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TOPOLOGICAL ENTROPY OF THE LOZI FAMILY
By
Izzet Burak Yildiz

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
IV'Iathematics

2010

ABSTRACT
TOPOLOGICAL ENTROPY OF THE LOZI FAMILY
By
Izzet Burak Yildiz

We study the t0pological entropy of a two dimensional map, called the Lozi map.
The Lozi map is a piecewise-affine analog of the Henon map, one of the most studied
examples in dynamical systems. In this area, it is extremely important to understand
the complexity of a given system and t0pological entropy is a nonnegative number
which measures this complexity. We investigate how the complexity of the Lozi map
changes depending on the parameters.

In particular, we study the monotonicity and discontinuity properties of topologi-
cal entropy of the Lozi maps. In 1997, Y. Ishii and D. Sands showed the monotonicity
of the Lozi family £09,, in a C’1 neighborhood‘of a-axis in the a—b parameter space.
We. show the monotonicity of the entropy in the vertical direction around a = 2 and
in some other directions for 1 < a S 2. Also we give some rigorous and numerical
results for the parameters at which the Lozi family has zero entropy.

Moreover. in 2009, J. Buzzi showed that the entropy map f —+ ht0p( f) is lower
semi-continuous for all piecewise affine surface homeomorphisms. The upper semi-
continuity of entropy was an open question for these maps. We prove that topological
entropy for the Lozi maps can jump from zero to a value above 0.1203 as one crosses a
particular parameter and hence it is not upper semi-continuous in general. Moreover,
our results can be extended to a small neighborhood of this parameter and hence
disprove a conjecture by Ishii and Sands which states that there are at most countable
number of points of discontinuity of the entropy map.

We conclude with numerical results for the entropy of the Lozi maps for a large
set of parameters which coincide with rigorous bounds given by Y. Ishii and D. Sands

before.

ACKNOWLEDGMENT

I am grateful to my advisor, Dr. Sheldon Newhouse, for his excellent. guidance, pa.-
tience, and providing me with a great atmosphere for doing research. I would like to
thank Dr. Martin Berz for his interest in my project and his helpful suggestions. I
would also like to thank to my committee members Dr. Kyoko Makino, Dr. Zhengfang

Zhou and Dr. Michael Shapiro for being very helpful.

This work would not be possible without the support of my parents. I thank them

for their encouragement and love.

iii

TABLE OF CONTENTS

List of Figures ................................ v
INTRODUCTION ............................... 1
BASIC CONCEPTS ....................... 11
1.1 Symbolic Dynamics ............................ 12
1.2 Topological Entropy ........................... 14
MON OTONICITY OF ENTROPY AND ZERO EN TROPY PA-

RAMETERS .......................... 19
2. 1 Preliminaries ............................... 19
2.2 Pruning Theory .................... - .......... 21
2.3 Results about the monotonicity of the entropy ............. 25
2.4 Extension of the results to l < a _<_ 2 .................. 30
2.5 Results about the zero entropy locus .................. 39
DISCONTINUITY OF ENTROPY FOR LOZI MAPS ...... 45
3.1 Lower Bound Techniques ......................... 48
3.2 Discontinuity of entropy for Lozi maps ................. 49
APPENDIX 61
Bibliography .......................... 64

iv

2.1

2.2

2.3

2.4

LIST OF FIGURES

Henon attractor for H1_4‘0.3 and Lozi attractor for £1,105. .....

This figure shows the directions "U-i = (N3, —1) and v3 = (N3, 1) along
which entropy is non-decreasing. 1.2 < a, S 2 and b small ........

This picture gives a summary of recent results about the entropy of
the Lozi family. When a > 1 + Ibl, entropy is positive. The light gray
area on the left gives the parameters for which entropy is zero. The
darker gray area with complicated boundary represents our numerical
results for zero entropy parameters. In the darkest gray region on
the right, the entropy is log 2. In the triangular region, Misiurewicz
proved the existence of strange attractors. Note that maps with |b| > 1,
up to affine conjugacy, are inverses of maps with |b| < 1 and so not
interesting. In the rest of the picture, nothing much is known.

This is a summary of our numerical results for the entropy of Lozi fam-
ily where 1 S a. S 3, —1 S b S 1 and the height is the entropy. Observe
the consistency with Figure 3, especially zero entropy parameters when
b > O and the maximal entropy parameters. ..............

Symbol space (I) > 0) and the sets 215,3), Z and C ...........
This figure shows the approximate monotonicity results for different
N: and N3 values where 1.2 < a S 2. The topological entropy is

non-decreasing in the direction of arrows. ...............

This figure shows the primary pruned regions, Day), of maps for given
parameters. The :r-axis represents Cs and the y-axis represents C“.

One can expect to find some elements of 7501, at the boundary of Dal)-

Primary pruned region, Dal» for original parameters studied by Lozi:
a=1.7 and b=0.5 .............................

10

28

37

2.6

3.1

3.2

3.3

3.4

The picture shows the unstable and stable manifolds of the right fixed
point of £1115. ..............................

The shaded region gives the parameters for which ht0p(£a,b) = 0.
This figure shows a portion of the left unstable manifold of the fixed

point p1. Note that all the points on the line segment connecting F1
to F2 are period-4 points of £ ......................

Trapping region R(gray) and images £4(R)(darker) and £8(R)(darkest).

53

Themset Wfihickest solid_ broken line) and the part. of the images
£4(W)(thinner) and £:(W)(thinnest) which stay outside of R. Note
that everything above EC is mapped into R under £4. .........

This figure shows the quadrangles N1 and NZ and their images(thinner

boxes). Notice the covering relations: N1 => N1, '1 => N2 and
A72 =' All ................................

vi

43

44

52

58

INTRODUCTION

In 1963, Edward N. Lorenz, a meteorologist and mathematician from MIT, published
a paper[21] which included a set of three dimensional differential equations that had
important implications for climate and weather predictions. He observed that the so-
lution of the system exhibits complicated behavior that seemed to depend sensitively
on initial conditions. In other words. small variations in initial conditions would cause
large variations in the long term behavior of the system, a phenomenon now known as
the ”butterfly effect”. Moreover, the solutions seemed to form a complicated picture
which was later called a strange attractor. These discoveries planted the seeds for the

chaos theory.

In 1976, Michel Henon, a French astronomer, introduced a simple two-dimensional
map, called the Henon map, that exhibits chaotic behavior similar to that in the

Lorenz system[11].

M. Henon studied:

:1: 1+y—ar2
H=Hagbz l—r , a.bER,b7$0
y bar

A simple affine change of coordinates put. this map in the form:

:1: 1—(r12-i-by
H=Ha,b: H 7 a,b€lR,b;£0

y .r
Originally, M. Henon used the parameters of a = 1.4 and b = 0.3. He observed
that an initial point. of the plane under the iterations of the map either diverges to

infinity or approaches to a set now known as the Henon attractor. This set seemed to

1

be locally homeomorphic to the product of an interval and a Cantor set(See Figure
1). Although the map is given by a very simple formula, the rigorous mathematical
analysis of it turned out to be very difficult. The existence of an attractor was proved

years later only in a small neighborhood of b = 0 [2].

In 1978, Rene Lozi introduced another two dimensional family of maps which is
very similar to the Henon family[22].

The Lozi family is given by:

:1: 1—a|.r|+by
£=£a,b3 t—r , a.b€lR_. b7é0.
y :1:

Thus, the quadratic term (1.32

in the Henon family is replaced by the piecewise
affine term a|r|. This results in a considerably simpler family of maps. For instance,
in [26], Misiurewicz proved the existence of attractors for a large set of parameters.

The triangular region in Figure 3 shows these parameters.

Past Work and Recent Progress

Symbolic dynamics has played an important role in the study of iterated maps. Milnor
and Thurston have developed a kneading theory to study the topological dynamics
of piecewise-monotone self maps of the interval[23]. In their study, the itinerary of
the critical point, kneading sequence, is one of the most important ingredients. They
proved that a continuous, piecewise monotone map of the interval with positive topo-
logical entropy is semi-conjugate to a continuous piecewise affine map with a constant
slope and the same entropy. For unimodal maps (one critical point) with negative
Schwarzian derivative and no periodic attractor, the kneading sequence gives the com-

plete classification up to topological equivalence[17]. Also, the set of all admissible

2

 

I

 

 

 

 

1.5

0.5 r

-0.5 ~

1.5

0.5

-0.5

 

 

 

 

1.5

0.5 -

-0.5 r

_1L.

1.5

0.5

-0.5

Figure 1: Henon attractor for H1_4,0_3 and Lozi attractor for £1,105.

sequences can be obtained from the kneading sequence. Applications of the kneading
theory lead to a proof of continuity of entropy and monotonicity of the kneading

sequences for certain families of one dimensional maps.

On the other hand, there is no general symbolic theory for two dimensional maps.
In [6], Cvitanovic, Gunaratne and Procaccia presented a two-piece partition of the
plane which leads to symbolic dynamics of two symbols (-1 and 1) for the Henon
map. If a symbolic sequence corresponds to an actual periodic orbit it was called
admissible. They introduced subsets of the symbol space {—1,+1}Z called ”primary
pruned region” and ” pruning front” and conjectured that they specify all the periodic
orbits (Pruning Front Conjecture(PFC)). In other words, if all the backward and for-
ward iterations of a periodic sequence under the shift map stay away from the pruned
region, then this periodic sequence corresponds to a periodic orbit in the phase space.
This way they measured how far the given map is from a complete horseshoe in which
case the pruned region is empty. They used this idea to obtain a numerical estimation

for the topological entropy of the HenOn map.

A Pruning Theory for surface homeomorphisms (in a more general setting) was

given by A. Carvalho and Toby Hall[5].

Monotonicity of the Lozi family

Rigorous mathematical justification of the symbolic dynamics and an analog of Cvi-
tanovic’s Pruning Front Conjecture for the Lozi map was given by Y. Ishii, following
suggestions of J. Milnor[12]. Ishii introduced a similar pruned region which distin—
guishes the sequences that correspond to a point in the non-wandering set(admissible
sequences) from the sequences that do not(non-admissible). This characterization has

powerful applications. For example, he gave the boundary of the region in the (l-b

4

parameter space where the Lozi map has maximal entropy log 2(See Figure 3). Also,
in a joint work with Sands, they proved the entropy increases monotonically with a

for small fixed values of b:

Theorem 0.0.1 ([14]). For every a... > 1 there exists I)... > 0 such that, for any fixed

b with |b| < In, the topological entropy of £a,b is a non-decreasing function of a > a...

The main step in the proof of this theorem is to show that when a increases, the
primary pruned region decreases, and as a result entropy increases. It’s natural to

ask the following:
Question 1. Is the entropy monotone in other directions?

In this dissertation, we partly answered this question by proving the following
theorem (for details see Theorem 2.1.3 below):

Let us define R2>1+ = (a, b) E R2|a>1+]b]}.

Theorem 0.0.2. For every 1 < a S 2 there exist NC], NE E IR+ and two lines 7172 :
(4513,6132) —> R2>1+’ 51,2 > 0, given by 71(t) =.(a+Nc]t, —t) ands/20‘.) = (a+N§t,t)
such that the topological entropy of £710) and £72m is a non-decreasing function of

t.

So we can get the monotonicity directions given in Figure 2.

Discontinuity of the topological entropy

A well known result. in one dimensional dynamics is Misiurewicz and Szlenk’s lap
number entropy formula[27]. According to the formula, topological entropy of a
piecewise monotone map of the interval, f, is given by: ht0p( f) = lim,,,_.oo%log£n( f)
where in (f) is the number of monotone pieces of the nth iterate of f. In [16], Ishii and
Sands give a similar lap number entropy formula for piecewise affine homeomorphisms

of the plane. In [15], they use this formula to obtain some rigorous upper and lower

5

 

 

 

C
l k |\ l \\_ l \\\\\\\ \\ \\\ l
l N l \I \l \\\\\\ I \\\l
1 2 1 4 1.52 I 6 l 88 2
16 : (10,1) 135 = (2:1)
/ a
v2 = (3,1)
\ C t
L , Dr... J l,» l / 1 /',/'//{/t/'4
I 1 If I / l /////’////// I
12 1.43 1.59 1.7 TIQ

Figure 2: This figure shows the directions 23—] = (N3, —1) and v3 = (N31) along
which entropy is non-decreasing. 1.2 < a S 2 and b small.

bounds for the entropy of the Lozi family. Analyzing these results, they proposed the

following conjecture for the Lozi family:

Conjecture 1 (Ishii and Sands [15]). There are at most countable number of points

of discontinuity of the entropy map (a, b) —* h(£a,b)-

Moreover, in [4], Buzzi proves a Katok-like theorem for piecewise affine surface
homeomorphisms which shows the lower semi-continuity of the entropy map, f —>

hmp( f) The following question was asked by Buzzi:

Question 2. Prove or disprove the upper semi—continuity of entropy for piecewise

afline homeomorphisms of the plane.

In this dissertation, we proved that the topological entropy of the Lozi maps is
not continuous depending on the parameters showing that it can jump up from zero
to a value above 0.1203 as one crosses the parameter (a, b) = (1.4, 0.4) (see Theorem
3.0.5 below). Moreover, similar jumps occur in a small neighborhood of this point

along the line a = 1 + b, disproving the above conjecture:

Theorem 0.0.3. In general, the topological entropy of Lozi maps does not depend

continuously on the parameters. For 61 > 0 and small and [62] small:

(i) The topological entropy of the Lozi maps with (a,b) = (1.4 + 62,0.4 + 62),

ll(£1.4+62.0.4+62), is zero.

(ii) The topological entropy of Lozi maps, h.(£(1'4+€1+62’0_4+62)), has a lower bound

of 0.1203.

Zero Entropy Parameters

In [15], Ishii and Sands give rigorous entropy computations for some rational values

of a and b. Unfortunately. their algorithm gives poor results when entropy is close

7

to zero. So, the precise shape of the set of zero entropy parameters is unknown. On

the other hand, it is possible to use Brouwer’s Translation Theorem to prove some

sufficient conditions[15]. The light gray region in Figure 3 gives these parameters.
In this dissertation, we modified the use of Brouwer’s Translation Theorem to

prove the following (see Theorem 2.1.4):
Theorem 0.0.4. In a neighborhood of the parameter (a, b) = (1, 0.5). hmp(£a.b) = 0.

The proof of this theorem can be extended to a larger set of parameters. But it.
gets more complicated especially as b gets close to 1. So, we give some numerical
results where our proof seems to be working and obtain a picture (See the medium

gray area in Figure 3) for zero entropy parameters.

Numerical Results

Besides the numerical studies of zero entropy parameters, we also introduced a numer-
ical algorithm to approximate the entropy of Lozi maps. Our algorithm measures the
growth rate of the number of intersections of the unstable manifOld of the right fixed
point with the y-axis. For most of the parameters, entropies we obtain are within
i005 of the rigorous computations by Ishii and Sands. For example, for the original
parameters studied by Lozi, (a. b) = (1.7, 0.5), our entropy estimate is 0.5146 where
as the rigorous upper bound obtained by them is 0.5087. Summary of our computa-
tions are given in the picture below. However, we are not able to yet obtain rigorous
bounds for accuracy of our estimations. Another interesting point is that when our
method is applied to the Henon family, one still gets very close approximations for
the entropy. For example, for (a, b) = (1.4, 0.3), we get 0.465 which is very close to a
recent rigorous lower bound by Newhouse, Berz. Makino and Grote[29]. We will not

go into any more details here.

 

 

 

 

 

 

Figure 3: This picture gives a summary of recent results about the entropy of the
Lozi family. When a > 1 + [b], entrOpy is positive. The light gray area on the left
gives the parameters for which entropy is zero. The darker gray area with complicated
boundary represents our numerical results for zero entropy parameters. In the darkest
gray region on the right, the entrOpy is log 2. In the triangular region, Misiurewicz
proved the existence of strange attractors. Note that maps with [b] > 1, up to affine
conjugacy, are inverses of maps with |b| < 1 and so not interesting. In the rest of the
picture, nothing much is known.

 

 

Figure 4: This is a summary of our numerical results for the entropy of Lozi family
where 1 S a S 3, —1 S b _<_ 1 and the height is the entropy. Observe the consistency
with Figure 3, especially zero entropy parameters when b > 0 and the maximal
entropy parameters.

10

Chapter 1

BASIC CONCEPTS

Throughout this section let f : X —+ X denote a continuous function where (X, (1) is
a metric space with metric d. For k E Z. f k denotes the k-times composition, i.e.,
fl" = fofo-nof (Ir—times).

Definition 1.0.5. The forward orbit of a point a is the set 0+(a) = {fk(a) : h 2 0}.
If f is invertible, then the backward orbit is defined by: 0'(a) = {fk(a) : k S 0}.
Them-the (whole) orbit of a point a is 0(a) = {fk(a) : k E Z}. If f is not invertible
we take f‘lfy) = {r = f(I) = y}. ' A

Definition 1.0.6. A point a is a periodic point of period n provided f "(a) = a and
fj (a) 79 a for 0 < j < 72. If a has period one, then it is called a fired point.

Now, let us give some fundamental definitions connected with convergence and

stability of periodic points.

Definition 1.0.7. A point q is forward asymptotic to p provided d(fj(q), fj (p)) goes
to zero as j —+ 00. If p is a periodic point with period n, then q is asymptotic to p if

d(fj"(q),p) goes to zero as j —> 00. The stable set ofp is defined as:

W3( p) = {q : q is forward asymptotic to p}

11

If f is invertible, then a point q is said to be backward asymptotic to p provided

d(fj(q), fj(p)) goes to zero as j —> -:>c. The unstable set ofp is defined as:
W" (p) = {q : q is backward asymptotic to p}

The next notion has a great importance in dynamical systems theory and this

document.

Definition 1.0.8 (Attractor). A compact region N C X is called a trapping region
for f provided f (N) C int( N) A set. A is called an attracting set provided there is a
trapping region N such that A = flkzo fk (N). A set A is called an attractor if it is
an attracting set and f [A is topologically transitive, i.e., given any two open sets U

and V in A, there is a non-negative number n such that f "(U) H V ¢ 0.

Remark 1.0.9. The existence of attractors for the Lozi family was proven by
lVIisiurevdcz[26] for the parameters given in Figure 3. For these parameters, he also
proved that the attractor is actually the closure of the unstable set(manifold) of the

right fixed point.

1 . 1 Symbolic Dynamics

Following [20], let us define a finite alphabet set N = {1, . . . , n}. This set is a metric
space with the metric d(j, k) = 1 — djk where (SJ-k is 0 if j 7E k and 1 ifj = k. The
topology defined by this metric is a discrete topology and N is compact. Let us
form two sequence spaces. The one-sided sequence space is 2,1,: = {1, . . . , n}N where
N = {0,1, 2, . . .} and its points are in the form (50. 51,52, . . .) where e,- E N. The two
sided sequence space is 2,, = {1, . . . .n}Z where Z 2: {...,—2,-—1,0,1,2,...} and its
points are in the form (. . . .54, 5-1. 50, 81,52, . . .) where e,- E N. We put the product

topology in each space. By Tychonoff’s theorem for products of compact spaces. both

12

spaces are compact.

A set of the form [70, . . . ,‘idt = {5:

(1)

t = 2'0, . . . ,5”; = if} is called a cylinder set
or a block. The cylinder sets are both open and closed and they form a countable ba-

sis for the topology in each space. Every open set is a countable union of cylinder sets.

1 — (5 I
I 9° 5'5.- .
The product. metric (1(5. 5 ) = ——2i—1—‘ on 2?; generates the topology. Sim-
1=0 '
1 — 6 I
I ' ‘ 5'5-
ilarly, the product metric d(s,5) = Z 21. I 7' on En generates the topology.
i=—oo

Note that both 2;: and En are homeomorphic to the standard middle-third Cantor
set. Next, let us define the shift map, a, on both spaces as 0(5),- : 5i+1- The shift

map is a continuous. onto and n-to—l on 2?,” and it is a homeomorphism on Zn.

Definition 1.1.1. The dynamical system (2,1; ,0) is called the one-sided shift on. 11
symbols or full one-sided n-shz’ft. Similarly, (2". o) is called the two-sided shift on n

symbols or full n-shift.

1.1.1 Subshift of Finite Type (SFT)

A subshift is a closed, shift-invariant subset of a full shift. Equivalently, let D be any
set (finite or infinite) of cylinder sets. The set SD of sequences that do not contain

any element of D is a subshift, and any subshift can be expressed in this form.

Example 1.1.2. Even shift: Let SD be the set of sequences consists of 0 and 1’s so

that between any two 1’s there are even number of 0’s. So, D = {1(0)2k+11 : k E N}.

Definition 1.1.3. A subshift of finite type ( SF T ) is a shift space that can be described
by a finite list of forbidden cylinder sets. Another classical definition is as follows:
Let A be a square {0, 1} matrix with its rows and columns indexed by {1, . . . , n}.

One can define a closed, shift-invariant subset 2A of 2;!” (or En) by selecting sequences

13

E E Z A with the rule that A5137.+1 = 1 for all 2' E N (or Z). The dynamical system
(2A, 0) consisting of this compact space E A and the restriction of the shift map is
the one—sided ( or two-sided) subshift of finite type defined by A or topological Markov
shift defined by A. The matrix A is called the transition matrix.

Example 1.1.4. The golden mean shift 2 A is given by the transition matrix A =
l 1

1 0

Remark 1.1.5. Although subshifts of finite type have very nice properties which we
will not discuss here, the Lozi maps which are discussed in this dissertation can not
be represented by SFT’s. In general, there are infinitely many forbidden cylinder sets.

On the other hand, they can be approximated by SFT's (See [15]).

1 .2 Topological Entropy

Topological entropy is a quantitative measurement of how complicated a map f is.
A rough interpretation could be given as follows: Suppose one can distinguish two
distinct points only if the distance between them is larger than a resolution 6. Then
two orbits of length n obtained by taking n-iterations of these points under f can
be distinguished provided that there is some iterate m between 0 and n for which
their distance is greater than e. Let r(n,e, f) be the maximum number of such
distinguishable orbits of length n. The entropy for a given 6, h(e, f), is the growth
rate of r(n. e, f) as 72. goes to infinity. Then the entropy h.( f) is the limit of h((:. f) as

the resolution 6 goes to zero.

Definition 1.2.1. Let f : X ——> X be a continuous map on a compact metric: space
(X, d) with a metric (1. Two distinct points at, y E X, 1? 3A y, are called (n, e) -sepanated

for a positive integer n and E > 0 if there is at least one 771, 0 S m S n, such that.

14

(l(f"‘(1r),f""(y)) > e. A set U C X is called an (n.e)-sepamted set if every pair of
distinct points I, y E U, r yé y, is (n, e)-separated.
Let r(n, e, f) be the maximum cardinality of an (n, e)-separated set U C X. By

compactness. this number is always finite.

 

l .. .
Define h(€.f) :: lim sup Og(7‘(:’€ f))

Tl —"00
defined as:

. Thcn topological entropy of f, h(f), is

h.( f) = lim h.(., f).

6—»0,€ >0

Note that this limit exists (can be infinite) because for 0 < 62 < 61, r(n, 61, f) S

r(n., (2, f), so h(e, f) is a monotone function of 6.

Remark 1.2.2. The original definition of topological entropy was given by Adler.
Konheim and McAndrew, [1], using a different idea involving covers of open sets.

The definition above was given by Bowen[30] and independently by Dinaburg[7].

Remark 1.2.3. Note that the Lozi map 5 is defined in R2 which is not compact. To
be able to investigate the topological entropy of the Lozi maps, we take one-point
compactification of R2 and extend the map continuously putting C(00) = 00. For

more details about this continuous extension see [16].

Now, we would like to summarize some important theorems related to topological

entropy. Most of the proofs can be found in [31].

Theorem 1.2.4. Let X be compact, f : X ——> X be a continuous function and

k 2 1 be an integer. Then, the entropy of fk is equal to k times the entropy of f,
W") = khm.

Remark 1.2.5. If f is a homeomorphism, then theorem becomes h(fk) = |k[h( f ) for

any integer k.

Note that we make use of the above theorem when we prove our discontinuity re-

1+\/(5)
2

sult. In other words. we show that. for some specific parameters h(£4) > log

1 1 1
and this implies h(£) = 4—h(£4) > Elog-—+§\/—‘(—52.

The next theorem says that the wandering orbits do not contribute to the entropy.

Theorem 1.2.6. ([30]) Let f : X —> X be a continuous function on a compact metric
space X. Let Q C X be the nonwandering points of f Then, the entropy of f equals

the entropy off restricted to its nonwandering set, h(f) = h(f|fl).

Another related theorem states that any map whose nonwandering set is a finite

set of points, has zero entropy:

Theorem 1.2.7. Let f : X -—> X be a continuous function on a compact metric space
X for which 9( f) is a finite number of periodic points. (For example Morse-Smale

difieomorphism} Then the entropy of f is zero.

We use the above theorem to prove the zero entropy results for some parameters
in the Lozi map. In other words, we show for some specific parameters, the nonwan-

dering set consists of periodic orbits only.

Now, let us give the definition for semi-conjugacy and conjugacy which will allow
us to make comparisons between the dynamical properties(such as entropy) of two

systems:

Definition 1.2.8. Let f : X ——> X and g : Y —-+ Y be two maps. A map k : X —+ Y

is called a semi-conjugacy from f to 9 if:
c k is continuous,

o k is onto and

16

o kofzgok.

The map k is called a conjugacy if it is a semi-conjugacy, one-to-one and has a.

continuous inverse. i.e. k is a homeomorphism.

The next theorem gives important relations about the entropy of semi-conjugate

and conjugate systems:

Theorem 1.2.9. Let X and Y be compact metric spaces. Let f : X —) X and

g : Y —) Y be tow maps and k : X —) Y be a continuous map with k 0 f = g o k
(i) Ifk is onto (i.e.. a semi-conjugacy), then h(f) Z h(g).
(ii) If k is one-to-one (not necessarily onto), then h(f) S h(g).

(iii) Ifk is onto and uniformly finite to one, then h(f) = h(g). Note that k can be

a conjugacy in this case.

Symbolic dynamics may be very helpful sometimes determining the entropy of a
given system. If one can find a conjugacy or semi-conjugacy between a subshift and
the given system, then we can use the above theorem to understand the entropy of
the system. This is actually the idea behind the monotonicity results given in this

document.

On the other hand, finding the entropy of an arbitrary subshift can also be difficult.

The next theorem gives a useful tool to compute the entropy of subshifts:

Theorem 1.2.10. (1') Let a : En —+ 2,, be the full shift on n symbols {one-sided
or two-sided). Assume E C 2,, is a closed invariant subset. 50, (EUR?) is a

subshift. Let wm be the number of words of length m in E, i.e.,

, I
j forO Sj < m for some 5 E Z}

(I)

L07" : {(50 '° '95711-1) 15-7-:

17

Then,

lo to
h(o|E) = lim sup M
1n—+OC m

(ii) Let Aux” be a transition matrix on n symbols. Let a A : 2A —+ X] A be the
associated subshift of finite type (one-sided or two-sided). Then h(oA) = log()\1)
where /\1 is the real eigenvalue of A such that A1 2 IAJ-l for all other eigenvalues
Aj of A.

Example 1.2.11. Using the second part of the above theorem, one can easily see

1+\/(5)

that the entropy of the golden mean shift (defined before) equals to log——2——.

18

Chapter 2

MONOTONICITY OF ENTROPY
AND ZERO ENTROPY
PARAMETERS

2. 1 Preliminaries

Since its discovery in 1976, the Henon map[ll] has been one of the most studied
examples in dynamical systems. It was introduced by M. Henon as a simple model
exhibiting chaotic motion. On the other hand, the Lozi map [22] which is a piecewise-
affine analog of the Henon map has been also important since it has a simpler structure

but similar chaotic behavior.

The Henon family is defined by:

.7: 1— 0.12 + by
H=Ha.b: 5—» , (1,1)ER, b7$0
y :1?

19

while the Lozi family is defined by:

x 1—a|x|+by
£=£(I,l): H I 0.,bER, bTéO'

'y I

2 in the Henon family is replaced by the piecewise affine

Thus, the quadratic term ax
term alxl. This results in a considerably simpler family of maps. For instance, in [26]
the existence of attractors is proved for a large set of parameters, while in the Henon
family, this is only proven for very small b 74 0(see [2]).

In this article we improve some of the entropy results obtained by Ishii and Sands
in [14] and give some partial results about the parameters at which the tepological

entropy of the Lozi family is zero.

The following result about monotonicity was obtained in [14]:

Theorem 2.1.1. For every a... > 1 there exists b... > 0 such that, for any fixed b with

|b| < b..., the topological entropy of £01, is a non-decreasing function of a > a...
Our results can be summarized in the next three theorems:

Theorem 2.1.2. For any fixed a* in some neighborhood of a = 2, there exist b’] > 0
and b; < 0 such that the topological entropy of £01, is a non-increasing function of b

for 0 < b < b’[‘ and a non~decreasing function of b for b; < b < 0.
Let us define 1R2)1+ = {(a, b) E R2 | a >1+|b|}.

Theorem 2.1.3. For every 1 < a S 2 there exist 1\"(],N3 E R+ and two lines 712 :
(‘51.2= 61.2) —) Riv—v 61,2 > 0, given by 71(t) = (0+Ngt, —t) and72(t) = (a+NEt,t)
such that the topological entropy of £71“) and £720) is a non-decreasing function. of

1‘.

Theorem 2.1.4. In a small neighborhood of the parameters a = l and b = 0.5,

topological entropy of Lab.- lltop(£a,b).- is zero.

20

Remark: The proof of this last result can be extended for other parameters as well.
But it gets complicated especially when b is close to 1. So we give some numerical
results for such parameters and obtain a picture (See Figure 2.6) for the zero entropy
locus H0 = {(a, b)| ht(,,,(£ay,) = 0} when a > 0 and b > 0.

Outline The remainder of this chapter is organized as follows. Section 2.2 gives an
introduction to the Pruning Theory and some results by Ishii and Sands that we are
going to use. Our monotonicity results are proved in Section 2.3. Then, Section 2.4

extends these results. Section 2.5 describes the results about the zero entropy locus.

2.2 Pruning Theory

The Pruning Theory was suggested by Cvitanovié[6] as a way of obtaining symbolic
dynamics for the Henon map. Certain conjectures were formulated which still remain
unproved. Motivated by this, and following suggestions of J. Milnor, Ishii[12],[13]
provided an analogous Pruning Theory for hyperbolic Lozi maps (ie. those satisfying
a. > 1 + |b|) and proved an appropriate ”Pruning Conjecture” which yielded a good

symbolic description of the bounded orbits of hyperbolic Lozi maps.

Let us recall the basic elements of this Pruning Theory:

Let 2 denote the symbol space {—1,+1}Z with product topology. Define the
shift map a : 23 —+ E which is a continuous map given as o(. . . 5-2,e_1 - 50,51 . . .) =
(...E_2,€_1.€0-51...). For any _5_ E Z we call ,2“ = (. . .e_2,e_1) the tail of; and
is = (50,51...) the head of ,2. Let C" and C" be the set of all tails and heads,
respectively. So 2 may be identified with C“ x C3.

Define p(. . . , 3-2, 5-1)(0, b) = 1— bs_2 + b23_28_3 — b33_gs_3s_4 + . ..

where 3,, is defined as

 

 

 

 

 

s" E (2.1)
b
——asn +
b
“(1571—1 'l'
b
_0'571—2 + —
Similarly define q(.—:0, 51 . . .) = r0 — foil + was — . . .
where f.” is defined as
1
7‘.” E (2.2)
b
0511 ‘l"
b
(15114.1 +
b

a'5n+2 + _-

Note that p(§"‘)(a, b) and q(§5)(a, b) are defined on C“ x R2>1+ and 05 x R2>1+’
respectively. In the rest of the paper, we identify p with poiru and q with q o in, where
77'” : 2 x Riv, —+ Cu x 1R2)1+ is the map (§)(a,b) —» (§”)(a, b) and 719:2 x Rift ——>
CS x R2)1+ is the map (§)(a,b) —+ (§'9)(a, b). So, we consider p and q as functions
p,q:ExR2>1+—>IR.

For the proof of the next lemma, see lemma 4.3 and 6.1 in [12].

Lemma 2.2.1. For fixed 5 E Z, the functions p(g), q(§), sn(§), rn(§) : R2)1+ —> 1R
are real analytic in (a,b). Moreover, p, q, 3", in and their partial derivatives with

respect to the three variables (a, b,§) are continuous.

Definition 2.2.2. We call

put) E {i E 2| (P- Q)(---5—2,5—1 '50.&‘1...)(a,b) = 0}

22

the pruning front of £01, and
Dab E {2’ E 2 I (I) - (1)(- --€—2,€—1 -€0,51...)(a,b) < 0}
the primary pruned region of £091, The pair (7901,, Deb) is known as the pruning pair

of [rab-

We call #40,!) :—: Z \ UnEZ onDafib = {9: E 2 I (p — q)(o"§)(a,b) 2 mm 6 Z} the

admissible set.
Definition 2.2.3. The set P“, E Pub 0 A“, is the admissible pruning front.

Let K = '1; denote the set of all points whose forward and backward orbits

remain bounded. For a point X E K we put 7r(X) = (. . . e_2, 5-1 - £0, 51 . . .) where

+1 if but), > 0
52'. E a: if £i(X)x = 0
—1 if but), < 0

where * can be both +1 and —1; and Yx is the x-eomponent of Y. An element of

77(X) is called an itinerary of X. So a point X can have more than one itinerary.

Now let us define the standard partial orders on C 3 U C'”:

Definition 2.2.4.

1. Let g" and Q“ be two distinct elements in CS. Then there exists the smallest
number i Z 0 such that 5,: ¢ 6,. W’e say :5 <,.-,. Q's if one of the following is
satisfied:

(i) The number of +1‘s in {0 . . . 5,-_1 is even and 5,: < (52-,

23

(ii) The number of +1‘s in so . . . 8,-_1 is odd and 5,: > 6,,

where order on the symbols is -—1 < +1.

2. Let g” and _6_'” be two distinct elements in C“. Then there exists the largest
number i < 0 such that 82' aé 6,. When b > 0 (resp. b < O), we say _e_“ <1, 6“ if

one of the following is satisfied:

(i) The number of —1’s (resp. +1’s) in 5,-_1 . . .50- is even and 5,: < 6,7,

(ii) The number of —1’s (resp. +1’s) in e.,-_1 . . .50- is odd and e,- > 6,, where

order on the symbols is —1 < +1.

See Fig.2.] for the case b > 0.

In [12], Ishii proves the following version of the Pruning Front Conjecture(PFC)

which was motivated by Cvitanovié et al [6].

Theorem 2.2.5 (the pruning front conjecture). Suppose that Lay, satisfies a > |b| +1
and let _5_ 6 {+1. —1}Z. Then there exists a point X 6 K5 such that g E 1r(X) if and

only if a"; does not lie in Due], for all n E Z.

Next, we will summarize the results of Ishii and Sands [14], which prove the mono-

tonicity of the entropy in the positive a-direction.

Recall that the tent map Ta : R ——> R is given by Ta(x) = 1 — alx].

Definition 2.2.6. An itinerary of a point x 6 1R under the map Ta is an element of
i,,,(x) E {55 E C" | 5,1203) 2 0 Vi Z 0}. We call [{(a) E i(,,( 1) the kneading invariant
of Ta,

A

Proposition 2.2.7. Suppose I < a S 2. Then 7r,,~('Pa.0) = [{(a) where rrs : Z —-> C8 is

the map g —> gs

24

Lemma 2.2.8 (Stability of ’P). Suppose a > 1 + [b]. Then for every neighborhood U
of Pay, there exists a neighborhood V of (a, b) such that Pa 8 C U for every (5., b) E V.

A

o o v “I I . I "
Defimtlon 2.2.9. We say that (”Pa (”Ad b) < (’P~ “A. .) if A,, b C A, . and P, . fl
‘ " ‘ a,b a,l) " a,b a..b
AaJ) : 0

The main step in the proof of the monotonicity in [14] is the following theorem:

Theorem 2.2.10 (Local Monotonicity). Suppose f : (—6, 6) —-> REM, 6 > 0, is C1

and

dip — q)(:)f(t)
dt t=0

 

>0

for all g E 73f(0)- Then there exists a C’1 neighborhood .7 of f and a neighborhood

I of0 such that for any C1 curve 9 E .7: the map t E I -4 (Pg(t),Ag(t)) is order

A

preserving: ift1.t2 E I and t1 < t2 then ( gltl)"Ag(t1)) < (Pg(,2),Ag(t2)).

A

It is also proven that if (”PayAab) < ('P . A. 5) then ht0p(£a.b) S ht0p(£

a,b’ a. (i,b)°
0(1i - q)(:)(a» 0)
do

they use local monotonicity to prove the following:

 

In [14], Ishii and Sands show that > 0 for any g 6 730.0. Then

Theorem 2.2.11. For every a... > 1 there exists b... > 0 such that the map a E

A

(a*, 00) —> (Pugh/4011,) is order preserving for all |b| < b...

So Theorem 2.1.1 follows from these facts.

2.3 Results about the monotonicity of the entropy

In [13], Ishii mentions that although we have monotonicity in the direction given
above. we do not know anything about the monotonicity in b direction. We look for

a solution to this question near the point ((1,1)) = (2. 0).

Now we want to concentrate on the point (a, b) = (2, 0). We will first figure out
the set 7320. Using the stability of ’P this will give us some information about P293,
for [b] small. After that we will use the local monotonicity by taking b-derivative of

(p — q) to show the monotonicity in b-direction around (2, 0).

Proposition 2.3.1. Let 6" = (+1,—1,—1, —1...). For (a, b) = (2,0) we have 152,0 =
77:1(63) = {6“ - +1,—1.—1,—1...|6_” 6 Cu} and D10 = Q).

Proof. First note that by Proposition 2.2.7, rig-(752,0) = [{(2) = (+1,—1,—1, —1...).
So. 75290 C 7r.;'1(6“‘). To prove rr;1(63) C 7520, we need to show that for any
6" 6 C” the sequence 6 = (6“ - +1,—1,—1.—1...) is in 7520 = A290 ('1 732,0, i.e.
(p - q)(o"’§)(2,0) Z O for n E Z and (p — q)(6_)(2,0) = 0. Note that for an ar-

 

1 . 1 1
bitrary _E_ E Z, p(§"’)(2,0) = 1 and fin = — and q(§"’)(2,0) = — — —2-— +
1 1 2511 250 2 8051
—.——— — + —1 7' + So, 53 2,0 is maximized at onlv
23505152 ( ) 2"+15051...en (IL )( ) .

i=oo i

- I

6‘“ = (+1,—1,—1,—1...) and its maximum value is Z (5) = 1. This shows that
i=1

for any (_5 E its—1(6“), (p — q)(6)(2,0) = 0 and (p - q)(o"_6_)(2,0) > 0 for n 75 0. This

proves 7Ts-l(é3) C P290 and also D270 = (ll. Cl

Lemma 2.3.2.
3(2) — t1)(§)(2~ b) 1

 

 

for g 6 P21).

Proof. Recall that
p(. . . , 5-2, €_1)(a, b) = 1-l).5‘_2 + b23__23_3 — b3s_gs_3s_4 + . . .

and
(1(50951- - .)((I.,l)) = 7:0 — 720721 + 72072173 -— . . .

26

where s" and fn are given by (2.1) and (2.2). Taking the partial derivative of p with

 

 

respect. I) we get:
53—15 2 —s_2 — b.S'I_2 + 2bs_23_3 + b2(.'_28_3)’ + - --
Since .9." are analytic V17. 5 —2 we obtain:
01) 1 1
.07) b=0 : —S_2lb=0 = (15-2 = 25-2
= (+1,—1,—1,—1o) we have

g such that g3

73%; first note that for
where :1: = (a — Va? + 4b)/2 (See 3.2.2).

Now for
b =
) (a. + :r) (b + :r.)
is continuous with respect to b; a calculation(See 3.2.3) shows that:

 

 

 

q(§)(a,
Since %
2
ob b=0 — (Ho 0b ’ 1).—.0 0b (a + 1:)(1) + x) ‘ a(a - 1)?‘
So for a. = 2 we have % b=0 = . D
— s 2.
0(1) Q)(_)( ml _ depends on 5-2.

 

(9b

The previous lemma says that. the sign of

Proof of the Theorem 2.1.2. First let’s define:

.€_3,_1,+1-50.81.€2-“}

Z {---€_3,+1,—1'50,51.E‘2"-}

Also define the curve f(t) by t E (—6, +6) —+ (2, t) E Ril‘t where 6 > 0.
a” 6 C“) by Preposition 2.3.1

Note that we have 7520 = {5‘ - +1. —1,—1,—1--

and D2,” is empty.(See Figure 2.1).
27

cu
...+++.

IIIfi+I
DID-+I

III+-+I

O I I-+-I

I O I++-I

 

CO

I '+'+++
l+£1114|

Figure 2.1: Symbol space (b > 0) and the sets XJ’, Z and C

28

(P — q)(§)(2, 1))
8b

 

Then by Ixamma 2.3.2, 0 is positive for g E 73230 n (X U Z) and

negative for g 6 ”P10 0 y.

By continuity with respect to g there exists a cylinder set C around 752,0 such that

0(1) -— q)(;)(2-. b)|
0b b=0

 

>0f0r§ECfl(XUZ)

and

0(1) — q)(:)(2~ b)

0b lb=0<0for§ECfly.

 

Again by continuity with respect to b, there exists a neighborhood B C (—6, +6)

3(7) — r1)(§)(2ab)
0b

 

around 0 such that if (2, b) E f(B) we have > 0 for g E C (W (X U Z)

and 0(1) — qgéélag b) < 0 for g E C H y.

 

Now we want to show that for b > 0 and small, 152,!) F) (X U Z) is empty.(See
Figure 2.3)
To do this, first observe that C“ x C is a neighborhood of 752,0. By stability of
fi(Lemma 2.2.8) there exists a neighborhood V of. (2, 0) such that V(a, b) E V 7501, C

C.
(9b
exists a neighborhood B C (—6, +6) around 0 where (p - q) (g) is increasing when b is

> 0 for g E C D (X U Z). This means there

 

We also know that

increasing. This implies there exists b’i‘ > 0 such that for every (2, b) where 0 < b < b’f

and for every g E C n (X U Z) we have

(P — (1)(§.)(Qa b) > (p - q)(§)(2= 0) 2 0

In particular this tells us that all elements of 752,1, are in C n y. But then we know

29

0(1) - (1)(§)(2J))
8b

is non-decreasing as b demeases to 0.

 

that for these elements < 0 and so using Theorem 2.2.10 the entropy

A similar argument applies for b < 0 and small where it can be shown that.

1321, C C H (X U Z) and that the entropy is non—decreasing as b increases to O.

2.4 Extension of the results to 1 < a S 2

In this section we would like to prove some monotonicity properties for other a values
as well. However, we are not able to prove the monotonicity in the vertical direction

because it is not possible to use local monotonicity when we move away from a = 2.

5(1) - <1)(_s_)(a.b) is

The reason behind this is the fact that for such (1’3 and small I), 0b

 

positive for some g E 7303;, and negative for some other 5 6 750,5.
So we prove the next best thing: Monotonicity in the direction of lines which make
some angle with the a—axis(See Figure 2.2). To prove this result we modify and use

some of the computations done in [14].

Lemma 2.4.1. (Lemma 11 in [14]) Suppose I < a. S 2 and g3 E n(a). Then

a3 + 2a2 — 6a. + 2 < 0(p — q)(§)(a., b) < a3 + 2a.2 — 6a + 4
2(12(a — 1) _ 0a (a..0) — 2a2(a — 1)

 

 

 

 

6(1) - (I) (i) (a: b) I

Z(\/§-1)/2>0'ifaZ\/§-
(9a

In particular;
(a .0)

Lemma 2.4.2. (Corollary 13 in [14]) Suppose 1 < a S 2 and g3 6 16(0). Then

00) - q)(§)(a, b)|
(

> 0.
(9a

(1,0)

 

Lemma 2.4.3. ( Corollary 7 and Equation 3.11 in [14])

30

Suppose I < a S 2 and is E K.(a). Then

Z(_1)i=‘0_--_:€L-_1 = 0 (2.3)
i=0 at
and
QC ; ,-5()...Ei+j 1'50...€i_1 .5
Z<*1>’+’7+7+T‘ =l—1l—aT—m) <24)
i=0

where we define the empty product so . . .s_1 to equal 1.

Now, we use these results and similar techniques to prove the following:

Lemma 2.4.4. Suppose I < a S 2 and _.-:_s E h‘.((l). Then

 

 

 

 

 

 

 

 

1 —2a2 + 7a — 2 < 0(p — q)(§)(a, b) ‘ 1 ——2a2 + 7a — 8
0.5.2 2a3(a. — 1) - 6b b=0 _ as_2 2a3(a — 1)
e .b 1
Proof. From the proof of Lemma. 2.3.2 we know that MI 2 . So we
(9b b=0 (151.2
6 5 .b
need to find some upper and lower bound for _(m—dbulb 0.
Remember that q(50, 51 . . .) = 1‘0 — fofl + 1:01:11“? - . . ..
00
Let’s write q(€0,51...) = TO—T1+T2—T3+. .. = Z (—1)"Tn where Tn = foal . . . an.
n=0
Now we have the following:
r | — ————l — l
71 _ — A _—
b—O as" + b7’-n,+1 (=0 (1
and
. 0(7‘ )(€)(a~~b) . - - 5
I n _ . 2 ,I _ +1
rrzlb=0 = 8b lb=0 = — r,,(rn+1+brn+1)|b:0 — — :3

Taking term by term derivative of q, we. get the following. Note that 50 = +1 and

0)

1=—1:

(1"

I J 1 1

g:

31

 

 

I I. €151 5052 1 8052
—-=— r+M‘ +—+——=—
1 (r 710 0 1): a3 a a a3 a4 a4
I . AI 52 50 805153
T2 = (7'07"1) + (7071)72 = —5 — I1?) -— a5

 

 

 

 

 

I 5253 505 5051 50515254
_T3= —(r0r1r2)r3+(r0r1r2)r r3: (16 + a6 + a6 + a6
A A . . A . . A . A 525354 505354 505154 505152 505152535:
T’:7‘T'I‘07"1‘~FTTT9T r'=— — — - - —
4 (()1 - 3) 4 ((ll _ 3) 4 a7 a7 a7 a7 at
(9 a,b
Note that. ‘1‘ 3: LE _ Ta— 7" + T'— --+(—1)"T.’.+
I I 1 n—l
Claim: Tn = (mrl...rn) = -—an+3(Z:OEO-~5i5i+1~~57! + €051...E,‘,5n+1),
, z:

n. 2 2 where 50 . . 3.5-1375,“ . . . en means 5,: is missing in the term.

5‘71.

Proof of the Claim: Note that T’-— — (Tn_1rn)’= ,,_1—+ (forl . ..en_1)7=;, =

E . S E . . . E _E , ,
T,’l_1—n— — O 1 n n+1. So the claim follows by induction.
a an+3 ~

 

32

 

.2538 mEESQmESU 2: E 953 23 mo 52m. .2: 385$ 33 ame— mfi “as... was: $2 .38 mzmcaoagtg .2: E

£53 2: mo .55 23 3 12:5 3. :CIVV:E:_8 $5 2: 5 £53 23. “5.5.“ 5.55.: miko=£ 2: E 2.23 .2: osiewao m: and

 

 

 

 

 

.5 —“2 I
v .
MN+I|I .m. .15. 2.3. m... r. $5.:va
NQCIu 8
n+2: n+2: n+ee Tree 2 2 v
I.I... I l .iii... I l ,I... | y... I), | .2 |
:.....I. 5:; _l .55; a wrist easel .
label hell he I Ibeln hi a.
nmmmmmfimcm mmfimcm «.mflmom «.mnmcm wmnmmm L.
.c o: o: a: m.
3.355 $8 3.6 Ed 1 I
we alel %I a
”cameo so we 1..
v: v: H
mmom fi \ I

 

 

33

Since the series which give the derivative of q is absolutely convergent, regrouping

 

 

 

 

 

the suitable terms together, we can write:
8q(5)(a. b) 1 5052 0C
1 ‘ = — + *. + S + R
db |b=0 a3 a4 Z n
1121
where
505152 50515253 5051525354
*1 = _ .- + - +
ao ab a7
_ Sofia 80:1525354 EOE—15535455
*2 — (i _ 7 + 8 _
a a. a

511—15715n+1 50- - ~5n—15n5n+15n+2
+ (—1)1)+1 .- + o a 0
“n+0

__ ._ ,150.
*n'—'( 1) “n+4

 

and
1 5. 5051
5:51—33 F‘ ”Ll—1) ai+5
and
5051553 8051-52554 j+150"'5j-1:5j+1
=:—- . . —- . i>
R (1.5 + (16 +( 1) aJ+3 J 2

First let’s start. with observing that by the equation (2.3):

 

 

_ 1 50 5051 505152 _
Secondly,
I 1 1 1
R<—rl+-+—+... :—
l I — (1" a a2 ) a4(a — 1)

34

For 2101:] *n, by the equation (2.4) we have:

i:

2 - 2-051 2
*1(-a 50:1) = (—1) 02 Tall)

 

9 505152 .
*2(-a‘€1€2)= (—1)3 T"(1>

T a.
and
*n(—(125-n_1€n) : (_1)n+li:1;_fiflnn+l(l)
SO
x 1 30 "0:1 ‘2' 1 '+2
,- - ...-‘_ g I
n=l i=0
Again by (2.3) we also have
00 _ l :x: V775051...€i_11i+2
Z *7, — ——4— EH) —i——(Ta <1) — a)
a. . a
17:1 2:0
1 T 1 - 1— T 1 -
for any a 6 IR. Let a = —+——a(—) and 0 = _—§£(—) = %. Since 73(1) 6 [Ta(1), 1]

for every 2' Z O we have —6 S T};( 1) — a S 6 for every 2? Z 0. Note. that by direct

calculation 7112(1) — a = 6(2a — 3). This gives us:

0° 1 5 s s
”glint = _le(Ta2(1)— Ct — f(T3(1)— a) + %(Tj(1)— 0') _ . . )

1 1 1 1 2a2 — 5a + 2
<——52.—3-(5— — — =——r—— 2.7
_ (14((0 ) (a.-+_cz.2_*-(L3+ )) 2a‘5(a—1) ( )
Similar calculations show that.
so 2 _
Z *n. 2 -2%-35;:£ (2-8)
12:1 a (u— )
Now. combining (2.5).(2.6),(2.7) and (2.8) we get the desired result. [:1

Proof of the Theorem. 2.1.3. By lemma 2.4.2 we know that for any 1 < a S 2,
0(1) - q)(§)(a—~ b) l

 

> 0. Also by the previous lemma for any such a,

 

Ba ((1,0)
' — 5 .1'
6(1) 051:.)(0, )) |( 0) has an upper and lower bound. So there exist NJ 6 R+ such
a. _

that

N1 80) - a)(:)(a, b)
‘ “ 6a,

_ 3(2) - q)(:)(a-, b)
((1,0) 81)

 

 

 

 

>
((1,0)

and NE E R+ such that

0(1) - q)(§)(a, 1))
((1.0) + 0b

 

 

N2 0(7) — (1)(§)(a. b)
i a 0a

 

 

(a,0)

This means that the directional derivatives of (p — q)(_s_)(a,b) in the direction vi =
(Ni, —1) and v2} = (NE, 1) are both positive. So by local monotonicity theorem,

result follows. E]

8(19 — (1)(§)(a,b) and
do

we can compute the directions in which the entropy is non-decreasing

 

Since we have explicit upper and lower bounds for both
(9b
(See Figure 2.2).

 

36

 

 

| X 1\‘ 1
l N l “I \l \\ \\\\ l \\\l
1 2 1.4 1.52 16 1.88 2
6’2 = (10,1) 172 = (2,1)
< 27' = (3,1)
(
l ...—————>l Ml /l 14/7V/r/l/ll
If! I f l / l / ////// l
12 1.43 1.59 1.7 (I2

Figure 2.2: This figure shows the approximate monotonicity results for different N01
and N3 values where 1.2 < a S 2. The topological entropy is non-decreasing in the

direction of arrows.

37

 

 

Cu
Cu

 

 

 

 

 

 

 

 

 

 

 

cs 03
(i) a=2 and b=0 (ii) a=2 and b=0.1
3 3
O 0
CS Cs
(iii) a=1.95 and b=0 (iv) a=1.95 and b=0.01

Figure 2.3: This figure shows the primary pruned regions, Dab, of maps for given
parameters. The af-axis represents 03 and the y-axis represents C". One can expect
to find some elements of Pa), at the boundary of Dab-

38

 

 

 

 

CS

Figure 2.4: Primary pruned region, Dab: for original parameters studied by Lozi:
a=1.7 and b=0.5

2.5 Results about the zero entropy locus

In this section we turn our attention to the parameters for which htop(£a,b) = 0.
Note that it is enough to consider the maps with |b| S 1 since the maps with |b| > 1

are. up to affine conjugacy, inverses of the maps with |b| < 1.

Let us first. review the following theorem:

Theorem 2.5.1 ([15]). If the Lozi. map £a,b satisfies either (2) —1 < b < O and
a. S b — 1, (2'7?) 0 < b 31 and a. S —b+ 1, then hf0p(£a,b) = 0.

Proof. If a _<_ b—l _<_ -—a then £0), has no fixed points. When b < 0, Lay, is orientation
preserving, so by Brouwer’s translation theorem[3] it has an empty non-wandering set
and therefore zero entropy, proving (i). When 0 < b S l and b — 1 S a S l — b,
there exists a unique saddle fixed point p = (1/(1 + a — b),1/(1 + a — b)) in the first
quadrant. Also note that there is no other period-two points. Now v8 = (A, l) where

A = (—a + Va? + 4b)/2 is a stable direction at p and W’flp) = {p+ vst E 1R2|t > O}

39

is invariant under £01,. Also 1&2 \ (111(1)) U {p}) is homeomorphic to R2 and £2 b has
no fixed points there. Since £39,) is orientation. preserving when b > 0, htop(£n..b) =

Now let us start stating our results by the following theorem:
Theorem 2.5.2. For a = 1 and b = 0.5, ht0])(£a.b) = 0.

Proof. First note that when 0 < b < 1 and 1 — b < a < b + 1, £031, has two
saddle fixed points: p1 = (1/(1 + a — b), 1 / (1 + a. — b)) in the first quadrant and
p2 = (1/(1 — a. - b), 1/(1 -— a — b)) in the third quadrant. Also there are two at-
tracting period-two points: m = (N, (1 — aN) / (1 — b)) in the fourth quadrant and
112 = ((l—a. ’)/(1—b), N) in the second quadrant where N = (l+a-b)/[(b—1)2+a2].
By a direct calculation of £31), one can check that there are no other period-four

points.

Now of = (Xi, 1) where A3 = (—a + Va.l + 4b) / 2 is a stable direction at M and
I'Vflpl) 2 {p1 + v‘ft E R2|t > 0} is invariant under £01,. Similarly. 22% = (—/\u, —1)
where N; = (a+ v a.2 + 4b) / 2 is an unstable direction at p2 and VVflpg) = {p2 +12%. 6

Rzl t > 0} is invariant under £01,.

The more challenging part is to show that the right and left parts of the unstable
manifold of p1 are attracted by 77.1 and n2, respectively. We will show this happens
when we consider £4. Now, let Z be the intersection of the line 61 = {m + "off 6
R2|t > 0} and the r—axis where vi‘ = (—/\'1‘,—1) and A“ = (-a — MHZ See

Figure 2.5.

Claim: For a. = 1 and b = 0.5, £3212) ——> 11.1 as m —+ 00.

4O

Proof of the claim: Let us use £135 2 1:. Let P be the polygon whose corners
are given by Z, £2(Z). 134(2) and 116(2). Since 58(2) m (1.223, —0.375) is in P,
£2(P) C P, i.e., P is invariant under £2. Now consider the Lyapunov function
V(.r., y) = (:1: — 771(711))2 + (y — 772021))2 where m :R2 -+ IR and 7r2 : R2 —-> R are the
projections to the :r-coordinate and y-ooordinate, respectively. It is not hard to see
(with the help of a computer if necessary) that V(£4(:r. y)) — V(:1:., y) < 0 ,V(a:, y) E
P \ {m}. This implies that,(see for ex. [8]) Z (actually every (23.3)) E P\ {n1}) is

asymptotically stable to 72.1 under £4.

Similarly it can be shown that £(Z ) is asymptotically stable to n2 under iterations
of £4. Now let W,-(p1) be the forward iterations (under £4) of the line segment
connecting p1 and Z. Similarly let Wg(pl) be the forward iterations (under £4) of
the line segment connecting p1 and C(Z). To complete the proof of the theorem. we
apply Brouwer’s translation theorem to £4. Note that R2\(l ’ i (p1)U{p1}UW1‘ (p2) U
{m} U Wr(p1) U {m} U Wg(p1) U {n2}) is homeomorphic to R2 and L4 has no fixed

points there. Since .6“ is orientation preserving ht L = hto £4 4 = 0. E]
0P P

Proof of the Theorem 2.1.4: The proof of the above theorem, using similar Lya-
punov functions, works for the parameters in a. small neighborhood of (a, b) = (1, 0.5)

as well.

Remark: When we move away from a neighborhood of (a, b) 2 (1.0.5), it is
sometimes the case that the unstable manifold of the right fixed point intersects with
the stable manifold of the same fixed point causing a homoclinic point and positive
entropy. The parameters for which £a,b is numerically observed to have zero entropy
is given in Figure 2.6. For more details see [32]. Note that since positive entropy
occurs as a result of a homoclinic intersection of the stable and unstable manifolds

of a fixed point (which are piecewise linear), the boundary of the zero entropy locus

41

is expected to be piecewise algebraic. But writing the equations explicitly requires
more work.

The case a=1+b: “'hen a = 1 + b and b > 0, it can be shown that the
portion of the line f : y = —.r + (1 — b2)/(a(1 + b2)) that stays in the region given by
1+aat+by Z O, 1 —a(1+a.r+by)+b;r S 0, :1: g 0 and image of that portion of the line
f under £0”), give all the period-four points except the fixed points of £01,. In other
words there are infinitely many period-four points that lie on two line segments. But
it can be again observed numerically that as long as there are no homoclinic points,
the unstable manifold of the right fixed point is attracted by these two line segments
causing the entropy to be zero. Note that when a > 1 + b, the period-two points
become saddles, so we can expect that some portion of the line a = 1 + b, b > 0 is a

part of the boundary of the zero entropy locus. See Figure 2.6.

42

Wis-(P1)

 

52(Z)

x

 

V

Figure 2.5: The picture shows the unstable and stable manifolds of the right fixed
point of £1.05.

43

 

 

 

//W

88888
0. 0. 0. 0.

 

 

 

//W

2222222

Chapter 3

DISCONTINUITY OF ENTROPY
FOR LOZI MAPS

There have been some recent developments in the study of piecewise affine surface
homeomorphisms. In [16], Ishii and Sands give a lap number entropy formula for
piecewise affine surface homeomorphisms and in [4], Buzzi proves that under the as-
sumption of positive topological entropy, there are finitely many ergodic measures
maximizing the entropy. He also shows that topological entropy is lower semi-

continuous for these maps. The following question was asked by Buzzi:

Question 3. Prove or disprove the upper semi-continuity of entropy for piecewise

afline homeomorphisms of the plane.

Also, Ishii and Sands, motivated by their rigorous entropy computations for the

Lozi family, made the following conjecture:

Conjecture 2 (Ishii and Sands [15]). There are at most countable number of points

of discontinuity of the entropy map (a, b) —> h(£a,b)-

Our goal is to answer Buzzi’s above question by showing that topological entropy

of the Lozi map is not upper semi-continuous at a given parameter. Moreover, our

45

results can be extended to disprove the above conjecture by Ishii and Sands.

Let us start with a review of the subject:

Piecewise afiine maps: Let f : 1R" —> IR” be a continuous function where n E
Z+. An affine subdivision of f is a finite collection U = {U ,..., UN} of pairwise
disjoint non-empty open subsets of IR” such that their union is dense in IR” and
f ] U,- = AilU, for each i = 1, . . . , N where A, : R" —> R" is an invertible affine map.
A piecewise affine map is a continuous map f : R” —+ R" for which there exists an

affine subdivision.

Example 3.0.3. Lozi maps are piecewise affine homeomorphisms of the plane given

by:

a 1—a|;r[+by
£=£a.b= H , a,b€lR,b7é0.

b :1?

Note that U = {U1,U2} where U1 2 {(13.31) E R I a: > 0} and U2 = {(13.11) 6
IR | 1' <0}.

Let us first review some of the related results in different dimensions. Throughout
this paper, we will denote the topological entropy of a map f by h( f ).

In one dimension, one can work with piecewise monotone functions. Let I denote
a compact interval of IR. A map T : I -—> I is called a piecewise monotone function
if there exists a partition of I into finitely many subintervals on each of which the
restriction of T is continuous and strictly monotone. Two piecewise monotone maps
T1 and T2 are said to be e-close, if they have the same number of intervals of mono-
tonicity and the graph of T 2 is contained in an e-neighborhood of the graph of T1
considered as subsets of 1122. It was proved by h‘Iisiurewicz and Szlenk[27] that the
entropy map f —+ h( f) is lower semi-continuous for piecewise monotone continuous
maps. They also gave upper bounds for the jumps up of the entropy. For unimodal

maps(two-piece continuous monotone maps) entropy is continuous for all maps for

46

which it is positive.

In higher dimensions, let C r(M ") denote the set of Cr self maps of an n-dimensional
compact manifold. It is a classical result of Katok[19] that the entropy map is lower
semi-continuous for CH“ diffeomorphisms on compact surfaces. Yomdin[33] and
Newhousc[28] proved that entropy is upper semi-continuous in C°°(.M") for n 2 1.
Combining these two results, one can get the continuity of entropy in C°°(M2). This
result does not hold for homeomorphisms on surfaces. Also, Misiurewicz[24] con-
structed examples showing that entropy is not continuous in C°°(.M") for n 2 4 as
well as examples[25] showing that entropy is not upper semi-continuous in CHM")
where r < 00 and n 2 2.

For piecewise affine surface homeomorphisms, the following Katok-like theorem(sce

[18]) is given by Buzzi[4]:
Theorem 3.0.4. Let f : M —+ M be a piecewise afiine homeomorphism of a compact
afline surface. Let S be the singularity locus of 111, that is, the set of points :1: which
have no neighborhood on which the restriction of f is affine. For any 5 > 0, there is a
compact invariant set K C M \S such that h( f IR) > h( f) — 5. Moreover f : K -—+ K
is topologically conjugate to a subshift of finite type.

The lower semi-continuity of the entropy follows from the above theorem. The
goal of this paper is to disprove the upper semi-continuity by showing a jump up of

the entropy in Lozi maps. Our results can be summarized as follows:

Theorem 3.0.5. In general, the topological entropy of Lozi maps does not depend

continuously on the parameters: For 61 > O and small and [62] small.

(i) The topological entropy of Lozi maps with (0,1)) = (1.4 + 62.0.4 + 62).

h’(£1.4+€2.0.4+62)2 is zero-

{ii} The topological entropy of Lozi maps, h(£(1.4+61+€2‘0.4+€2)), has a. lower bound

of 0.1203.

47

3.1 Lower Bound Techniques

There are some computer assisted techniques to give rigorous lower bounds for the
topological entropy of maps like Henon and Ikeda. They were first introduced by
Zygliczyi’iski [34] and developed in [10] and [9]. There are also more recent methods
by Newhouse, Berz, Makino and Grote[29] which gives better lower bounds for the
Henon map.

Let us review the following ideas which were used in [9].

Let f : R2 -—+ R2 be a continuous map and N1, N2, . . . Np be p pairwise disjoint
quadrilaterals. Note that we can parametrize each N,- with the unit square 12 =
[0,1] x [0, 1] by choosing a homeomorphism h,— : I 2 -+ Ni. We call the edges h,({0} x
[0,1]) and h,({1} x [0,1]) ”vertical” and the edges h,([(), l] x {0}) and hi([0, l] x {1})
”horizontal”. We define a covering relation between two quadrilaterals in the following

way: (See Figure 3.4)
Definition 3.1.1. “’0 say N,- f -covers NJ- and write N,- = Nj if:

(i) For each p 6 [0,1], f(h,({0} x {p})) and f(h,-({1} x {p})) are located geomet-

rically on the opposite sides of Ni.

(ii) For each p 6 [0,1], there are two numbers t},,tg 6 (0,1) such that f (h,({t},} x
{p})) lies in one of the vertical edges of Ni and f (hi({t?,} x {p})) lies in the

other vertical edge of NJ- and Vt; < t < t?,, f(h,({t} x {p})) E Nj.
(iii) For 0 S t < t}, and t?, < t S 1, f(h,,-({t} x {p})) 0 Ni is empty.

If one can show the existence of these quadrilaterals and associated cover relations,

they can be used to give rigorous lower bounds for the topological entropy of f:

Theorem 3.1.2. ([9]) Let N1. N2. . . . N], be pairwise disjoint quadrilatemls and f :

48

R2 —> R2 be continuous. Let A = (an) be a square matrix where 1 _<_ i. j S p and

1 if A} 2 .‘Vj
a,j =
0 otherwise
Then f is semi-conjugate to the subshift of finite type with transition matrix A. In
particular, h( f) _>_ log()\1) where A1 is the largest magnitude eigenvalue (A1 2 [Ajl

for all eigenvalues of A )

Note that there is no easy way to detect these quadrilaterals. They are usually
found by trial and error. In [9], Galias introduces 29 disjoint sets around the non-
wandering set of the Hénon map and covering relations between these sets. The
transition matrix obtained gives a lower bound of 0.43 for the topological entropy of
the Hénon map. Note that these bounds also hold in a small neighborhood of the

studied parameter. Later, this bound is improved in [29] using different teclmiques.

3.2 Discontinuity of entropy for Lozi maps

Since Lozi maps are piecewise affine surface homeomorphisms, topological entropy of
these maps are lower semi-continuous[4]. In other words, if parameters are slightly
changed, entropy of the map can not jump down. There are also some monotonicity
results(see [14] and Theorem 2.1.2 and Theorem 2.1.3 above) about the entropy of
these maps around the parameter b = 0. It is also known that the topological entropy

is continuous for all Lay, where a > 1 and b = 0.

We first prove that. the entropy jumps from zero to a positive value if parameters
are slightly changed from (a,b) = (1.4, 0.4) to (a, b) = (1.4 + 6,0.4) where 6 > 0 and

small.

49

Theorem 3.2.1. For 6 > 0 and small:
(i) The topological entropy of Lozi maps with (a. b) = (1.4, 0.4), h(£1,4_0,4), is zero.

(ii) The topological entropy of Lozi maps. h(£(1.4+6.0.4)), has a lower bound of

0.1203.

Proof of the Theorem 3.2.] (i).

Let’s denote £14114 = .C. We will prove that h(C‘l) = 0. Note that £4 has the
following fixed points: (i) p1 = (1/2, 1/2) and p2 = (—5/4, —5/4), (ii) the closed line
segment (’1 which connects (0.15/29) to (~20/29,35/29) and (iii) £(l’1).

Note that p1 is a saddle fixed point and cf = (A3, 1) where A? = (-7 + My 10 is a
stable direction at p1 and WT: (p1) = {p1 + 1ft 6 RQIt > 0} is invariant under £(or
£4). Similarly, p2 is a saddle point and cg = (—/\‘2‘, —1) where )32‘ = (7 + x/8_9)/10 is

an unstable direction at p2 and W}: (p2) = {p2+u"2‘t E RQIt > 0} is invariant under £4.

Let’s call the left and right parts of the unstable manifold at pl; 14"}; (p1) and Wr(pl),
respectively. If we can show that Wg(p1) is attracted by 61 and W}(p1) is attracted
by C(61) then we can use the Brouwer’s translation theorem in U = R2 \ (Wi(p1) U
{m} U Wfflpg) U {m} U Wr(p1) U (’1 U Wg(p1) U C(61)) which is homeomorphic to R2.

Since £4 has no fixed points in U and it is orientation preserving, h(£4) = 4h(£) = 0.

W} (p1) is attracted to £1: Now, let Z be the intersection of the line m = {p1+r[’t E
RQIt > 0} and the x-axis where 211‘ = (—)\'1‘,—1) and [\‘1‘ = (—7 - J89)/10. In
other words, Z is the first intersection point of VIE-(p1) with the :r-axis. Note that
Wg(p1) = U30=0 £4n({p1 — aft I 0.1 > t > 0}), i.e. forward iterations of a small piece
in the unstable direction. Let the portion of 1475(p1) which connects [1(2) and £5(Z)

be called W. It is not hard to see that Wg(p1) = iii-3c £4"(VV). We want to show

50

that every I e W(so every :1: 6 WA 121)) is attracted to £1.

Trapping Region: We introduce a trapping region R around 61 such that any point

:r E R is attracted to a point in £1. Let:

R1 = (—20/29, 35/29 + 0.2)
R2 = (—20/29 + 0.1, 35/29 — 0.2.3)
R3 = (0, 15/29 — 0.25)

R4 = (—0.2, 15/29 + 0.5)

Let’s call the left and right end points of 61; F1 and F2, respectively. Let R be the
hexagon with vertices R1,F1,R2,R3,F2 and R4. The sides F1 R2 and F2R4 are parallel
to each other with slope —5/ 2 and they are stable directions at F1 and F2, respec-
tively. Since R1 is in the stable manifold of a point in 61, it is attracted to £1 under
iterations of £4. Similarly, R4 is attracted to F2 since it is in the stable manifold of
F2. So, the quadrilateral with vertices R1,F1,F2 and R4 is mapped to thinner‘and
thinner quadrilaterals for which one of the sides is always 51 = F1F2. Similarly, the
quadrilateral with vertices F1,R2,R3 and F2 is mapped towards €1(See Figure 3.2).

So, R is a trapping region.

we want to show that more and more portions of W is mapped into R under forward
iterations of £4. Let’s start with the part of 11" which connects C(Z) and 163(2). The
image of this line segment(under L4) is the portion of Wg(p1) which connects 165(2)
and £7(Z)(See Figure 3.1). Let’s call this portion W. £5(Z) and £7(Z) are both
in R but there is a part of W which is still outside of R which we denote by W, ie.
Wis the closure of W \ R. Note that [’6 : y = 1 — 1.4(1+ 1.411: + 0.43)) + 0.41: is a

critical line for £4 around F1. ie. images of lines which transversally intersect (c are

01

 

 

 

 

A
‘ 7

Figure 3.1: This figure shows a portion of the left unstable manifold of the fixed point
191. Note that all the points on the line segment connecting F1 to F2 are period-4
points of £

52

’QQ

R1

 

 

 

 

¢ >117

 

Figure 3.2: Trapping region R(gray) and images £4(R)(darker) and £8(R)(darkest).

53

broken lines. Let 7:; = £4(r:(,). Also, let W n RIF] = WRI F1, W n 82F] = W32 F1,
W D I: = ”ch and the intersection point of W and 6}; which stays below I: be ch'
W consists of two parts: The line segment which connects ”"31 F1 and W35 and the
line segment which connects W7C and W R2 F1.(See Figure 3.3). It is not hard to see
that W2: is mapped into R in the next iteration(under £4) so all points on the line
segment connecting W31}:1 and WE; is mapped into R, too.

On the other hand, W120 is mapped to a point on 2;. So, the line segment connecting
Wine and Wt: is also completely mapped into R under £8.

The only part left is the portion that connects l/Vgc and W’R2 F1. But note that
Ill/321:1 is on the stable direction so forward iterations move towards F1. Wgc is
mapped between WE and F1. So, one can repeat the same argument to this line
segment connecting 31(ng F1) and £4(VV[C).

This analysis explains that forward images of W consists of some parts which is
mapped into R and some parts which stays outside of R. However, the parts outside

of R gets shorter and shorter attracted by F1(See Figure 3.3).

Now, for the other portion of W(connecting £3(Z) and £5(Z)) similar arguments can
be done while this time the critical line 8c is the 1 —axis and the parts outside of R are

either mapped into R or attracted by F2.

Also. note that H’Apl) is attracted to [’1 implies that Wr(p1) = £(ll’g(p1)) is attracted
to £(I'1).D

Proof of the Theorem 3.2.1 (i2).
We want to show that for any 6 > O and small, there are various subsets which
factor onto symbolic systems and so give lower bounds for the map £(1.4+€.0_4) by

Thm. 3.1.2.

I
I

Q
Ell

 

 

Figuie 3.3: The set Wflhickest solid broken line) and the part of the images
£4(W)(thinner) and £8(W)(thinnest) which stay outside of R. Note that everything

above E is mapped into R under £4.

55

Fix an 6 > 0 and denote £5 = C . Note that the line se 'ment connecting
(1.4+e,0.4) 8

F1 2 (—20/29, 35/29) and F2 = (0,15/29) consists of period-4 points of £04.04).

Now, let N1 be the quadrilateral given by the four vertices:

A = (0,15/29 — 6)
B = (6,15/29 + (my)
c = ((5/2)e,.15/29 + (5/2)()

D = ((3/2)€, 15/29 — 26)

Also let N2 be the quadrilateral whose vertices are:

E = (—3e., 15/29 + (7/2).»

F

(—26, 15/29 + (5/6)e)
G = (0,15/29 — (1/2)e)

H = (~15, 15/29 + (13/6)€)

For Nl. let. the sides AB and CD be ” vertical” and the other two sides be "hori-
zontal”. Similarly for N2, let EF and GH be ”vertical” and the other two sides be
"’ horizontal”. Note that the images of N1 and N2 under £3 are also quadrilaterals and
vertical edges are contracted since they are chosen very close to the stable directions

around (0, 15/29) and (—20/29.35/29).

By direct calculation, it can be shown that the images of the vertices under the map

56

£2 are given by(See Figure 3.4):

 

 

 

 

 

53(4) = (ii—2:6 0(3), $3 — $36 + 0(3)) z (1.686, g —1.756)
£2(B)=(:(13::6 + 0(8), é; — 173—21596 + 0(8)) x (1.706, 2% — 1.816)

52(0) = (—:::36 + 0(62), g- + %36 + 0(3)) m (-3.286, é—g + 2.926)
52(0) = (— 13252155036 + 0(8), :3 + %e + 0(8)) R1(—3.316,-;% + 2.986)
£3(E) — (—19—82182£56 + 0(62), é—g + £36 + 0(62)) 2: (—0.516,-;-g + 0.426)
£2.1(F) = (—:::g:€ + 0(62), g + €67;ng + 0(3)) a: (—0.426, g + 0.346)
53(0) = iii—2:6 + 0(52), % — gig—:6 + 0(3)) :9 (2.006, :4; — 2.066)
£?(H) = 151433578546 + 0(62), é—g — €3,17—g6 + 0(62)) m (2.086, é—g — 2.106)

It is not hard to see that we have the following covering relations: N 1 2 N1,

Nl :5 Ng and N2 => Nl. So the transition matrix is given by:

where the largest magnitude eigenvalue is

J5+1
‘2'—

process 11(in): 1h(flf) 2 ilog 5+1 > 0.1203 by Thm. 3.1.2.

Now, we can extend our results from (a, b) = (1.4, 0.4) to (a, b) = (1.4+62, 0.4+62)

where lfgl is small:

Proof of the Theorem 3.0.5 .

Let L denote £(1.4+€2.0.~’l+62)'

57

. Since we are using £21 during the

All

 

 

- £201)

I / J
13(3) 41“”
MA)

< ’11?

 

 

Figure 3.4: This figure shows the quadrangles N1 and N2 and their images(thinner
boxes). Notice the covering relations: N1 => N1, N1 => N2 and N2 ==> NI

58

(i) The entropy is zero for [3:

For IEQI small and fixed, we still have two line segments of period-4 points: the
1 — (0.4 + (.2)?

(1.4 + €2)(1 + (0.4 + 62)?)

the image of this line segment under (I. So, we can still find a similar trapping region

 

. . 6 f 9 ,6
hne segment connecting F22 = and F12 = £“(F22) and
using the vertical lines and the stable directions at F? and F52. The rest of the

proof is the same as in the case of (a, b) = (1.4, 0.4).

(21') The lower bound for (a, b) = (1.4 + 61 + 62, 0.4 + 62):

Let £51 = £(1.4+E1+62_'0_4+62). We need to find two boxes as in the case of
(a, b) = (1.4, 0.4) which give us the covering relations. We slightly modify the points
we used before:

For 61 > 0 and small, let N1 be the quadrilateral given by the four vertices:

£1 = (0.1”;2 — q)

B = ((51:17;52 + (7/2)€1)

6* = (camel. F;-2 + (ES/an)
D = ((3/2)q, Pg? — 261)

Also let N; be the quadrilateral whose vertices are:

E = (—361.F;2 + (7/2)(1)

F- : (—2€1,F;2 +(5/6)61)
6: = (or? — (1/2>e1>
I? = («1.17:2 +(13/6m)

59

In other words, 6 is replaced with 61 and 15/ 29 is replaced with F262. Although
finding the images of these points under £31 looks difficult, it is not hard to see the
differences between this case and the case (a, b) = (1.4,0.4). For example, 3641(3)
consists of terms including 61 and some others not including 61. Observe that if 61
equals zero then F52 is a period-4 point, so the terms not including 61 in [3211(3)
add up to Fg2. On the other hand, the terms including 61 can be made arbitrarily
close to the terms including 6 in the (a, b) = (1.4, 0.4) case by choosing small enough
(2 values and letting 61 = 6. So, our new boxes also satisfy the previous covering

relations giving the same lower bound (0.1203) for the entropy.

60

APPENDIX

Proposition 3.2.2. For g with 5" = (+1,—1,—1,—1...) we have

where :r = (a — V a2 + 4b)/2.

q(§)(aa 1)) =

 

b
(a + .r)(b + 11:)

Proof. First note that since :3 = (+1, —1,—1, —1 . . . ).

 

 

 

 

1 1 b
a) = = g + ‘ where :1: =
b a ‘L b
a + 7 —a +
b b
—a + — —a + —
b °'
_a + —
b
Note that the continued fraction for a: can be written as :r. = + and this equation
—a. 1'

gives two solutions. We choose .r = (a — Va,2 + 4b) / 2 as in [12].
1

 

 

 

, A .’L‘
Also note that rn : for all n 2 1. So rn = 3 for n 2 1.
b
—a +
b
—a +
b
_a + _
, . - A A , 1 :c I2
Now we have. q(§)(a.b) = r0 — rorl + rorlrg — = (1 —— — + — — . . ) =
a + .1; b b2

61

 

a + 1(1§0(—‘3)n) = (a + :L')(b + :17). D

 

8_(1| 19:1_

Lemma 3.2.3. For 1 < a _<_ 2 (”Id § ’5 752.07 8b 1:1)“0 (91—
1:0"? ’

 

“m ah b )_H

11—.0017 (1+r)(b+.r.) —a(a-—1)2

For a = 2 we have lim 8— =0.

1,...0db

Proof. Note that by the above proposition, we have

01
8b b:

 

 

 

0=1im—gzlim-(I?-( b )
b—+0 0b 17—.0 8b (o.+:r)(b+:1:)

0q_ (a+:r.)(b+.r)—b[1’(b+x)+(a+:r)(1+1:’)] 6g
N — —.
etc that 0b_ (a + 1;)2(b+ 1:)2 .To find lim 00b

we need to apply L Hospital’s Rule twice. Applying L’Hospital’s Rule the first time,

 

0g
lim — after cancelation. becomes:

b—~0 0b

—b[1:”(:b+:1)+:1:’(1+17')+x'(1+a:')+(a+:1:)1:"]
b—~>0 2(a+.r)1: ’+(b+:r)2 (a+1:)22(b+1:)(1+:r’)

 

which equals:

l'm —b[.1;”( a+b+2.r)+2:r’(1+.r’)]
bl—.02(a+r)(b+.r)[(r’(b+;r) +(a+x)(1+1")]

 

Applying the L’Hospitals’ Rule. again, the b-derivative of the numerator becomes:

III

—[-.r”(a+b+21:)+21:’(1+:r')]—b;r[ .’”((1+b+21:)+;r”(1+2.1:')+21"”(1+I’)+2.7:r]

and the b-derivative of the denominator becomes:

62

2[(1"(b+ 1:) + (a. +r)(1+1‘.'))(:1:'(b+ :r) + (a. + 1:)(1+.r'))+(a.+ w)(b+1:)(.r”(b+ 1:)+

27:’(1 + :r') + (a + r)-7:")l

Now, taking the limit of the numerator and denominator as b goes to 0 gives the

 

, , a. — \/ a2 + 4b 8.1: 1
result. Note that 11m 1: = 11m = 0 and — = 1:’|b_0 = —— and
b—.0 b—+0 2 0b 0:0 - a
8233' = x/ll : 3.
(Dbl? b=0 (’20 a3

 

63

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66

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