w 1 + 1---wn ^

w- W e h a v e wi+i , , , wn -

w. . . .w ., since w00 is shift maximal. It suffices to 1 n-i

consider only,

Thus assume (#*) holds. Again, since w00 is shift maximal,

17

we have

wn-i+T ' ,Wn - W1' ' *wi *

First, if w . ,...w < w,...w., then the parity of

n-i+i n i x

wi+l*''wn i s o d d' s i n c e wi+l* *"wnwl"* *Wi -

W1' " *wn-iwn-i+l" * "Wn = W' T h e r e f o r e ' w i + i " -w n < W"

Next, if w n_ 1 + 1---wn = w ^ - . w ^ t h e n

W = " j . - V i V u r - ^ n " M i + r - - W - - V T h u s- w i s n o t

primitive and therefore w = v with v odd and primitive.

Hence, v = wi + 1--*wn a n d therefore Wi+1 *Wn < W"

THEOREM 2.3. Let m € N and w « w 1'**wm ^ {

R» L) m b e shift

maximal and primitive. Then for each j, 2 < j < m,

w....w w,..«w. , is not shift maximal. 3 m l j-i

Proof. Suppose there exists a j , 2 < j < m , so that v =

w ...ww ...w. , is shift maximal. It suffices to assume j m l j-1

2(j-l) < m. Let k • [m/(j-1)]; then k > 2.

First, w and v both shift maximal imply the following:

(1) W1 - W y W2 = W j + 1, ... ,Wj_1 W2(j-1),

(2) the parity of is odd,

(3) wx = w t ( j_ 1 ) + 1, ... 'wj-i = w(t+l)(j-1),

for t - 1,2 k-1, if k > 3, and

( 4 ) wk(j-l)+l = W1 Wm = wm-k{ j-1) '

18

The parity of ... is odd, since w is shift

maximal. Note that k(j-l) < m, since w is primitive.

However now,

w < w!-••*j>1w1***

wm_k(j-l),

with w 1...w j_ 1w 1...w m_ k ( j_ 1 ) a right shift of w. Hence v can

not be shift maximal.

THEOREM 2.4. Let m € N, and w € {R,L}m be primitive. Let C

be the set of all cyclic permutations of w. Then there

exists exactly one word in C that is shift maximal.

Proof. Theorem 2.3 implies that there is at most one shift

maximal word in C. Assume m > 1. Now, m > 1 and w being

primitive imply that both R and L appear in w. Let

q = max { n € N j B v € C so that v begins with RLn >.

Note that q > 1. Let ^ be the set of v in C that begin with

RL^. If jCjl « 1, then that one element is shift maximal.

Assume that | J > 1, and that each v in Cj is not shift

maximal. Let v » v„...v € C,. Then Theorem 2.2 implies 1 HI "1

that v00 is not shift maximal and so there is some j, 2 < j <

m, with vj•••vmvi•••vj_i > v* L e t v l = vj•*•vmvl'*"Vj-1*

Then v1 > v implies that v1 € Similarly, there exists

v 2 € Cj so that v2 > v1. However, is a finite set. Hence

19

there is exactly one word in C that is shift maximal. This

completes the proof of Theorem 2.4.

Now, for each n € N let

p(n) = I{ w I w is shift maximal, w € {R,L}n,

and w is primitive}j

Then for n € N,

(*) p(n) = 1 I (i(d) 2 n / d, n din

* r

where p{m) is the Mobius function: ^(1) = 1» 0 = (_1) if

m is a product of r distinct primes, and /i(m) ® 0 otherwise.

To see this, for each s € N let P(s) be the number of words of length s that are primitive. Then 2 n = E P(d), and by

din

Mobius inversion,

P(n) = E /t(d) 2 n / d. d|n

Theorem 2.4 now gives us (*).

THEOREM 2.5. For n € N we have that,

2|MSSn| - jM S S

n / 21' i f n i s e v e n '

p(n) = \ 2|MSS^I, if n is odd.

Proof. First, Lemma 2.1 implies that p(n) < 2|MSSn| for all

20

n € N. For n odd Theorem 2.2 immediately gives equality. So

assume n is even.

The case n = 2 can be simply checked, so assume n is even

and greater than two. We now let,

B - { B • b 1...b n_ 1C | B € MSSn,

and either b, . . .b 1L or b,...b -R is NOT primitive }. 1 n-l i n-i

It suffices to show that |B| • | M S SN/ 2|*

Let b 1-..b n_ 1C € B. Then bi••- bn/ 2

bi* **bn/2 i s s h i f t

maximal and therefore bi*** bn/2 * s maximal. Thus,

b l " b n / 2 - l ° 6 M S Sn/2 ^ so | B | < |HSSn/2|.

Let D - • • dn/2-i°

€ M S Sn/2. c h o o s e dn/2 f r o m ( R , L ) S O

that * s P ri mi ti v e* Since D € M S S n/2' di***dn/2 * S

shift maximal. We claim that *"'dn/2 i s a l s o

shift maximal. For if not, there is some j, 2 < j < n/2, so

that

V " * d n / 2 d l * ' " d n / 2 > dl**- dn/2 dl*• , dn/2'

.00 However, dj/** d

n/ 2 maximal implies that * s

shift maximal and therefore

dj' * *dn/2dl* * *dn/2 "* dl* * *dn/2dl" * ,dn/2-j+l

In particular, d^ . . ' ' 'dj-l 85 dl"'*dn/2' w b * c b

21

contradicts <*1---dn/2

b e i n 9 primitive. So our claim holds.

Thus, dx..-dn/2

di-•,dn/2-lC € - a n d I - I " lMSSn/2

Theorem 2.5 makes the proof of Theorem 2.6 a simple induction

argument.

Ir THEOREM 2.6. If n • 2 (2m-l), then

—i i—1 MSS I = E 2 p(n/ 2 )

n l 1=1

Colorings of Necklaces

For each n € N, partition the elements of {l,-l}n into

equivalence classes, where equivalence is determined by

C x S . Here C is a cyclic group of n elements, namely n 2 n

cyclic permutations, and S 2 is the permutation group on two

elements. Thus, two elements w and v of <l,-l>n, are said to

be the same iff there exists some 7 € C R x S 2 so that ?(w) =

v. Each equivalence class containing primitive elements

gives a distinct coloring for a necklace consisting of n

beads, where there are two possible colors for each bead.

Let CL denote the collection of such equivalence classes. n

Arbitrary members of a class will be used to represent it.

For example we can express CL^ as, CL^ = { <-l,l,l,l>,

<-l,l,l,-l> }. Of course, among others, <1,1,1,-1> and

22

<-1,-1,-1,1> are equivalent to <-l,l,l,l>. The coloring

<1,-1,1,-1> is not primitive. Gilbert and Riordan [17] give

a formula that yields |CLn| for each n € N, and they computed

JCL j for 1 <n < 20. Their values match those in MSS table 2

[30] and the table given by Myrberg [35]. The following

algorithm gives a bijection between MSSn and CLn, for each n

€ N.

DEFINITION 2.7. Let n > 2 and B » b 1 —bn _ i

C € MSSn- Then

define h(B) as follows.

h(B) = <elfe2,...,en>,

where e^ = -1, = 1- and

ei-l' l f bi-l * L e. =

I -.a if b a R i-1' i-1

Thus h : U MSS_ -» U {-1,1 }n.

, 3 2 n n>2

THEOREM 2.8. For every n > 2, h| M S g i s a bijection onto n

CL . n

COROLLARY 2.9. | M S Sn| = |

C Lnl'

f o r e a C h n € N <

Corollary 2.9 follows immediately from Theorem 2.8,

23

Theorem 2.8 is proven with the next four lemmas.

LEMMA 2.10. Let n > 2 and B = bi** , bn-l

C ^ M S S n- Then h(B)

is primitive.

Proof. Suppose h(B) is not primitive. Then we can express

h(B) as

h(B) = , ...

where n/p > 2. Thus B = b1-..bp ... bi*-*bp-iC a n d

therefore, using Theorem 2.2, n/p is either one or two.

Assume n/p - 2 . We have two cases.

CASE 1: Assume the parity of bj.-.b j is odd, i.e., there

are an odd number of R's appearing in b]_,,*bp_i* Then B =

b b b,...b ,C € MSS„ implies that b - L, for otherwise 1 P i P""1 11 P

b ...b b ...b is less than its right shift b1...b . Thus a 1 p 1 p i f f

• -1. This implies that = 1, since the parity of

h ...b 4 is odd. However, a - -1 by the definition of h. 1 P-1 A

Thus n/p B 1 and h(B) is primitive.

CASE 2: Assume the parity of b^-.b j is even. The

argument is similar to case 1.

LEMMA 2.11. Let n > 2, and B £ D in MSS^. Then h(B) and

h(D) are inequivalent colorings.

Proof. Let B = b„...b .C and D • d....d C. Suppose that 1 n-l l n—i

h(B) and h(D) are equivalent colorings. Express h(B) as h(B)

24

<e ,...,en>. Then 3 j, 2 < j < n, so that,

h(D)» . . . , e n , e i r . . . o r

h(D) - — 6 . i • •• i ^ 1' * * * ' e j-i> *

In either case,

D - V - b n - l b b l - V 2 C '

where b is L if e n - e± and R if eR t e f Observe that both

w = k^...bn_^b and v = b^. . "k^^bb^ • * j—2^j — 1 a r e shift

maximal. Now, w is primitive or has minimal subperiod n/2.

If w is primitive we contradict Theorem 2.3. If w is not

primitive, then j must be (n/2)+l and therefore B = D.

Hence, the result holds.

DEFINITION 2.12. Let n > 2, and <elf...,en> € {l,-l}n. Then

we define f(<e%,...,eR>) as follows.

4 (< e j , . . . , ) b j . . . bn_ ,

where

C R, if e t e b. - 1 1 1 + 1 . 1 2, and e = <e1#...,en> € {-1,1} be

primitive. Then,

25

(i) f(<elf...,en,e1>) is either primitive or has minimal

subperiod n/2.

(ii) If «(<e1 e

n'ei > ) = bl"- bn/2 br*- bn/2' t h e n t h e

parity of bj/*,bn/2 *

8 0<id*

Proof. Let w - b r - -bn =f(<e1

en'

ei > )' SuPPos® <*>

does not hold. Express w as,

w = b,...b b....b ... b1...b . 1 p 1 p P

Then e = -e,, since otherwise e would not be primitive. p+1 1

Thus, e now looks like

e = <e.,...,e ,-e ,...,—e ,e.,...»®_»••• 1 p i p i P

depending on the size of n/p, and therefore e n is either e p

or -e . Recall that b = b is obtained by comparing e to p n p 11

e r or ep to e p + 1. Thus, e p + 1 = -e1 implies that e n = -ep.

However, this implies that e is not primitive. Thus (i)

holds.

Next, (ii) is shown to hold. Assume the parity of

b ...b is even. Then e primitive implies that 1 n/2

e = <el en/2'~el'''"'~®n/2>'

However, the parity of b 1**, b

n/2 b e i n9 e v e n iraplies that

there must be an even number of "sign changes" in

el en/2' en/2+l'

Thus, = e n / 2 + 1- Now we have ea = * n / 2 + 1 ' ^ •

Therefore, (ii) holds.

LEMMA 2.14. Let n > 2. Then h| M s g is onto CLn-n

Proof. Let e • <e,,...,e > 6 {-l,l)n be primitive. We have 1 n

26

two cases.

CASE 1: Assume that f » • • • ' en '

e1 > ) = w = b i , , , b

n i s

primitive. If w is shift maximal, then B = € MSS^

and h(B) = e or h(B) = -e. If w is not shift maximal, then,

by Theorem 2.4, there exists some j, 2 < j < n, such that

b-.-b b4...b. , is shift maximal. Then D = 3 n 1 j-i

b^.-.b b4...b4 „C is in MSS„ and h(D) is equivalent to e. j n 1 j-2 n

CASE 2: Assume that f(<e^,...,en,e^>) has minimal period

n/2. The argument is similar using Lemma 2.13 (ii) and

Theorem 2.4 applied to

2 Periodic Points of f(z) = z -2

2

In this section the periodic points of f(z) - z -2 are

considered and are related to MSS sequences.

THEOREM 2.15. f n has exactly 2n distinct real fixed points

each of which is between -2 and 2.

Proof. Fix n and let z be a real valued fixed point of fn.

Then, z| < 2 implies that there is a u € [0,*] so that

z = 2cosu. Hence, fk(z) = 2cos2ku, k>l, and therefore one

need only solve cos2nt = cost for t € [0,*]. One finds,

n— 1 2* m / (2 -1) , m = 0,1, 2 , . . . , 2 /

2t m' / (2n+l), m' = l,2,...,2n-1 .

This completes the proof of Theorem 2.15.

27

2

I now classify the periodic points of f(z) = z -2. First

notice that for any real valued z and positive integer n.

n-l . ( f n ) ' (z) - 2 n II f ( z ) .

i=o

2

Thus, if w is a periodic point of f(z) = z -2 of period ra,

then the orbit of w is called positive if (fm)'(w) > 0 and

negative if (fm)'(w) < 0.

Now to each periodic point of f(z) = z -2 associate its

period. Then for each n € N, partition periodic points of

period n into orbits and orbits according to the sign of the

derivative. Hence, for each n € N, we have positive and

negative orbits of order n. Let q(n) be the number of such negative orbits of order n, n € N.

Fix n G N and let z be a fixed point of fn. One can show

that (fn)'(z) < 0 iff z « 2cost for t of the form

23T m'/(2n+l). Thus, q(n) < 2n_1/n. More accurately,

q(n) » ( 2 n 1 - f R ) / n »

where,

€ n =

n an odd prime,

n = 2m, m = 0,1,2,...

n E q(n/d)/d , otherwise. din

d odd d>l

28

Note that I have derived q(n) in much the same way as

Myrberg [35], and include the derivation to more clearly

present my own work that follows.

I will first show that q(n) = |MSSn| for all n, using a

number theoretic argument. For n an odd prime, Theorem 2.5

gives that |MSSn| • p(n)/2. Thus,

MSSj « [2n 1 -l]/n = q(n).

Similarly for n a power of 2, the argument requires no

induction. I now consider the "otherwise" case.

THEOREM 2.16. For every s € N the following holds. If k > 0

and n = 2kp ...p , where the p. are odd primes not I S

necessarily distinct, then |MSS^| = q(n).

Proof. Induct on s. First assume n is of the form n » 2 p1,

k > 0. One only need consider k > 1. Now,

k+1 . i ,. k+1 1p—i+l MSS | - E 2 p(2 p.) = (l/2n) E P (2K P l).

m i = 1 1 i = 1

We must show that

k+1 t _ u 1

(*) E P (2 p,) - [ E P (s) ] - 2c . i=l ~ s|n

However,

29

k+1 fn = n 5 q(n/d)/d = (n/p ) IMSS . 1 = 1 T P(2 ).

d n 1 1 2 2 i=l d odd d>l

It now follows that (*) holds.

The induction step is straightforward and patterned after

the case s = 1. Hence I omit it. This completes the proof

of Theorem 2.16.

Next I give a bijection from MSS^ onto the negative

orbits of order n. The following theorem is stated in order

to help clarify how the negative orbits are going to be

represented.

THEOREM 2.17. For each n € N, define p from the fixed r n

points of fn Into <l,-l)n as follows:

'n(2) " <eo'el en-l>f

where

1. if fa(z) > 0 e, = 1 )-1, if f1(z) < 0.

Then pn is a bijection.

The following two facts follow from Theorem 2.17.

(1) If w is a periodic point of f of period m, then p (w) r m is primitive.

30

(2) If a, ft € {l,-l}m are primitive and (I is a cyclic

permutation of a, then P m~1(h is in the orbit of

Cm"1**)'

Thus if e = < eQ,...#em_1> € is primitive and

m-1 II e. < 0, then e or any cyclic permutation of e can be used i=o 1

to represent the negative orbit of Fm_1(e). For example if m

= 3, one can use <1,-1,1>, <-l,l,l>, or <1,1,-1> to represent

the negative orbit of p--1(<1,1,-l>). O

THEOREM 2.18. Let n € N, n > 1. Define g from MSS into the n

negative orbits of order n as follows.

n-1 <ej,...'e

n_1,-l>f if H e^ ^ 0 ,

9 ( b1 * * "bn 1C> = i 1 n-1 ) n-1

<e ,...,e ,1>, if II e. < 0 , i=l 1

where e. = J 1' bi " L

' -1, b4 - R.

Then g is a bijection.

Proof. Let B - b i " - bn _ i

c € MSSn. Then both ^...b L and

bl'--bn_l

R are shift maximal. Thus, using Theorem 2.2. and

n the fact that II [g(b. . . .b C) ] . < 0, it follows that

i=l 1 n i l

g(bl••-bn_i

c) i s primitive. Hence, the range of g is as

31

claimed.

Suppose D 6 MSSn with D f B. Then Lemma 2.1 and Theorem

2.3 imply that g(D) is not a cyclic permutation of g(B) and

therefore that g is 1-1. That g is onto follows from Theorem

2.4, and Lemma 2.1. For n = 1 one simply sends C to <-l>.

CHAPTER III

UNIQUENESS FOR H (P)

Chapter three is divided into two sections. The main

theorem of this chapter is stated in section one and

necessary background for the theorem is given. Also it is

indicated that an extension of the main theorem to include

all aperiodic shift maximal sequences would prove that the

set (defined and discussed in Chapter one) is dense in e

[0,1]. A proof of the main theorem is given in section two,

Section One

For convenience, throughout Chapter three all MSS

sequences will be expressed without the terminal C; even

though one must use the C when working with the

parity-lexicographical order. The extended

parity-lexicographical order is not used in Chapter three.

For example, the MSS sequence RLLRLC would be written RLLRL.

Recall that to say an MSS sequence P has odd (even) parity,

or more simply to say P is odd (even), means that an odd

(even) number of R's appear in P.

DEFINITION 3.1. Let P = PjPg.-.Pfc € {R,L}k be an MSS

sequence. Set,

32

33

H (P) L H (P), if H ,(P) is odd, H,(P) = { n 1 11-1 n~ 1

Hn_l<

p) R Hn_i(P)» if Hn-l*P* i s e v e n '

where

HQ(P) = P.

Then H n(p) is called the n**1 harmonic of P.

For example, if P = RL then H ^ P ) = RLLRL. Note that if P €

{R,L}k, then Hn(P) € {R,L)2 ( k + 1) _ 1. T h e following are three

well known facts about the harmonics [9,11,30]. Let P be an

MSS sequence.

FACT 1. Hn(

p) is shift maximal for all n.

FACT 2. Let n > 0. There does not exist an MSS sequence Q

so that

H„(p) < Q < Hn+1(P).

in the parity-lexicographical order.

FACT 3. Let n > 0. Then

V p » <

REMARK 3.2 A historical comment is in order. Beyer and

Stein [4] remark that Feigenbaum [15] and others call H (P) n

the "subharmonics" of P. However, Beyer and Stein retain the

term harmonic. Myrberg's work [35], which preceeds the MSS

34

paper [30], involves the MSS sequences and the harmonics,

although he does not use this terminology.

DEFINITION 3.3. Let P be an MSS sequence. Define H (P) to 00

be the unique element in {R,L}N that is the common extension

of the harmonics of P.

For example,

Hq(R) = R

H1(R) = RLR

H2(R) = RLRRRLR

H3(R) = RLRRRLRLRLRRRLR,

and H (P) begins as

H (R) = RLRRRLRLRLRRRLR

One can easily see that if P is an MSS sequence, then H (P)

is shift maximal.

LEMMA 3.4. Let P be an MSS sequence. Then H (P) is 00

aperiodic, i.e., there does not exist a positive integer s

and A € {R,L}S so that H (P) = A°°.

Proof. Suppose there exists a positive integer s and A €

S (B

{R,L} so that H^fP) = A . Without loss of generality we may

assume that A is primitive. Express P as P « P ^ g • • • P^ and

A as A = A1 A« ... A . A simple comparison of H (P) and A00 A & S (X)

shows that s does not divide (t + 1).

CASE 1. Assume (t+1) > s. Comparing

35

^ ~ ^1^2 * * * * * * 1 ••• Aj ...A^Aj...Aj j ...

H (P) = P P P R P P n L n® i r' *1*2 *•• t L 1 2 Pt R ' *

we find PJ = A^ with 1 < j < s.

Thus, A j • • • Ag = Aj . . ,AsA1.. . This contradicts A being

primitive.

CASE 2. Assume (t+1) < s. Let n » min {i | 21{t+l)-l > 2s>.

Note that n > 1. Now, Hn(P) = Hn_1(P) £ ^ ^ ( P ) , depending

on the parity of P. For simplicity express H n j(P) as

**n-l(P) = Hi H2 ••• ( k = 2n 1(t+l)-l ). Then s < k+1 <

2s+l, and as before s can not divide (k+1). First, if k+1 <

2s+l, by comparing

A =A 1A 2...A gA 1 ... A^ . . .A g lA sA 1. . .A^j^. . .

Hm(P) - «iH2 ... H k ^ H l ... H k £

we find that A^ = Hj for some j with 1 < j < s. Since k > s,

we have

( ) ... Ag •* A j ... Aj A . . • A j ^,

which is a contradiction. Lastly, if k+1 = 2s+l then again

(•) holds, with j = 2. This completes the proof of Lemma

3.4.

MAIN THEOREM. Fix e 6 (0,1/2). Let P be an MSS sequence.

if Then there exists a unique >1 € [0,1] such that I e(i) •

H (P) . oov

36

REMARK 3.5. Let me comment on my interest in this theorem in

general terms. It is widely believed, as noted in Chapter

one [18, 9, pp. 31, 69], that * is dense in [0,1]. For the

family is(x) it is known [9, pp. 13; 12, pp. 74] that there

exists at most one stable periodic orbit for each X .

Moreover, is(x) has a stable periodic orbit if and only if

/Is I {X) is finite or periodic [9, pp. 69]. Thus v> consists

8

of precisely those ,1 such that is has a stable orbit. The

following similar results hold for the family ^fe.

(1) If i £ e, then -if has at most one stable orbit. V?

Moreover, if X > e and if X f & has a stable orbit,

then the orbit is superstable.

(2) Let i > e. The map if has a superstable orbit or

a finite orbit containing either e or 1-e if and

i f

only if I e ( X ) is finite or periodic.

Statement (1) and the "only if" part of statement (2) follow

from elementary observations. To see the "if" part of 4 f -

statement (2), suppose that X > e and that I [X ) is

periodic of period k. The map (if is monotone on k ^ f

e v [Uf e) (X), >1] for all j > 0, since I ( (if ) (i ) ) = ,if e

0 j

I (-1). If for some j > 0 we have that {>1 f eJ i ) € [e,l-e]

or that (if )^(i) = X, then we would be done. So, assume

not. Then, (recall that X/e > l) there exists a positive

integer m so that the diameter of the interval A, which is

the image of [(ife)^(i), X] under (if^)™, is greater than

37

one. This is a contradiction.

Thus, as with the family /Is, consists of those values e

of >1 such that if has a stable orbit. e

Let g be a unimodal map on [0,1]. One way to try to

establish that is dense in [0,1] is the following. It is

/I g shown (in Lemma 3.10) that if 0 < Jl < Jl < l with I (J ),

1 2 1 2g N

I distinct elements of (R,L) , then there is some X € * o

^ o®

s o that I (/lQ) is finite. Hence, we have the

following theorem.

THEOREM 3.5. Let g be unimodal. If for each aperiodic shift

maximal sequence A there is at most one scalar J in [0,1] \ a

such that I 9(J) « A, then is dense in [0,1].

Now, suppose further that g is Lipschitz continuous,

round-top concave, and exhibits uniqueness. Let A be an

aperiodic shift maximal sequence. Then, Theorem 1.3 implies

that there exists some X 6 [0,1] with I*9(JI) = A. (Theorem

1.3 does not require uniqueness; however uniqueness is used

in Theorem 3.12.) Next, the remaining lemmas and theorem in

this section establish that

*A,g = { X 6 [ 0' 1 ] I ) = A }

is either a singleton or a closed interval. (This fact will

be used in the proof of the Main Theorem.) Thus if $• were A, g

38

shown to be a singleton for all aperiodic shift maximal

sequences A, then the set would be dense in [0,1]. Of

course, both s(x) = 4x(l-x) and f are unimodal Lipschitz

continuous round-top concave functions. My work involves the

function f^, not s. Possibly some of the techniques

developed here will prove useful in showing that ? s is dense

in [0,1]. Theorem 3.8 is taken from BMS [2].

REMARK 3.7. The cardinality of the set of aperiodic shift

maximal sequences that are not of the form H (P) for any MSS 00

sequence P is C • In fact, the sequences generated from the

tree given in figure three of Chapter one are all aperiodic

and are not infinite harmonics.

THEOREM 3.8. Let f be a unimodal Lipschitz continuous

function that has a continuous derivative in a neighborhood

of x — 1/2. Suppose 0 < X 1 < \ 2 < 1 and A is a shift maximal

sequence other than L , C, R00, or RL00. Suppose further that

I (^x) < A A > I (j2). )

In the proof of Theorem 3.8 BMS prove the following lemma.

39

LEMMA 3.9. Let be a unimodal Lipschitz continuous function

that has a continuous derivative in a neighborhood of x =

X f 1/2. Suppose /lQ € [0,1] is such that I ° (J ) is finite.

XQf °

SaY» 1 ° U Q ) 9 PC, where P € {R,L}n for some n. Then there

exists an open interval U C [0,1] containing X so that if )i o

is in U, then

I i f ( X ) € < PC, (PR)®, (PL)00 }.

REMARK 3.10. Let f be a unimodal Lipschitz continuous

function that has a continuous derivative in a neighborhood

of 1/2. Moreover, assume that f exhibits uniqueness. Then,

by Theorem 3.8, exactly one of the following hold for all i ,

^ 2 € [0,1].

M M (1) If 0 < < J2 < l and I

1 (Jj), I 2 (X2) are

V V both finite, then I {X ±) I *

For f • f , (l) holds.

LEMMA 3.11. Let f be unimodal,

0 < ''i < K < 1.

V V N

and I (J ), I (/12) be distinct elements of {R,L) . Then

the following holds. X f

There is some € Uj.Jg) such that I ° (XQ)

40

is finite.

X.f X.f Proof. Let k be the first index where I (X i ) and I (X )

differ. Set,

T « sup ( J € (J 4.J 2) | I H ( J ) agrees with I 1 (J )

in the first k positions >.

Then

< 7 < ^ 2'

7 f

and I (7) is finite, since otherwise the definition of 7 is

contradicted.

THEOREM 3.12. Let g be a unimodal Lipschitz continuous

round-top concave function that exhibits uniqueness, and let

A be an aperiodic shift maximal sequence. Then,

* k t Q = U € [0,1] | I^g{i) = A }

is either a singleton or a closed interval.

Proof. Let

u - sup { X € [0,1] | I^9(JI) - A >,

and

1 = inf { X € [0,1] | I*g{X) = A }.

Suppose that 1 < u. Then Lemma 3.4 and Lemma 3.9 imply that

both I 9(u) and I^9(l) are not finite and therefore must both

be equal to A. If I * g ( X ) - A for all jl € [l,u], then we are

done. Suppose there is some X € [l,u] such that I^9(/l) # A.

Then, by Lemma 3.11, there exists some X € [l,u] such that

Xo g

I ( X Q ) - Q is finite.

41

CASE 1. Suppose Q > A. Then, using Lemma 3.11, there is

some 4 > u so that

Q > I ^ U ) ,

i G

and I (<l) is finite. This contradicts Remark 3.10.

CASE 2. Suppose Q < A. The argument is similar to case one.

This completes the proof of Theorem 3.12.

Section Two

To begin with, some notation is defined. Given e €

(0,1/2) and an MSS sequence P, let 4 p denote the number in

[0,1] so that

V e I P e U p ) = P.

Fix e € (0,1/2) and let P be an MSS sequence. First note

^pf

that 1/2 is a periodic point of /l_f with I *(>!„) = P. p e P

Then as we increase A , /lH is the next value of /I such

that 1/2 is a periodic point of X t , X „ is the next value e 2

after such that 1/2 is a periodic point of f#, and so

on. Let,

Then,

X = lim . CD H (P)

n-to n1 '

- VP>"

We will show that X is unique, i.e., if J f. X then I e(X) CD 00

£ **00(?) . In order to show this we construct a sequence of

MSS sequences { Dn(P) ) n > 2 such that the following hold:

42

U ) < ••. < D4(P) < D3(P) < D (P), and

(2) if Q is an MSS sequence so that Q # D n(p) f°r

all n and Q > H ^ P ) , then there is some m > 2

such that D (P) < Q.We let m

*oo * l l m ( P )' n-too n

and show that

Note that

X = 6 CD 00

6 f , oo e

U ) CD

= H (P) 00

and that if X^ < X < S^, then I e(J) = H (P).

Once we have that X = S , we will know that X is unique 00 00 00

since is either a singleton or a closed interval. oo e

See Figure four for a picture of the above.

Fig. 4 Geometric view of X and S oo oo

DEFINITION 3.13. Let P be an MSS sequence. For each n > 2

set,

43

D (P) = n*

H (P) L H (P) , if H (P) is odd, n—l n—2 n-l

Hn-l { P ) R Hn-2(P), if H , (P) is even. n— i

THEOREM 3.14. Let P be an MSS sequence. Then D n(p) i s a n

MSS sequence for all n.

Proof. Fix n > 2. It suffices to show Dn(P) is shift

maximal. There are two cases.

CASE 1. Assume that H ,(P) is odd. Then, n-l

H (P) = H _(P) R H _(P) L H „(P) R H „(P) n n-2 n-2 n-2 n-2

D (P) = H n(P) R H 0(P) L H ft(P), n n-2 n-2 n-2

with H _„(P) even. We need to show that D (P) is greater n & n

than or equal to all of its right shifts. For simplicity,

let a1a2...a_t denote H

n_ 2(p)' w h® r® € {R,L} for all i.

Three cases are considered, the others are clear..

CASE 1, a. We have

Dn(P)° > ... atC

for 1 < j < t, since H n(P) is shift maximal.

CASE 1,b. Suppose,

odd D (P)C = a ... a R1 a ... a .. L a ... a C n l t l ii x v shift = a. ... a.La, ... a. . ... a^C,

, 3 t 1 J-l_, t even

for some j, 1 < j < t. Then, since H n_ 2(p) i s even» the

44

parity is as shown and therefore

al a2 • • • ^t ^ ^ ®j • • • 3^ Ei cij • • • 3 .

Thus Dn(P)C > shift , since Hn(P) is shift maximal.

CASE 1,c. Suppose,

^n^P)C R ••• 3^ L 3^ ...

shift — 3j • • • • «3j

for some j, 1 < j < t. Then Dn(P)C > shift, since Hn(P)C is

shift maximal and a ^ is either L or (exclusive) R. This

completes case one.

CASE 2. Assume that Hn_^is even. The argument is similar to

case one. This completes the proof of Theorem 3.14.

REMARK 3.15. Let P be an MSS sequence. One can easily check

that (1) and (2) hold for the sequence ^Dn(

p))n>2*

Fix e € (0,1/2), and let P be an MSS sequence. Both e

and P are fixed throughout the rest of Chapter three.

DEFINITION 3.16. For each /I in (0,1] set

-1

and

Fjl > R<X) = l-(e/i )x,

-1 FJI ,L{K) ~

45

for x in [0,1].

See Figure five for a picture of the two functions

defined above.

1

l-£ -t

-I

4 L. x,\-

X 1 x

- l - l Fig. 5 Graphs of F^ R and Fj L

DEFINITION 3.17. Let n € N and A - A.A. ... A € {R,L}n. X il

Then,

(i) |A| = n,

(ii) p(A) = { j € N | A » R },

(iii) aj(A.x) -

for each X € (0,1], and

-1 ( F» , (x) )

* n • ) ) ,

(iv) gA(^) - Qx(h,l/2), X e (0,1].

THEOREM 3.18. Let A = A ^ ... A r € (R,L}nand X € (0,1]

46

Then

Gj(A,x) = S ( - 1 ) j€p(A)

l<"\ •VI 1 ( e / \ )

j-1

j / > ( A ) | | Aj + (-1) (e/^) x.

Proof. Simply induct on the length of A.

A f

Notice that I e(/l) = L00 for X < 1/2. For the rest of

Chapter three we will assume that X € [1/2,1].

LEMMA 3.19. For every e > 0 there exists a positive integer

Mf so that if n > M then

g Hn(P)

for all } in [1/2,1].

Proof. Recall that,

U ) - g (J) | < e Dn(P)

Hn(P)

and

D (P) nx '

Hn-l L Vl(P>

V l R Hn-:l

Hn-l(P» 1 H„-2

V l < p » R V 2( p )

if H (P) Is odd

if is even,

if i s o d d

if Hn_i(P) i s even.

Let X be in [1/2,1]. Then,

47

j (4) - g U)l < Hn(P) Dn(P)

P n H J 3-1 (1/2) ((eAl)' n i + (eAl)' n l ) + E (e/A)

j€p(Hn(P))

j>|Dn(

p)

Thus,

g U ) - g U ) | < £ (e/^)J

H n ( P ) V P > J>|Dn<P)|-1

j < E (2e)

3>|Dn(P)|-l

£ 3 The result now follows, since E (2e) < oo . This

j=o

completes the proof of Lemma 3.19.

The following four lemmas are technical lemmas for the

proof of the Main Theorem.

LEMMA 3.20. Fix n € N, n > 2. Then,

g (J ) - X > 0 D (P) H . (P) H , n n+k n+k(P)

for k = 0,1,2,3,....

Proof. We will show the case when H (P) is odd. The case n

48

when Hn(P) is even is similar. For simplicity, since P is

fixed, we will write H for H (P) and D for D (P). We have m m m m

D » H , R H ., n n-1 n-2

since H n is odd. Note the following.

Hn " V l R »n-2 L Hn-2 = Dn L Hn-2'

"-i, * H_ L H - D L H „ L H , n+1 n n n n-2 n

Hn+2 = H n +1 R Hn+1 = Dn L "n-2 L H „ R Hn +1

For each s > 0 let be such that H n + g = D r Q . For

example, if s = 0, then Q = LH Notice that Q starts o n-2 s

with an L for all s. For simplicity, express Dn as

d ^ g . - . d ^ j. Next, fix k > 0. Then,

^H = 9H ( ) - GJ ( H ,1/2 ) n+k n+k Hn+k „ n + k

n+k

Thus,

\ + k = % < D- 9 k' 1 / 2' = X ' D°'GJ„ < Qk' 1 / 2 ) > n k n+k n+k Hn+k

Hence,

49

1 V , J P ( dl d1)l"1 3'1 H ^ *e/^H ^ n + k j€p(Dn) n+k

|/®<D )| | D I + (-D (e/J )' n 6. ( Q 1/2)

"n+k H , k

n+k

while

l/o (d, ... d .) I -1 j-i gD H J = S (-D j (e/J- )

n n + k j€/» (D ) n + k

P(D )| ID I + (-1)' n ' (eA»„ )' n (1/2)

n+k

Thus, 9JJ ( JJ ) - /l„ is equal to n n+k n+k

k(Dn}l lDnl (-1) (e//,H_J ( 1/2 ~ Q

/l„ (Qk>l/2) ). n+k "h .

n+k

„ fe Now, I n + k (J„ ) . H , (J f ) (ijj ) not In

n+k n+k n+k [e,l-el for 3 ' 0,1 |Hn+k| - 1, and g (J ) =

n+k n+k ^„ imply that n+k

a, (Qk,i/2) = (j fe)|Dnl(JH ).

n+k n + k "+k

However,

50

' X + k ^ ' ' D Hn+k' ' 1 / 2'

Thus,

gn ) " *u > °-n n+k n+k

LEMMA 3.21. Fix n € N. Then,

9 u »I ) ^ ^ u

n n+k n+k for k = 1,2,3,....

Proof. Again, we will show the case when Hn(P) is odd. The

case Hn(P) even is similar. As in the previous proof write

H m

for H (P). m Recall that

Hn+1 = Hn L H n

Hn+2 = Hn L H R n H L

n H n

Hn+3 " H„ L H R n

H L n H L H L H R H L H , .

n n n n n

For s > 1 let Rg be such that H n + g - Rg Hn> For example if s

= 1, then Rj = HnL. Fix k > 1. Then,

, " gH . (iH ] = G,l ( Rk Hn' 1 / 2 ) = n+k n+k n+k H , K n

n+k

% " V X (Hn.l/2)>.

n+k n+k

Notice that

< \ + k f e , ; l ( \ + k ) ' [ e' l" e I f° r 3 ' 0 , 1 IHn+kI " l-

•J H fe

I n+k (. a n d

n+k n + k

U H ) < for j - 1,2 IH .. I-1. n+k e n+k n+k 1 n + k l

51

Thus,

gH (/lH } = Gi (H ,1/2) n n+k H . n

n+k

|R. I

n+k e n+k n+k

LEMMA 3.22. Fix n € N and suppose that >1 < S . Then, oo oo

9jj ^ ( ) ^ n n

for \ € [/I ,6 ] . CD CO

Proof. Suppose that \ e ^oo'^oo1 l s s u c h t h a t { X ) * ^ * n

i , Then U f e ) (>l ) is not in [e,l-e] for all j, since I e(/l ) =

Hoo(P) w h l c h l s aperiodic. However, g„ U ) = J and (Jf n e

\f not in [e,1-e] imply that I ®(i) = H . This is a

n

contradiction. Similarly one sees that g D (J) f } for /I in n

00 00

LEMMA 3.23. Fix n € N and suppose that \ < $ . Then 00 CD '

g D (^) > J and g ( \ \ n n

for all /} in [J ,6 ] . oo 00

Proof. Suppose there exists a Jl in ] such that

g D U ) < * , n

we know g (A) f } by Lemma 3.22. Now, n

- + X ,

n+k k-to 00

52

g D continuous, and n

gn ^ h ) > ^ u f o r k = 0,1,... n n+k n+k

imply that there exists a J c [Jl ] so that g_ {\ ) = H . O 00 00 D o o

n

This contradicts Lemma 3.22. The argument is similar for the

function g H (X ). n

MAIN THEOREM. There exists a unique ^ € [0,1] such that

/If

Proof. Recall that

and

Suppose that

I e U ) = H J P ) .

= " m JH n-to n

s°> " ^ ''d ( p )

n-to n

X < 6 . 00 00

Then, by the previous lemmas,

gD (P) ^ ^ ^ and g^ (pj ^ ^ ^

for all J in Let e > 0 and M a positive integer

that n > M implies

K ( p ) ^ ~ 9d {P)( iH < e/2 ' nx nx '

for all ^ in [1/2,1]. Hence,

g H (P) " g D (P) ' loo ~ L ° n n n-too

on I W - Thus- II 9h <•" - J IL -1 0 on ' W -n n-Joo

Let {h±}±>x denote H q o(P) . Define

53

|^(h1h2 ...h^)| - 1 j-i 9 ^ ) = S (-1) J (e//|,

jGl^P)

for X in [1/2,1]. Then

I9*'0 ~ gH (P)(/I)l ^ S (e/J)3 1

n J^(Hoo(P))V(Hn(P))

|H (P) + (l/2)(e/J) n 1 , for X in [1/2,1]

Thus,

j-1 1 £ (2e)

n' |9(^) ~ 9jj (P) M — ^ (2e)

j€^(Hoo(P))V(Hn(P))

iHn(P)l + (1/2) (2e) , for X in [1/2,1]

In particular, on [J 1 we have CD 00

g " gH (P) I loo iL °* n' n-»oo

Hence,

g U ) = X

o n This is a contradiction, since after making the

change of variable t = e/X we would have a power series equal

to a constant on a closed interval. Thus X = S . This CD 00

completes the proof of the main theorem.

CHAPTER IV

BIFURCATIONS

Introduction

Bifurcation theory studies the changes that maps go

through as parameters change. For an introduction to this

theory see Devaney [12] or Guckenheimer and Holmes [20].

Chapter four is concerned with two specific one-parameter

families; namely, >ls(x) = Ux(l-x) and if (x), >1 € [0,1] €»

(throughout Chapter four it is assumed that e 6 (0,1/2) is

fixed).

4-

X \

Fig. 6a Regular period doubling

Fig. 6b Reverse period halving

Block and Hart [5] have shown that only period doubling

or period halving bifurcations can occur for ^-continuous

families of maps on compact one-dimensional spaces. Thus the

54

55

family /Is can exhibit only period doubling or period halving

bifurcations. Nusse and Yorke [36] point out that the family

Jlg(x) = M l - (1/2) (x2 + x4)) With J € [0,2] exhibits both

regular period doubling and reverse period halving 2

bifurcations. See Figure six. The map g(x) = 1 - (1/2)(x +

x 4) does not have an everywhere negative Schwarzian

derivative, where as s(x) = 4x(l - x) does. If ( x0"* 0)

i s a

3 § bifurcation point for the family ^s with = - 1'

then (since the Schwarzian is negative) only a regular period

doubling or regular period halving bifurcation can occur (see

Guckenheimer and Holmes [20], or Whitley [46]). It is

believed [36] that the family ^s exhibits only regular period

doubling bifurcations. However, Nusse and Yorke [36] show,

by example, that an everywhere negative Schwarzian derivative

is not sufficient to guarantee that only regular period

doubling bifurcations occur. Section two of Chapter four

establishes that only period doubling or period halving

bifurcations can occur for the family ^ ^ [0,1].

A relationship between uniqueness and the type of

bifurcations that can occur has not been established. The

following question naturally arises. To my knowledge this is

an open question.

QUESTION. Let f be a unimodal map that exhibits uniqueness.

Does uniqueness imply that only regular period doubling

bifurcations can occur? If not, does there exist a parameter

56

* *

value X such that if X > X then only regular period

doubling bifurcations can occur?

2 4

If the one-parameter family ^g(x) = (1 - (l/2)(x + x )) with

i € [0,2] does exhibit uniqueness, then obviously the answer

to the first part of the question is no. Uniqueness has not

been established for the family ig.

In the literature [4,18,26] there exist computer

generated bifurcations diagrams for the families >ls and

X € [0,1]. When analyzing the bifurcation diagrams for these

families the following property, which will be referred to as

property G, appears to be present.

Let g represent the map or the map s. If P is an MSS

sequence and /ip and X H ( p ) are such that I ( X p ) - P

VtP)9

and I 1 (JH ( p )) - H1(P), then there exists a unique

1 in (ip,JH ( p )) so that for X in the map Xg has

a stable periodic point of period the length of P and

for X in ( 7 ^ H ( p )) the map ig has a stable periodic

point of period twice the length of P.

Some results of Beyer and Stein [3,4] on the bifurcation

diagram of f deal with property G above. In particular,

their work on what they refer to as the contiguity of the

57

harmonics appears to establish property G for fg. However,

there is some confusion concerning the completeness of their

proof. In section three I will attempt to point out the

difficulty in their argument, as I understand it, and to

present partial results of my own. I have been able to

establish that property G holds for P of the form RLn, n > 0.

I am unaware of any results for the family is.

Bifurcations

The following lemmas and theorem establish that only

period doubling or period halving bifurcations can occur for

the one-parameter family /lf0, X € [0,1]. The lemmas and

theorem have been taken from Block and Hart [5] and modified

slightly, since f is not a C1 function. First note the

following:

(i) if Jl € [0,e) , then 0 is an attractive fixed point

that attracts [0,1],

(ii) if X = e, then e is a fixed point that attracts

[e,1-e], each q € [0,e) is a fixed point that

attracts 1-q, and

(iii) if X € (e,l-e], the i is a super stable fixed

point that attracts (0,1).

DEFINITION 4.1. For each i > 1 let

F. - { (JfJ 1 I € (1-e, 1] }, JL © I

and

58

F = U F. i>l 1

The proof of the next lemma follows from elementary

observations.

LEMMA 4.1. Fix 1 € N and X € (1/2,1]. Let x € [0,1]. Then

exactly one of the following hold.

(i) ( U fe )

a ) ' ( x) d o e s n o t e x i s t

(ii) ( (JfJ 1 )'(x) " °-

(i ii) | ( (J^) 1 )'(x) | = (i/e)1 > I-

LEMMA 4.2. Let f € 9 and (p^Pg. ••• 'Pk> b e a Periodic

orbit of f of period k where k > 2 and p 1 > 0 a n d f' < Z ) "

-1 .

Proof. For some integer m with 1 < m < k, either *(Pm) ~ Pj

or f<pm> - Pk. If t(pm) - p r then there exists y 6

( EV pm+l' s o t h a t f' ( Y ) * °' I f £ °*

Next, let i be the smallest element of {l,2,...,k} with

f(p^) < p^. Then i > 1, f(P^) £ Pi-i' an(* ^^i-1^ ~ ^i*

Thus there is some z in (Pj^j/Pj^ s o ***at ^ < kemma

one implies that f'(z) < -1.

LEMMA 4.3. Fix i € N. Let f and fR, n > 1, be elements of

59

9^ such that

<*> II fn " f I loo 1 °' a n d

n-ta) (ii) for each n, x r is a periodic point of f R of period

k, where k is a fixed positive integer with k > 2.

Let x € [0,1] be such that x -• x. n-to

Then x is a fixed point of f^ but not a fixed point of f.

}£ Proof. By continuity, x is a fixed point of f . Thus we

only need show that x is not a fixed point of f.

Suppose that f(x) = x. For each n let pn denote the

smallest element and q n the largest element of the orbit of

xn« By taking subsequences if necessary let p,q be such that

D -4 p and q -f q. There are positive integers l,m n . n „ .

n-to n-to

such that (fn)1(xn) = P n and (fn)

m(xn) = q n for infinitely

aJL IXfc

many n. Hence, by continuity, f <x) = p and f (x) = q.

However, f(x) = x implies that x = p = q. Lemma 4.2 implies

that for all n there are points Yn»zn i n fPn'

qn-' s o t h a t

f «(yn) > 0 and f

n'(z n) < -!• Next, choose 6 > 0 so that

exactly one of the following hold for all z with |z - xj < 8.

(1) If X € (1-e,1] and ( Ufg) 1 )'(z) exists, then { Uf e)

A )'(z) > 0. (2) If I € (1-e, 1] and ( {If J 1 ) ' (z) exists, then

©

( Ufg) 1 )' (z) < o.

Lastly, choose n large enough so that |pn -x|, |qn - x| < 6.

Then we can not have *n* (Yn) > 0 a n d f

n'^Zn^ - - 1* Tllis i s a

contradiction. Thus f(x) £ x.

60

THEOREM 4.4. Let i € N. Let f and fn# n > 1 be elements of

such that

1 1' " fH°° „ 1 °' a n d

n-»oo (ii) for each n, x is a periodic point of f of period

n n,

k, where k is a positive integer.

Let x € [0,1] be such that some subsequence of ( xn) n>i

converges to x.

If k is odd, then x is a periodic point of f of period k and

if k is even, then x is a periodic point of f of period k or

k/2.

Proof. There are three cases.

CASE 1. Assume that k is odd and k > 3. If k « 3, then the

conclusion follows from Lemma 4.3. Assume that k > 3. Lemma

4.3 implies that x is a periodic point of f of period r,

where 1 < r < k and r divides k. Hence, k = rs where s is an

odd positive integer.

Let g = (f )r and g = fr. Then g^ -+ g and for each n n n n-to

n, x n is a periodic point of g n of period s. Suppose that s

> l. Lemma 4.3 implies that x is a fixed point of gS but not

of g. However, g = fr and x a periodic point of f of period

r imply that g(x) =» x. This is a contradiction. Thus s is

not greater than one and therefore s = 1 and k = r. Hence,

case one holds for all i e N.

CASE 2. Assume that k = 2s for some integer s > 0. If k =

1, 2 then the conclusion is immediate; assume that k > 4.

61

Let g R - (fn)

k / 4- T h e n f o r e a c h n' x n l s a P e r i o d l c point of

of period 4. Lemma 4.3 implies that x is a periodic point

of g of period 2 or 4. Hence, x is a periodic point of f of

period k or k/2. So case two holds for all i € N.

S V CASE 3. Assume k = mr, where r = 2 for some s > 1 and m is

odd with m > 3. Then (f„)r and xR is a periodic n-*»

point of (fR)r of period m, for all n. Hence, by case one, x

is a periodic point of fr of period m.

Similarly, (f )m -» fm. Thus, x is a periodic point n n-to

of fm of period r or r/2, by case two.

Let t denote the period of x as a periodic point of f.

Then, (1/2)r and m are relatively prime, (l/2)r and m divide

t, and t divides k. Hence, t » k or t - k/2. This completes

the proof of Lemma 4.4.

Uniqueness and Bifurcations

Fix e € (0,1/2). One way to show that property G holds

for the family Jlf is to establish the following.

Fix x € [e, 1-e] and P - a n M S S sequence of

length n. Then there exists a unique >l(x,P) € [0,1] so

(f) that x is a stable periodic point of ^(x,P)fe of period

i (x, P) f

n with the property that I ^x^i = Pi f o r 1 ~

1,...,n-l.

62

Whether or not (f) holds for f is an open question [27].

Beyer and Stein [4] used the above in their arguments. They

state that the above "will be shown elsewhere". However, ("f)

has not yet been shown to hold. I have only been able to

show that ("J") holds if P is of the form RLn, n > 0. In what

follows I present my results.

For convenience for each MSS sequence P let Jp denote the

V e

value in [0,1] such that I (.Ap) = P. Also, for an MSS

sequence P - PjPg . . . o f length n let

P„Prt P ,R, if P is even _. . , 1 2 n-1 P(n) -

PjP2 ... Pn_jL, if P is odd.

The following definition and lemma will be used.

DEFINITION 4.5. Let n € N and P 1^ 2 ••-pn e {R,L}n. Then

hD o D <*) = x fD <xfP ( ••• x fP ( x ) •••))' 1 2 ' * n n n-1 F1

where fD(x) = (l/e)(l-x), fr(x) • (l/e)x, and x € R. K JU

LEMMA 4.6. Let n > 2 and P, ... P„ € {R,L}n. Then, ~ i n

xhp p p (x) — hp p (x) 1 2 * # n 12" * n

63

n-2 , n + S ( X j ) ( h _ p ( x ) ) ( H f p ' ( h p p (X))

j= l 1" ' n - j s = n - j + l P s *l'mm s - 1

n + x n h p •(x) H f p ' ( hp p (x)) ,

P 1 s=2 P s * V , , 4 s - l

where i f n = 2 the middle sum i s assumed t o be z e r o .

P roo f . Induct on n . Cases n = 2 ,3 can be e a s i l y checked.

Assume the r e s u l t ho lds f o r a l l k jC n . Then

h_ 0 (x) = x f (hp ( x ) ) , and so 1" ' n+1 n+1 1 " n

hp p Mx) = f p (h p (x)) + 1 ' ' n+1 n+1 P l " n

x ( f p ' ( h p p ( x ) ) ) ( h p p » ( x ) ) . n+1 1" " * n 1 * * * n

Thus,

xh p 1 (x) « h p (x) +

P l " n + 1 1 * " n+1 ( x f p ' ( h p (x)) )( h (x) +

n+1 1 * * n p l , , , p n n-2 . n

S ( x 3 hp p (x) n f p *(h (x)) ) j= l 1" ' n - j s - n - j + 1 r s r l * * - r s - l

n + x n hp ' ( x ) n f p ' ( h p p ( X ) ) )

P 1 s=2 p s • h (x) + ( x f ' ( h p (x)) ) (h p (x))

1' * n+1 n+1 1 * * n P l " - P n n-2 . n+1

+ £ ( x ^ 1 hp p (x) n f ' (h p (x)) ) j= l 1 * " n - j s = n - j + l ®s * l - m t r s - l

n+1 + x n + 1 hp ' ( x ) n f p ' ( h p ( x ) ) .

P 1 s»2 p s p l , " t s - l

Now

x h •(x) » h p (x) 1 " n+1 1'* n+1

+ * «P ' " V P <*>> bp P < x )

n+1 I n I n

64

n-1 . n+1 + S ( x K h p p (x) n f p ' ( h p p (x)) )

k = 2 ' V ' ^ n + l - k s = n + 2 - k s r l " , r s - l 1 n + 1

+ x n + 1 h p ' ( X ) n f p ' ( h p p ( X ) ) . P 1 s = 2 p s p l " - p s - l

Finally,

x h •(x) - h p p (x) P l " n + 1 1 " n + 1

n - 1 . n + 1 + E x* h p (x) H fp ' (h p (x))

k = l 1 * ' ' n + l - k s = n + 2 - k s 1 " " " s n + 1

+ x h '(x) n f '(hp (x)). 1 s = 2 s 1 *"* s - 1

This completes the proof of Lemma 4.6.

Let P » P.P„ ... P ,C be an MSS sequence of length n > 1 2 n-i

1, and A = ,„*)• If the following four properties, ^ n^irj

which will be referred to jointly as property (*), are

established, then (f) will hold and therefore property G will

hold.

(i) If i is in A and 0 0 and z

€ [e,l-e]. Also, if z is a stable periodic point

of some if then z must be in [e,l-e]. e

(iii) If P is odd (even), then h _ 1(x) < 0 1 2 n-1

(> 0) for all x in A.

(iv) Figure seven is gotten from (i) through (iii),

65

depending on the parity of P. Let 7 be as shown in

Figure seven. Then (.1 ffi)2n( 1/2) = hH for 4

and h. in (r^H i { P )] »Hi(P) '(x) > 0 ( < 0 ) for all x

in ( P )) if P l s o d d (even).

I --

l - e

•/a. -

U.i-e)

_i I L.

> * > P H,(P)

1 ->

\-e

e

V

a,e>

_1 ! l_

> * * p *\

Fig. 7a Graph of hp(x),

when P is even.

Fig. 7b Graph of hp(x),

when P is odd.

I have been able to show that (i) through (iv) hold for

MSS sequences of the form RLnC, n > 0. One can easily check

the case P = RC; so assume that n > 1. Lemma 4.7 is an

elementary observation; however I include the proof for

completeness. To simplify notation, if w = WjW2

{R,L)n for some n € N then let <r(w)

w n

W2W3 W 9 ( W ) =

W3W4 VYV etc Again, note the following:

(1) if w € {R,L}n is primitive, then ff**(w) =£ w for j =

1,2,...n-1, and

66

(2) if w 6 {R,L}n and ff1(w) - ff-'(w) - w with 1 < j 0.

LEMMA 4.7. Let n € N and P € {R,L}n. If there exist j,k,l,m

€ N such that

(i) 1 > k and jk - ml « n,

(ii) P - P 1 ... P x

m times

P« ... Pfc, and

j times

(iv) P i s primitive,

then k divides 1.

Proof. Suppose that k does not divide 1. Then there exist

l_gk j?

r,s so that 1 = sk + r with 1 < r < k. Thus § (P) = ? (P)

= P. However, this contradicts Pj ... Pfc being primitive.

LEMMA 4.8. Let P = Pj ... b e a n M S S sequence, and -I €

[An,Jlu Then (Xtm)i(l/2) is not in [e,l-e] for 0 < i <

F ii j I F ) e

n.

i f e

Proof. First, from previous work, note that I U ) =

(P(n))00. Now, by Theorem 2.2, P(n) is primitive or has

period n/2 with Pj ... P n / 2 both primitive and odd. Suppose

there is some i, 0 < i < n, such that (if ) (1/2) G [e,l—e].

67

There are two cases.

CASE 1. Assume that P(n) is primitive. Let q = lcm(i,n) and

let k be the period of PjPg ... P i (so k divides i). Then

If _ I (Jl) « (P{n)) implies

P(n)

(q/i) times (q/n) times

Now Lemma 4.7 implies that k divides n. This contradicts

P(n) being primitive.

CASE 2. Assume that P(n) is not primitive. Then P(n) has

period n/2 with P1 ... P n^ 2 primitive and odd. However,

since either one or two below must hold, we see that case two

cannot occur.

^ ( 1 H i V i R P n / » r " V i °

even even

P(n) = Pj ... P n / 2-i R Pn/2+l Pn-1 L

(2) P -'*1 ••• ••• pn-l °

odd odd

P(n) - P, ... P^J.J L P n / J + 1 ... R


68

LEMMA 4.9. Let P = P1 ... b e a n M S S sequence with n >

1, and ^ (P)^* Then (Jffi) (1/2) *= hp^ ^ p ^(4).

Proof. This is immediate from Lemma 4.8.

LEMMA 4.10. Let P = P ± ... b e a n M s s sequence with

n > 1, and i € [J»p,H ( p )]. Then (Jfe)1(l/2) € [e,l-e] with

0 n. ©

Suppose that i £ n, 2n. Let k be the period of (P{n))m; so k

< i and k divides i. However P(n) is primitive or has period

n/2. Hence, k — n or k ® n/2. Now k = n, k divides i, and n

< i < 2n give a contradiction. Next consider k = n/2.

Recall that the word (P(n))2 is shift maximal in the extended

parity-lexicographical order, and that Pj ... **as 0 <^

parity. Now k = n/2 implies

(P(n))2 = Pj ... P n / 2Pi ••• Pn/2P1 Pn/2P1 Pn/2

< P1 '* * Pn/2P1 "'* Pn/2'

even

in the extended parity-lexicographical order. This is a

contradiction. This completes the proof of Lemma 4.10.

Lemma 4.7 through 4.10 establish (i) and (ii). In what

69

n*"" X

follows we restrict our attention to P of the form RL C

with n > 2. Recall that for P • PjPg ... P m € {R,L}m, p(P) =

{ i | Pi = R }. Thus, |?(P)| denotes the cardinality of /HP) •

REMARK 4.11. Let P » PjP2 ... P R € {R,L}n, n > 2, and Pj =

R. Then, using Lemma 4.6,

xhp p '(x) — hp p (x) 1'* n 1 " n

n 2 | p {P j ,. < • • P_) I j + S ( (-1) (x/e)3 hp p (x) )

3=1 1 " ' n-j

|^(P ...P )|

+ (x/e) (-1) (l-2x). Simplifying further one

gets

xh p '(x) « h (x) 1 * * n 1 * * n

n-1 Ip (P_ .,1...P )I

+ £ < t"1* 3 hP....P n 1L ... L ( x ) > J = 1 1 n" 3 j times

n. ..I',<P---P-- x (x/e)"(-l)' 2 n

THEOREM 4.12. Let P = R L n l with n > 2. Then hp'(x) < 0 for

x € (P)3•

70

Proof. Remark 4.11 gives that

xhp'(x) = hp(x) + (n-l)hp(x) - x(x/e)n.

Or using the definition of hp(x),

xhp'(x) = (x/e)n ( n - (n+l)x )

Thus if x > n/(n + 1), then hp'(x) < 0. One finds by

setting hp(x) = 1/2 and solving for x. Now, hp(x) =

(x/e)n(l - x) - 1/2 gives that 2x n + 1 - 2xn + en = 0. Setting

r(x) = 2x n + 1 - 2xn + e11, one sees the following:

(1) r(1/2) < 0,

(ii) r(1) > 0,

(iii) r'(x) < 0 on the interval (0, n/(n + 1)), and

(iv) r'(x) > 0 on the interval (n/(n + 1),od).

Thus > n/(n + l). This completes the proof of Theorem

4.12.

Figure eight is a picture of hp(x) on the interval ^ <1

[^p,JH (P)3 for P of the form RL , n > 2.

The following two lemmas will be used to show that

hH (P)'(X) > 0 o n ^P'^H (P)] W h 6 n P 1 S ° f t h® f° r m R L

> 2 .

, n

i -

i-e

!6

H,(P)

Fig. 8 Graph of hp(x)

LEMMA 4.13. For n > 3,

71

(*) 2n + 1 (

2(n + 1 } n + 1 ) 1 / n > 1 / 2 .

Proof. When n = 3, {*) holds. Both 2n + 1

2(n + 1) and

( 1/n converge up to 1 as n approaches oo. Thus (*) n + 1

holds for n > 3,

LEMMA 4.14. For n > 3,

X RL

n-1 2n + 1

2(n + 1)

Proof. As In the proof of Theorem 4.12, to find ^ we RL

set h (x) RL

n+1 „ n , n 2x + e . 1/2. Again set r(x) = 2x

Then, using the proof of Theorem 4.12, one gets Figure nine.

72

rtx*

(i,en)

'/z. Ct/lW-H* I

Fig. 9 Graph of r(x)

Now,

. 2n + 1 . _ n r( ) = e -

2(n + 1)

2n + 1 .n ( t •

(n + 1) 2(n + 1)

Lemma 4.13 implies that r( 2 n + 1 ) < 0. The result 2(n + 1)

follows.

THEOREM 4.15. Let P be of the form (RLn 1), n > 2. Then

hp1(x) > 0 on [i n_ l f ^p]• RL

Proof. Note the following:

2n+l (i) xhp'(x) = rxhp(x) - (n + l)h 2n(x) + x(x/e)

RL

(ii) h «_(x) » (x/e)2n+1(1 - x), RL

(iii) hp(x) = (x/e)n_1( (x/e)(l - (x/e)n( (x/e)(l - x)) )

= (x/e)n - (x/e)2n+1(l - x), and

(iv) xhp'(x) = n(x/e)n - n(x/e)2n+1(l - x)

73

- (n + 1)(x/e)2n+1(l - x) + x(x/e)2n+1

n{x/e)n + (x/e)2n+1(2(n + l)x - (2n + 1))

If n > 3, then, using Lemma 4.14 and (iv) above,

hp'(x) > 0 for x in the interval [J n-l'^P^' RL

Assume that n = 2. Then,

xhp'(x) = (x/e)2 (2 + (x/e)3(6x-5)),

and is the root of r(x) = 2x3 - 2x2 + e2 that lies in RL

(1/2,1). One can check that again hp'(x) > 0 for x in the

interval [/I ,<lp]. This completes the proof of Theorem 4.15. RL

Property (*) is now established for MSS sequences of the

form RLn, n > 1, and therefore property G is also

established.

Let P be an MSS sequence of the form RLn with n > 0. A

portion of the bifurcation diagram for the family /I f is

shown in Figure ten part a. By establishing that property

(f) holds for P, I have shown that Figure ten part b can not

occur for the family The computer generated bifurcation

diagrams for the family js that I have seen are all like

Figure ten part a, i.e., they do not have any occurrences of

the type that distinguish Figure ten part a form Figure ten

part b. I do not know of any rigorous results along this

line for the family ,1s.

74

I -•

'/a

"> >

V\CP)

Fig. 10a Bifurcation diagram.

\ ••

Vz

"> >

Fig. 10b Bifurcation diagram.

CHAPTER V

HAUSDORFF DIMENSION OF BASIN BOUNDARIES

Introduction

The purpose of this chapter is to establish rigorous

results concerning the Hausdorff dimension of the boundary of

the basin of attraction of the stable orbit for a map

i f P 0

where P is an MSS sequence and I U p ) ® P. For an

introduction to Hausdorff measures see Rogers or Falconer

[14,39]. Throughout Chapter five it is assumed that e is

fixed in (0,1/2). As before, for each MSS sequence P let /lp V e

be the scalar in [0,1] such that I (/lp) = P.

DEFINITION 5.1. Let P be an MSS sequence of length n. Let

<yp(1/2) denote the orbit of 1/2 tinder the map ^pfe- Note

that 1/2 is a stable periodic point of ^ pf e of period n. Set

5Sp - { x € [0,1] | *(x) « 0p( 1/2)}.

(Here u(x) is the u-limit set of x, see Walters [45;p.l23].)

Then, 56p is the basin of attraction of the stable orbit for

the map ^pfe* Set, <€p = $(ap) .

REMARK 5.2. Let P be an MSS sequence. Then x € » p iff there

exists some q € N so that (^pfe)^(x) = 1/2. Also note

75

76

that * is an invariant repelling set, and that =

[0,1] \ ®p.

In section two an algorithm is given that for each MSS

sequence P defines a graph directed construction Gp so that

the construction object is precisely In section three

some examples and conjectures are given. The rest of section

one is a brief introduction to graph directed constructions.

Mauldin and Williams [29] introduce the idea of

geometric constructions in Rm that are determined by a

directed graph G and by similarity ratios that are labelled

with the edges of this graph. Results of Moran [31] are

obtained from a specific type of graph directed construction.

Thus the results of Mauldin and Williams are more general

than those of Moran. For each graph directed construction G

there is a number a that is the Hausdorff dimension of the

construction object. The Hausdorff measure of the

construction object in its dimension s is always positive and

^-finite. The order structure of the strongly connected

components of G determines whether the object has finite

measure in its dimension.

The remainder of section one is taken from Mauldin and o

Williams [29]. Assume that G C {l,...,n) is a directed

graph. For each subgraph H of G, let V(H) be the vertex set

of H. The following notation is established. Set G(l) =

n}, and for each integer m > 2 set

77

G(m) = € {1 n}m | U(j) j+D) € G, j = l,...m-l}.

Also, set

G* = U ® G(m) m=i

and

G00 = <* € {1 ,n}N | € G, j = 1,2,3,...}.

* The length of e 6 G is denoted by

H C G Is a cycle means H is a directed graph such that

for some o € G , which is a closed path passing into every

vertex exactly once, it is true that every edge of t is an

edge of H. A directed graph H C G is said to be strongly

connected provided that whenever each of x and y is a vertex

of H, then there is a directed path from x to y. For a graph

G, a strongly connected component of G is a maximal subgraph

H of G such that H is strongly connected. A path component

of G rooted at a vertex i consists of all vertices j such

that there is a directed path from i to j. To say that a

path component is a cycle means that the subgraph of G over

that component is a cycle. If a path component is a cycle H,

then H is a strongly connected component of G.

A geometric graph directed construction in Rm consists of

(1) a finite sequence of nonoverlapping compact subsets of

Rm: J , ... ,Jn such that each ^ has a nonempty

interior,

(2) a directed graph G with vertex set consisting of the

integers l,...,n and similarity maps T^ ^ of Rm, where

(i,j) € G, with similarity ratios ^ such that

78

(a) for each i, 1 < 1 < n, there is some j such

that (i,j) € G,

(b) for each i, {Ti;J (J^) | (i,j) € Q } is a

nonoverlapping family,

U { | (i'3) e G > C JA

and

(c) if the path component of 6 rooted at the

vertex ii is a cycle: [,...,i^,*q +i =

then

q n t. . < 1 . k=l k k+1

For each o G G with j J = q ^ 2, set

T* = T(r(l) ,ff(2)° °T<r (q-1) ,<r(q) ,

and

J • T (J , . ). <r a r (q)

The construction matrix A = AQ associated with a graph

directed construction is the nxn matrix defined by

A = [ti.j]i.3<n'

where we make the convention that t. . = 0 if (i,j) is not in J- 9 J

G. For each # > 0, let A^ - AQ ^ be the nxn matrix given by

aa . . = t1? .. Also, let t(0) be the spectral radius of A,. p i x, j i, J y

According to the Frobenius-Perron theorem, is the

largest nonnegative eigenvalue of A^. An nxn matrix AQ is

irreducible iff its directed graph G is strongly connected.

79

Each geometric construction determines a compact subset K

of Rm. This set, which is called the construction object, is

pieced together by following the graph G and applying the

maps coded by the edges to the corresponding sets. The

construction object K can be expressed as

oo K = n ( u^ € G ( m ) Jr >.

m=l

Let SC(G) be the set of all strongly connected components

of G. Each H in SC(G) defines a graph directed

subconstruction. This subconstruction is based upon the sets

JA such that i € V(H), has directed graph H, and the

similarity maps are those from the original construction.

For each H € SC(G), let «H be the number such that 4(fi) »

1. Partially order SC(G) by stating that Hj < H 2 provided

there is a path y = ( g ^ - - ^ ) € G* such that g1 € Hj and gfc

€ Hg. Listed below are two main theorems from Mauldin and

Williams.

THEOREM 5.3. For each graph directed construction such that

G itself is strongly connected, the Hausdorff dimension of K,

the construction object, is a, where ^(a) - 1. Moreover,

the Hausdorff measure of K in its dimension a is positive and

finite.

THEOREM 5.4. For each graph directed construction, a -

max { <*H J H € SC(G) } is the Hausdorff dimension of the

80

construction object K and K has positive fl'-finite Hausdorff

measure in its dimension. Further, the measure of K is

finite iff no two distinct elements of { H € SC(G) j = a }

are comparable in the partial order on SC(G).

Graph Directed Constructions for Basin Boundaries

Let P = P P„...P ,C be an MSS sequence with n > 1. In 1 2 n-1

section two, P is fixed and n stands for the length of P. I

assume that n > 1 since •CQ = {0,1}. Let be such that ^ D

f«

I r (4p) = P. For convenience, set ~ * ( since both e

and P are fixed ), and for x € [0,1] let I(x) =

I (x),1,(x),I„(x) ... denote If(x). In what follows a graph O 1

directed construction is defined so that the construction

object is precisely <ep. For our purposes it will be convenient to index the vertex set with elements from {R,L}

n

t - e -

0^,e)

Fig. 11 Graph of F ,1 and

DEFINITION 5.5. Set

•L /JF f^fx) - (e/JD)x and f^lx) = 1 - (e/^p)x.

81

for x € [0,1].

- 1 - 1 The graphs of f_ and f_ are shown in Figure eleven.

L K

LEMMA 5.6. Let x,y G [0,1] with x < y. Then I(x) < I(y).

Proof. Suppose that I(x) > I(y). then, by Lemma 1.2, x > y.

This is a contradiction.

DEFINITION 5.7. Let A = € {R,L}N. Set,

yk(A) = (*!>!>£+!» f o r k = 0,1,2,3,

DEFINITION 5.8. Set

P,P0...P ,R, if P is odd p = 1 2 n-1 < .

PjP2• • .PJI-iL, if p *s even.

Note that (P<)°° < P.

LEMMA 5.9. Let A - {a1>1>1 € {R,L}N be such that ^ ( A ) <

(P<)00 for all j > 0. Then there is some x in [0,1] so that

I(x) = A.

Proof. Set

R a = { x € [0,1] | I(x) > A }, and

LA = { x € [0,1] | I(x) < A }.

To produce an x in [0,1] such that I(x) = A, it suffices to

show that R a and L^ are open.

Let y 6 R a and set I(y) = I 1I 2I3 L e t ® = m l n < 1 e N

Ii £ ai )• If Iffl £ C, then (using continuity) there is

82

some open interval U containing y such that 0 C Rft. So

assume I = C. Then I(y) > A implies m

r L, if • 8 1

(*) a = { L, if m > 1 and a....a„ , is even y ' m J 1 m-1 L R, if m > 1 and a,...a „ is odd.

l m-1

Let Aj be an open interval around y so that if z € Aj then

I(z) takes one of the following forms: * 11 0)

*am-lC'

P — a^.. ,am-lRPlP2'',Pn- i0' ° r

T • •• •am-lLPlP2* * *Pn-ic'

where if m = 1 it is understood that a,...a , l m—l

is the

sequence. By assumption ajcajc+1 ... < (p<)°° A and therefore ^ 1 C Rft.

Thus R^ is open. One can similarly show that LA is open.

Thus there is some x € [0,1] so that I(x) = A.

REMARK 5.10. If x € <ep, then I(x) € {R,L}N. Also, (using

Lemma 1.1) if y € [0,1], then I(f^(y)) 0.

83

DEFINITION 5.11. Let V p = {a = € {R*L}n | if i €

<2,3,...,n}, then < p o r *itti+1•••*n =

P1P2'**Pn-i+l D e f i n e a directed graph G p on Vp as

follows:

Let a = ffj...# , b • e vp' n o t necessarily

distinct, then (a,b) € G p (i.e., there is a directed edge

from a to b) iff

(1) ffj = ^j-1 f o r 2 - ^ - n' a n d

(* #)

(2) b < P.

LEMMA 5.12. If a € Vp, then there is some b € V p so that

<a,b) € Gp.

Proof. Let a = «,...<» € V_. There are two cases. 1 n P

CASE l. Assume that a^ = P ^ for j - 2,3,...,n. Thus,

a„a a- = P . . . ^ Set §. » a. for 1 < j < n-1, i.e., n l n—l j j+i 2 3

^1^2* ' *^n-l = "2*3* * ,<rn* S e t

L, if P is odd

n 1 R, if P is even.

Then, by Lemma 2.1, b = ji^02*••Pn = P1P2"**Pn-l^n * s

maximal in the extended parity-lexicographical order, and

therefore b € Vp. Also, b such that ±

PJ.J, i.e.. «2...«n * Pi--.Pn.l- S e t f i ' ")*1 f o r 1 * 3 -

n-1. Then, since a € Vp, regardless of whether is an L or

84

an R it will be true that b = n

 3. There are two cases.

CASE 2a. Assume that a....a ...P . . for each j € 3 n 1 n-j+1

<3,...,n>. Set = L. Then a € V p implies that b » n

€ V p and therefore (a,b) € G p.

CASE 2b. Assume there is some j € {3,...,n} so that

= P, ...P .... Let j be the least such j. Then, <r. ...<* = 1 n-j+1 o n

P l " - P n - 3 +1" S e t K " pn-j +2' T h e n ' *<>' 3 < J < o

®j...®n £ Pi , , , Pn-j+l an£* therefore, since a G Vp,

= "3 P' Note that < P °r = Pl'

and that 0. = a jrt-l--*"n-l"n "

ujrt-l---®n^n ~ P1*'*Pn-30+l

Pn-j0+2*

Lastly if j < j < n, then

P l " " PJ-J 0+ 1

"* ff j * • • • J Q J

Pn-j 0 +lPn-j 0 +2

"n pn-j +2' a n d

o

j

'j-l

n n-jQ+2

P 1 p 'n-l n-

V 2

Thus < P or = Pi -'* Pn-j+2 ( s l n c e p l s

shift maximal). Hence, b € V p and therefore (a,b) € G p.

85

Again, my goal is to give a graph directed construction

for <€_. In the next definition some notation is set that is P

analogous to that set in section one.

DEFINITION 5.13. Set Gp(l) = Vp and for each integer k > 2

set,

Gp(k) - { ^ I ^si,si+l^ ^ GP

for i = 1,2,...,k-l >

Also, set

* 00

G p = U G (k) v k=l

and

GP = { {si}i>l € VP I (8i'8i+l> € GP' 1 ° 1 ' 2 ' 3 " " >

DEFINITION 5.14. For each <r = e Gp' w h e r e si "

s. ...s , set i,l i,n

I (^ ) s S 1S1 0 • . . S1 8ft Sq S . S- • • • ; 1,1 1,2 1,n 2,n 3,n 4,n o,n

so 1(0 € {R,L}N.

LEMMA 5.15. Let <r = 6 G®. Then y k(IU)) 0.

Proof. Express I(<r) as I(<r) = Then, by Definition

5.14,

sj = ' n+j-l* f o r ^ - 2'

Now (sj,s^+1) € Gp for all j > 1 implies that sj + 1

 1. Thus yk{I(ff)) 0.

86

REMARK 5.16.

(i) Notice that Lemma 5.9 and Lemma 5.15 imply that for

each <r € G® there is some x„ € [0,1] such that I(x ) = I(<r) P 9 v

(note that if Q is shift maximal and Q < P, then Q < (P<)°°) •

(ii) In particular, if a = ar ...<r € Vp then there is

some x a € so that I^(x) = ®j+1 f o r 3 = 0 , 1 , . . . , n - 1 .

(iii) Also note that if b = ^ { R' L) n \ vp» then

(using the definition of Vp and the second part of Remark

5.10) there does not exist a y € [0,1] so that !j(y) = ^j+i

for j = 0,l,...,n-l.

DEFINITION 5.17. Set $ n = { x £ [0,1] j there is an open

interval A containing x such that if y € A then fn(y) = X X

fn(x) € cr ( 1 / 2 ) } .

REMARK 5.18. First $ n C [0,1] \ '€p. Also, [0,1] \ $ n is

composed of finitely many pairwise disjoint nondegenerate

2 n

closed intervals. By considering the graphs of f,f ,...,f

one finds that if x,y € [0,1] \ ® n are in the same closed

interval, then I(x) = I(y) for i = 0,1,...,n-1. Thus (recall

Remark 5.16) the pairwise disjoint intervals that make up

[0,1] \ 5&n can be indexed by the elements of Vp.

DEFINITION 5.19. For each a = e vp s e t

J = { x € [0,1] \ » I I .(x) = a. for j = 0,l,...,n-l } a n I j j+i

Then { | a € Vp } is a set of pairwise disjoint closed

87

intervals such that U J = [0,1] \ ® < r,. a n

a€Vp

DEFINITION 5.20. For each (a,b) € Gp, where a = s e t

Ta,b<x)

fL1(x), if a1 = L

f^fx), if a1 = R,

for x € J^.

LEMMA 5.21. Let (a,b) € G p and x € Jfe. Then x < >lp.

Proof. First, (a,b) € G p implies that b < P. Thus I(x) < P

= I(/Ip). Now Lemma 1.1 implies that x < <lp.

LEMMA 5.22. Let (a,b) € Gp. Then T fe: Jfe -+

Proof. Let = b a n d *i*,,<rn = a* l t s u f f i c e s t o s h o w

that T . maps the interior of J. into J . Let x be in the a,b e b a

interior of J^.

CASE 1. Assume that «1 = L. We have that 0 < T a b(x) =

f~X(x) < e, since x < /lp. Thus, f(TQ b(x)) = I f T

a,b*X*

€ a , then there is an open interval A, containing T K(x) so n * «# o

that A, C T K(JK), and so that if y € A, then fn(y) =

l a, D D i

fn(Ta b (x)) 6 0(1/2). However, fn(Ta,t,(

x)) ~ fI* 1 <f ( T a,b( x ) J

= fn-1(x). Now, fr(Ai) is an open interval containing x such L 1

that fL(Aa) C jfe. Let z € f^Aj). Then f n _ 1(z) = fn(fL1(z))

= fn(T K(x)) = fn _ 1(x) € (1/2). Thus x € ® . This is a

contradiction.

88

CASE 2. Assume that = R. The argument is similar to case

one.

LEMMA 5.23. Let a € Vp. Then { T a b(Jb) | <a'b) € Q p } is a

nonoverlapping family.

Proof. This follows immediately from Lemma 5.22, Definition

5.19, and Definition 5.20.

DEFINITION 5.24. The above work establishes that Vp, Gp, and

{ T a b | (a,b) € G p } define a graph directed construction.

Set

oo K - n ( u j ).

m=l <r€Gp(m)

Recall the definitions of J and T given in section one. 9 V

THEOREM 5.25. K = * p.

N Proof. For convenience, if r = ^ti^i>i ^ Vp' t h e n ri m

t«...t for m > 1. 1 m

For each k € K, let * k € Gp be such that for all m > 1,

k € J,| (for m = 1 set 3 | = (i) * • F o r e a c h k € K a n d m

> 2,

(1) k 6 T^k(l),*fc(2) 0 0 Tffk(m-1),#k(m)

(J^k(m)]

For each k € K and m > 2 let z , £ Jf . . be such that - m,k k(m)

89

(2) k - T„k(i) ,<rk(2) 0 0 T<fk(m-1) ,^k(m)

( zm fk)

Set z . = k for all k € K. Thus for each k € K, 1 § K.

- 1 - 1 ( 3 ) zm,k = T,k(m-l),fk(«) <••••«

T' kU).» k<2 )

for m > 2. Now, z . € J , . (2), and Lemma 5.21 imply - m,K r

that z m . < for all k € K and m > 2. m,K f

CLAIM l. Let k € K. Then z m = fm (k) for all m > 1 (f

is by definition the identity map).

Proof of claim one. Induct on m. Claim one holds for m = 1

Assume claim one holds for all q ^ m and consider m+1. By

definition,

<4> zm+l,k

— 1 ~ l "" l T#k(m),»k(»+1)

0 ° " • ° T' k(1).' k(2)

( k )

Thus,

(z ^ = z» i, = f { k ) ( 5 ) T<r

k(m) ,#k(m+l) m+1 , k Zm,k

Recall that 2 m + l k < ^p-

90

CASE 1. Assume that ,,k(B+1) - f^1. Then

'L1<*.•!,*> = V * * ni~~ l in

implies that f (k) < e. Thus z m + 1 ^ ~ f (k) •

CASE 2. Assume that T . . , . .,. = f"1. The argument is ,^^(m+i) R

similar to case one. Thus the proof of claim one is

complete.

CLAIM 2. K C "€p.

Proof of claim two. Suppose there is a k € K and m € N so

that fm(k) « >lp, i.e., k £ -ep. Then claim one implies the

following:

(6) ylp f (k) = z m + n + l fk ~ T^k(m+n) ,fk(m+n+l)

Zm+n,k^

-1 , -m+n-1.... T*k(m+n),^k(m+n+l)

T^k(m+n),^k(m+n+l)*1/2* ~ Jp/(2e).

This is a contradiction. Thus claim 2 holds.

CLAIM 3. <cp C K.

Proof of claim three. For each x € <6p form € Vp as

follows:

91

(i) for m > 1 set a (m) = a , where f (x) € J , v - x m,x m,x

and

(ii) ' x - Cx(«)>.2i-

Fix x € «p.

SUBCLAIM 3a. For each m > 1, (am,x'

am+l,x* € GP a n d

f m _ 1(x) - T a ( fm(x) )• m,x m+1 ,x

Proof of subclaim 3a. Fix m > 1. Then x € <Cp Implies that

f m _ 1(x) is either f^(f m(x)) or f^l(f"(x))- Note that * u + l f X

 1, since fm(x) < X p . Express a m + l x as a m + l f X

= «1«2*'*ffn* S e t'

Then a m + 1 (X < p a n d a„i+i,x 6 VP l m p l y t h a t a 6 VP a n d

<a'am+l,x» 6 V T h a S '

(7) - T„ „ (fm(K)) with f m _ 1 (x) e J . m+1 ,x

Recalling that f m - 1(x) € J and that { J a | a € Vp } is a ra,x

pairwise disjoint collection of sets, we have that a = a m > x-

Thus, (aB,x.aB+1,x) 6 V Moreover, see (7),

fm"1(x) - T a (f"(x)). m,x' m+l,x

92

This completes the proof of subclaim 3a.

SUBCLAIM 3b. € Gp.

Proof of subclaim 3b. This follows immediately from subclaim

3a.

SUBCLAIM 3c. For all m > 1 x € J i and therefore x G K. xl m

Proof of subclaim 3c. First suppose that m > 1. Then

,«» t _ t „ t (J ) with ( 8 ) J i = T o • • • o a

xlm l,x' 2,x m-l,x m,x m,x

f m - 1(x) 6 J . m,x

By repeated use of subclaim 3a (the second part) we have

T _ (f(x)) • x € J al,xa2,x" 'xim

Lastly if m = 1, the » X U ) " a

1 > x w i t h x G Ja * T h i S

' 1 i X

completes the proof of subclaim 3c.

Subclaims 3a,3b, and 3c prove claim 3. This completes

the proof of Theorem 5.25.

Examples

In the examples below, exact values for the Hausdorff

dimension of <ep are given for P of the form RC, RLRC, RLC,

RL2C, and RL3C. Conjectures are given for P of the form RL ,

n > 4. In general, the directed graph G p associated with an

93

MSS sequence P will not be strongly connected. For each of

the examples below the directed graph Gp is not strongly

connected. As seen in Theorem 5.4, if Gp is not strongly

connected then the measure of *€p in its dimension is finite

or infinite depending on the order structure of the strongly

connected components of Gp. Examples are given where the

measure is finite and where the measure is infinite.

H-dim(*£p) denotes the Hausdorff dimension of 'Cp.

EXAMPLE 1. Let P - RC. Then G p is a directed graph on four

vertices with two comparable strongly connected components

each with exactly one vertex. Thus H-dimf'Cp) = 0, and the

measure of *£p in its dimension is infinite and 9—finite.

Hence <ep is countable. See Figure twelve for Gp.

Fig. 12 Directed graph for RC

REMARK 5.26. Let P » RC. Observe that Up/e)/(l + ).p/e) is

94

the unstable fixed point of Jpfe- If x € *p, then exactly

one of the following hold:

(1) x = 0,

(2) x = 1,

(3) there exists n € N such that

U pf e)n(x) = (Jp/e)/ (1 + >lp/e).

To see this consider Gp. If x € <ep, then (according to Gp)

j P e(x) is eventually R°° or eventually L00. We consider two

cases.

CASE 1. Suppose that y € [0,1], ^ € (l-e,l], and there

Xf exists k e N such that I ^(y) = L for j > ky. Set z =

k ^ f (Jlf ) Y(Y). Then I ®<z) = L®. Thus z = 0, since X / e > 1

and Uf e)j(z) = U/e) jz for j > 1. Hence y = 0 or y = 1.

CASE 2. Suppose that y € [0,1], X € [^p,l], and there exists

Xf k k € N such that I ®(y) = R for j > k . Set z - (Xf ) Y(y)• y 3 y

Xf Then I e(z) = R , and therefore

lit = (JI/e) + + (Ve)(z-1)]

e ' i + X/e

for j > 1. Hence z = (X/e)/(l + X/e), since otherwise there

is some m € N such that (^fe) (z) > 1.

EXAMPLE 2. Let P = RLRC, the first harmonic of RC. Then Gp

has three strongly connected components and H-dim(<ep) = 0,

with its measure infinite and ^-finite. Again <ep is

countable. See Figure thirteen for Gp.

95

RLLL

RLLR

LLLL <

LLLR

LLRR

RLRL LLRL

RRRL LRRL

RRRR LRRR

RRLR \ LRLR

Fig. 13 Directed graph for RLRC

REMARK 5.27. For each n > 1 define

g (x ) 8n l

2 n n+1 1 + x + x + . . . + X - x

96

Then

(x-1) gn(x) = 2xn+1 - x n + 2 - 1, n > 1.

Let r be the unique root of 2xn+1 - x n + 2 - 1 that lies in n

(1,2). Thus, 9 n(rn)

= °- Note that for each n, rR is a

simple r-number as defined by Parry [37;38].

For completeness we give a definition of a simple

T-number. Renyi [38] has proven that if T > 1, then every

non—negative number x has a T—expansion:

x = f Q(x) + €i(x)/t + Z2WI

t2 + •••

where £q(x) = [x], ^(x) = [r<x>], e2 = [ r < r < x » ] etc. Here

[x] denotes the integral part of x and <x> the fractional

part of the real number x. Those f with a finite » -expansion

are called simple r-numbers by Parry [37].

EXAMPLE 3. Let P = RLC. Then Gp has exactly one strongly

connected component Hp with — H-dimCtfp). The

characteristic polynomial tp of the matrix AJJ ^ is given by

f(7)!=,)'3(k2 + k T - 7 2 )

where k = (e/Jtp)^. The spectral radius of AH^ ^ is

k(l + sqrt5)/2. Thus we have H-dim^p) « InCrj)/ln(<Jp/e) .

Note that r± = (1 + sqrt(5))/2, the golden mean. The measure

of *Cp in its dimension is finite. See Figure fourteen for

V

REMARK 5.28. Example three contradicts a statement made by

97

Metropolis and Louck [27,pp.14]. Their statement implies

that t n r is countable. If t o r were countable, then RL K"

H-dim(<e_r) would be zero. We also point out that (using RL

Sarkovskiis1 Theorem [40]) the map ^ R Lf e has periodic points

of all orders, not just of order 1, 2, and 3 as stated by

Metropolis and Louck.

RLL

V' LLR RLR

^ RRR LRR

RRL LRL

Fig. 14 Directed graph for RLC

EXAMPLE 4. Let P = RL C. Then again G p has exactly one

strongly connected component Hp with = H-dim^p). The

characteristic polynomial p of ^ is given by

f ( f ) - ?10( k 3 + 7k2 + ? 2k - ? 3 ) ,

where k = (e/Jp)^. We have

H-dimCep) = ln(r2)/ln(Jp/e),

98

and the measure of <ep in its dimension is finite. The graph

Gp is not shown due to its size.

3

EXAMPLE 5. Let P = RL C. Again, there is exactly one

strongly connected component Hp with = H-dim('Cp) . The

characteristic polynomial y for AH ^ is

F( 7) = r25( * 4 +?k3 + r2^2 + 7 3* - r4 )>

where k = (e/ip)^. Here,

H-dimt'Cp) = ln(r3)/ln(/lp/e),

and the measure of <Cp in its dimension is finite. Again, the

graph of G p is not included due to its size.

CONJECTURE. Let n > 4, and P = RLnC. Then,

H-dimUp) = ln(rn)/ln(>lp/e) .

Moreover, the measure of *Cp in its dimension is finite.

CHAPTER VI

QUESTIONS

The following is a list of some of the questions that

have arisen in my study. Some or all of them may be trivial.


impose on a unimodal map g to guarantee that

?> « {X € [0,1] I I g(J) is an MSS sequence 9 1

t*a

or is infinite and periodic}

is dense in [0,1]?

QUESTION 2. Fix e € (0,1/2). Does [0,1] \ have positive e

Lebesgue measure? If the answer is no, what is the Hausdorff

dimension of [0,1] \ T^ ? "e

QUESTION 3. Fix e € (0,1/2) and fix x € [e,l-e] different

than 1/2. Let P be an MSS sequence. Does there exist a

unique JI(P,x) € [0,1] so that x is a periodic point of

A(P,x)fe of period the length of P and so that

I e(J(P,x)) = Pro?


impose on a unimodal map g to guarantee that g exhibits

99

100

uniqueness?


impose on a unimodal map g so that for every shift maximal

sequence P there exists at least one i(P) € [0,1] such that

(P)9(j (p) ) a P ?

QUESTION 6. Let g be unimodal and A be an aperiodic shift

maximal sequence. What are necessary and sufficient

conditions to impose on g so that

« = {X € [0,1] I I*S(i ) = A> A,g 1

consists of at most one point?

QUESTION 7. Let A = <ai>i>i b e a n a P e r l o d i c shift maximal

sequence. Does there exist a strictly increasing sequence of

positive integers {n1>i®1 and a corresponding sequence

(D (A)}.n« of MSS sequences so that n^ i£l

(i) for each i, D (A) < D (A), ni+l i

(ii) the length of Dn (A) is nA for all i, and

(iii) if D n (A) - d1d2. .-d^.jC, then a j « for

1 < j < ni-l?

QUESTION 8. Does the family -Js(x) = 4ix(l-x), X G [0,1],

exhibit only regular period doubling bifurcations?

101

QUESTION 9. Let f be a unimodal map that exhibits

uniqueness. Does uniqueness imply that only regular period

doubling bifurcations can occur? If not, does there exist a * *

parameter value i so that if J > A , then only regular

period doubling bifurcations can occur?

QUESTION 10. Fix e € (0,1/2). For each scalar X G [0,1] set

h(JI) = H-dimCe )

What type of function is h?

I e(A)

QUESTION 11. Let P be an MSS sequence and fix e 6 (0.1/2).

Is

H - d i m ^ jpj) = H-dim(<ep)

for all n (using the map fe)? Can anything be said about the

corresponding measures?

QUESTION 12. Let P be an MSS sequence and s(x) = 4x(l-x).

If x,y are distinct elements of the boundary of the basin of

attraction of the stable orbit for the map /lps, is it true ^ ps ^ ps

that I r (x) ± I (y)?

QUESTION 13. In 1964 Sarkovskii [40] gave the positive

integers the following order:

1 < 2 < 4 < 8 <...<...<...2*7 < 2•5 < 2*3 <...< 7 < 5 < 3.

He then proved that if f:R -» R is continuous and has a point

102

of period n,then f has a point of period m for all m < n in

the above ordering. Let g:[0,l] -» [0,1] be unimodal.

Proposition 2 . 2 of Li et al. [22] gives that if I € [0,1] is

such that I^g(4) starts as RLL, then -Ig has a periodic point

of period three and therefore periodic points of all orders

by Sarkovskii's theorem. Does there exist an MSS sequence P

so that

(i) if Jl € [0,1] is such that < (PH)°°» (PL) ,

then >lg does NOT have a periodic point of period

three, and

(ii) if I € [0,1] is such that ) > (PR)00* (PL)00,

then /lg does have a periodic point of period

three?

QUESTION 14. For each positive integer n let denote the

minimal, in the parity-lexicographical order, MSS sequence of

length n. Let g be a unimodal map that exhibits uniqueness.

We know that the map g has a stable periodic point of (n)

period n. Is it true that if m > n, in Sarkovskii's order,

then the map JlD g does NOT have a periodic point of period P(n)

m?

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