"2.19
AJO.^QI)
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 < i < n.
n>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 < i < n-1
LL' i f ®i 53 ei+i
n LEMMA 2.13. Let n > 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 < I (i2).
Then there exists some /I € (i ) so that A wL
I i f (J ) = A.
V M ( The theorem also holds if I (i^ > 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 * ( X 2 ) .
X,f X0t (2) I f O < J 1 < i 2 < i and I Uj), I (^2) are
M M both finite, then I (j^) > 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<P> L Vl(P>
V l < p > R Hn-:l<p>
Hn-l(P» 1 H„-2<P>
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 <P> 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 < P 2 < ••• K Pfc-
Then there exist y,z € [Pj.P^] t h a t *'<*> > 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 £ < P m ) = Pfc' t h S n t h e r e
exists y € [Pm_j_'Pm3 s o > °*
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 < i < 2n, then (if )i(l/2) €
[e,l-e] implies that i = n or i = 2n.
(ii) If X is in A, then (if )n(l/2) » hp p p (/I). e 12'' n-1
Note that (/If )1(z) = (if )1(l/2) for i > 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 *\<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 < i
< n, then = w if (i - ffj) > 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
This completes the proof of Lemma 4.8.
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 < i < 2n implies that i is either n or 2n.
Proof. Suppose there exists some i, 0 < i £ 2n, such that
(Jlf )1(l/2) € [e,1-e]. Then Lemma 4.8 implies that i > 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<)°° < p for k
Let z € A j. Then,
A " al***am-lamam+lam+2-* *
even
a - a r . 0)
B 1 M O
f = a ^ . •am-lR P1 P2 * •' Pn-lC
7 = ai • • ' "an-lL P1 P2 • •• Pn-lC'
Recalling (*) we have that I(z) > 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)) < P = l U p ) for all j
> 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 < P by the definition of 0^. Thus
(**) hold and (a,b) € Gp.
CASE 2. Assume there is some j € {2,3,...,n> 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
< P and therefore (**)
will hold. We need to define so that b € V p. If n = 2,
set fi = L. One can easily check that b € V p and therefore
that (a,b) G G p. So assume that 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.
This completes the proof of Lemma 5.12.
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)) < P for all
k > 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
< p ^ o r
all j > 1. Thus yk{I(ff)) < P for all k > 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
< P for i > 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.
QUESTION 1. What are necessary and sufficient conditions to
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?
QUESTION 4. What are necessary and sufficient conditions to
impose on a unimodal map g to guarantee that g exhibits
99
100
uniqueness?
QUESTION 5. What are necessary and sufficient conditions to
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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