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dissertation entitled

Relations Among Conditional EntrOpy,
Topological Entropy and
Pointwise Preimage EntrOpy

presented by

Wen-Chiao Cheng
has been accepted towards fulfillment

of the requirements for

Ph . D . degree in Mathematic 3

@AA

Major professor

Date April 29. 2003

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6/01 clelRC/DateDue.p65.p. 15

 

RELATIONS AMONG CONDITIONAL ENTROPY, TOPOLQGICAL
ENTROPY AND POINTWISE PREIMAGE ENTROPY

Bv

\l

WEN-CHIAO CHENG

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of
DOCTOR OF PHILOSOPHY
Department of Mathematics

2004

ABSTRACT
RELATIONS AMONG CONDITIONAL ENTROPY, TOPOLO GICAL
ENTROPY AND P OINTWISE PREIMAGE EN TROPY
By
Wen-Chiao,Cheng

Entropy was introduced as a conjugacy invariant for measure-preserving transforma-

tions and continuous transformations- However, in 1995 Hurley, Nitecki, and Przy—
tvcki introduced several other entTOPY‘like invariants for non-invertible maps. The
Purpose of this dissertation is to define and StUd)’ two new invariants for non—invertible
maps. Our new invariants are motivated by the some Of those presented by Hurley~
NitECki. and Przytycki.

In Chapter 2 and Chapter 3 we introduce the standard notions 0f measure'

theoretic entrOpy and t0pological entropy and we recall their basic properties. See\::\:.

We a\so describe two of the preimage entropy invariants StUdied by Hufle‘j‘fi:\flhiCh

y thell

and Przytycki. After that we introduce new invariants of nonqnvertibie ma
we call the upper pre-tmage entropy and the metric pre—z'mage entropy and Stud
PrOperties. Among other things we obtain analogs of the well—known Variational Pr ill-
Ciple f rTo olo ical Entro v and Shan - ' - '
. o p g p‘ non McMillan Breiman theorem fOr these new
Invariants. The proofs require adaptation and modification of a numbe

r techniques

in the literature of ergodic theory and topological dynamics.

 

 

ACKNOWLEDGMENTS

First of all I would like to thank my academic adVisor, Dr. S. Newhouse for h '

guidance, help and encouragement throught my Studies at Michigan State Un’iversitzglr.S
I am especially grateful for his excellent lectures and the freedom he granted me in
preparing my thesis. His knowledge and mathematical insight have been invaluable-
1 W38 Constantly inspired by his enthusiasm and professional approach to research-

I Would like to thank Dr. Z - Nitecki for providing useful materials regarding the
aziera

pmlmage entropy and my dissertation committee members Professor MiChael Fr
their

Professor William Sledd, Professor Cliff Weil and Professor Zhengfang Zhou for

valuable suggestions and their time.

iii

Contents

1 Introduction

2 Measure-Theoretic Entropy
Measure—Preserving TranSfOImation ...............

2.1

2.2 Partition and Entropy
2.3 Entropy of a Measure—Preserving Transformation

2.4 Some Methods for CaICU1ating h(T) ........ .

2.5 Bogolioubov Theorem

3 Topological Entropy

3.1 Definition Using Open Covers ...........

3.2 Bowen’s Definition
3.3 Calculation of Topological EntrOpy

3.4 The Variational Principle . .
3.5 Measures with Maximal Entropy ---------------

4 Pointwise Preimage Entropy
4 - 1 Definitions of Pointwise Preimage EntrOpy .........

4 ~ 2 Forward Generator . . . .
4 - 3 Metric ,0 Compatible with the Topology ............

5 I\.<'I:()dified Pointwise Preimage entropy

E. _ 1 New Definitions .......................
5 _ 2 Variational Principle ................... .

iv

ooU‘Wu

\l

13

13

15

18

19
21

30

36
36
45

Chapter 1

Introduction

‘ - (I this
In 1958 Kolmogorov introduced the concept of entropy into e1— godlC theory, an

has been the most successful invariant so far. For example,

in 1942 it was known

that the two-sided (%,-§- -shift and the two-sided (é, %, %)-shift both have countable

Lebesgue spectrum and hence are spectrally isomorphic, but it. Vvas not knOWn Whether

they are conjugate. This was reSOI‘led in 1958 when KOImOgQI-OV Showed that they
had entropies of log2 and 1083’ respectively, and hence are nOt Conjugate. The no-
tion of entropy now used iS slightly differently from that 11 sed by KOImogorov-the
improvement was made by Sinai in 1959-

The topological entropy h(T) of a continuous map T of a compact, metric Space to
itself is a measure of its dynamical complexity. It was first defined by Adler, Konheim

and McAndrew, and later given several equivalent definitions by OWen an

. . l d Others
(see [2] for an exposition) and these definitions led to resu ts Conn ecu”

and measure-theoretic entropies. Roughly speaking, the topologica e::::010gical

measures the exponential growth rate with n of the number of different forWar: of ‘7‘

segments of length n that can be distinguished to at least some finite toleranCe Orbit
When the mapping T under consideration is a homeomorphjSm, then .

. e"tendin

this procedure into the past instead 0f the future results In the eIltrOpy h(T‘l)
. Of

the inverse mapping, which equals h(T). However, when the map 18 not hlVertibI
e,

different ways of “extending the procedure into the past lead to several new entrop

like invariants for non-invertible maps.

More recently, the preimage relation entropy hr(T) of a compact metric space
was introduced by Langevin and Walczak (See l8l) and Shown to be a new tool for
studying the topology and dynamics of compact metric Spaces. Later, Hurley and
Nitecki, Przytycki (see [5] and [9].) introduced several other entropy—like invariants
for non-invertible maps. One of these, which we Call preimage branCh entropy fit-(T),
is closely related to h.r (T). The other pair of entropy invariants is based on how map:
branches of the inverse of the iterated map T—n at a point :1: can be distingh‘s ed
by measurements of finite accuraCy. We call them pointwise preimage entropies :1 S
denote hp(T) and hm (T). In [5] and [9], Hurley established the following relationST )1:
among these five invariants: hp(T) S hm(T) g h(T) S Ill-(T) + hm(T) _<_ h,(

h (T) where T is a continuous mapping on a compact metric: Space.
m ’

In this dissertation we concentrate on hp(T) and hm(T), Since in the context of
pointwise preimage entrOpy, the definitions of hp(T) and hm (T) are in some sense

. In
Chapter 2 and Chapter 3 we introduce measure-theoretic and t op Ological entropjes

analogous to (and were motivated by) Bowen’s notion of “IOC a] entropy” (See [2])
and show the variational principle as conclusion. That is to Say When X is a compact
metric space and T : X —-> X is continuous, the theorem Sh Ows that the supre-
mum of measure-theoretic entropy h”(T) is equal to the topological entI‘Opy h(T)
of T. After that, we investigate basic properties of pointwise preimage entrOpies
hp(T) and hm(T), such as forward generator and metric p co Impatible With the t0p01_
ogy. Finally, we modify the definition of those two invarian ts in Order to Show the
preimage S-M-B theorem. We also show the relationship between metric Mel-m
entropy and upper preimage entrOpy. The main technique used is the ConstruCt. age
. . . . . . 1011 of
the Variational Pr1nc1ple by M. Misiurew1cz. Finally, we follow the method jutmd
by Shannon, McMillan and Brieman and use it to Show the asymptOtic uced

' ' ehaVior
metric preimage entropy and ergodic decomposmon. of

Chapter 2

Measure-Theoretic Entrapy

In this chapter we discuss measure—preserving transformations, measure-theoretic en-

tropy and some of their basic properties. Finally we will show the existence of invari-

ant measures. We refer to Peter Walters’ book for this chapte r. Seei14l-

2.1 Measure-Preserving Transformat ion

Definition 2.1.1 Suppose (X1, Bl,m1), (X2,82,m2) are probability 8
- Da ,
(a) A transformation T : X1 ——> X2 IS measurable if T-1(B2) C B ces
1 .

(b) A transformation T : X1 —-> X2 iS measure-preserving if T is
measurable and
mliT—1(BZ)) -_- m2(B2),VB2 E 32.

(c) We say that T : X1 —-> X2 is an invertible measure-

Dr‘eser ‘

' .. . 1 . vmg transformat'

if T is measure-preservmg, bijective, and T‘ 18 also measure~pregerv, 1011
L In .

Remarks:

(1) We should write T : (X1,Bl,m1) _, (X2132, m2)

since the meas
Ute- r .
property depends on the 8’s and m’s. p eservm

(2) If T : X1 —> X2 and S : X2 —+ X3 are measure-preserving so is SOT _

X1 fi
(3) Measure-preserving transformations are the structure preserving In X3.
3‘98

. (mo -
phlsms) between measure spaces. F

(4) Let (X,, 53,171,) denote the completion of (X,, 8,, m1),z’ = 1, 2. HT . (X
. 1,

31,m,)\)
3

(X2,B2,m2) is measure-preserving, then so is T: (X1 81 m1) _) (X B — )
’ ’ 2, 2im2 '

(5) We shall be mainly interested in the case (X1 [31 m1) (X B ) since we
, ’ = 23 2,7712

wish to study the iterates T". When T : X —_) X is a measu g tranSfor-
re-

preservin
mation of (X, B , m) we say that T preserves m or that m is T-invariant-
In practice it would be difficult to Check, using Definition 11, whether a given
transformation is measure-preserving or not Since one usually does not have explicit
knowledge of all the members of 8. However We often do have explicit knowledge of
a semi-algebra T generating B. (For example,when X i8 the unit interval ’T may be
the semi—algebra of all subintervals of X, and when X is a d ijrect product space ’T
may be the collection of all measurable rectangles.) The following result is therefore

desirable in checking whether transformations are measure-pre Serving or not-

Theorem 2.1.1 Suppose (X1a812m1)1 (X2,32,m2) are probability Spa-ces and T .
X, —-—> X2 is a transformation. Let 7; be a semi-algebra which generates B Iff
2- or
each A2 6 7} we have T‘1(A2) E 81 and m1(T“1(A2)) 2 m2(
A .
2) ’ the" T 18 measure-

preserving.

Examples of Measure-Preserving Transformations
(1) The identity map I on (X, B, m) is obviously measure.preserving
(2) Let K be the unit circle and B be the a-algebra of Bore] Subsets of K
let m be Haar measure. Let a be any fixed point in K and define T , and
. K 1, K

The

by T(z) : a. . 2. Then T is measure-preserving since m is Haar
re.

transformation T is called a rotation of K.
(3) The transformation T (as) = a - a: defined on any compact gr
011
is a fixed element of G) preserves Haar measure. Such transformat. 0 (Where a
10118

rotations of G. a

re Called
(4) Any continuous endomorphism of a compact group onto itSelf Preset

measure. For example T(z) = z" preserves Haar measure on the unit circl‘IeS. Haar
any non-zero integer. e If n is

(5) Any affine transformation of a compact group G preserves Haar m
easure A
' n

4

affine transformation is a map of the form T (11:) = (1 Ala?) Where a. Is any fixed element

1n G and A G —+ G is a surjective endomorphism It follows that T 18 measure
preservrng because it is the composition of a rotation and an endomorphism- When
dealing w1th affine transformations as measure-preservmg transformatlons we alwayS

assume the measure involved is normalised Haar measure

(6) Let k > 2 be a fixed integer and let (p0 P1,- :17). 1) be a probability vec—
tor with non—zero entries (i.e., p,- > 0 each 2' and 21:01‘0
denote the measure space where Y— — {O 1

sure pi- Let (X. B. m) '—

) Let, (KY2 u)

‘ 1k — 1} and the point i has “‘33“

niooo(Y,2Y,#). Define T : X __) X by T({$n}) 2: {ya}

where ya -— rn+1. If f denotes the semi-algebra of all measu. Iable rectangles, then
m.(T—1 A) =

m(A),VA E .7. By Theorem 2.1.1, T is measure‘ Preserv

ing. We call T
the two-sided (110,111, ~ ~-

,pk 1) Shlft. This is an example of «an invertible measure-
preserving transformation. We sometimes use the notation (- .

a x’lfoxla' ") for a.
pomt of X (the indicates the O-th position in the product) and then T Can be tt
Wl'l en

T((. . . ’$_1f0$1, - . . )) : (. . . ,fE—lflioivliliza ) The SCt Y iS Called the St t
a e S
the shift. pace 0f

2.2 Partition and EntrOpy

Throughout this chapter (X, B, m) will denote a probability Space,
Definition 2.2.1 A partition of (X, B, m) is a disjoint collectmn

Elem
whose union is X. Guts of 5’

Here we shall be interested in finite partitions. They will be d e
Date by

letters, e. g., (= {A1, A2, - ,.Ak} If C Is a finite partition of (X B m) d yG’reek

t
lection of all elements of B which are unions of elements of g is a finite b.6311 the c 01
Su -
of B. We denote it by A(C). a algebra

Definition 2. 2. 2 Suppose C and 17 are two partitions of (X, B, m) We w
rite C S7)

to mean that each element of C is a union of elements of 77. We have C
<

ACME/1(7))- ”r

Definition 2'2'3 Let C = {Ah/12’ "°’A"}’n = {011C2,...,Ck} be two finite parti-
tions of (X, B, m). Their join is the partition

<V"={A‘”Ci‘1 SiSn,l_<.j<lc}

Definition 2.2.4 Suppose T : X -> X is a measure-preserving transformation. If C

:{A1,A2, ...,Ak}, then T’"C denOteS the partitiOn {T-nAl . . T'"Ak} and if A is a,
sub-a-algebra of B, then T ”"A denotes the SUb-U-algebra {T‘na : 0. EA }

Definition 2.2.5 Let C = {A1,A2, ..., Ale} be a partition of (A, 3,111) The entropy
of C is the value H(C) = - 226:1 m(Ai)10gm(Ai).
Remarks:
(1) If C =2 {A1, ...,Ak} where m(A,-) = l,vz' then
’° 1 1
H“) : ‘ék'IOgE = logk.

We will find that logic is the maximum value for the entroD
y Of a part“. .

1011
sets. With k
(2) H<<> .>. 0-
(3) If T : X -—> X is measure-preserving then H(T‘1() : H ( <).
Conditional entropy is useful in deriving properties of entr0py, and We d'

ISCUSS it now

before we consider the entropy of a transformation.

Let A and C be two partitions on (X, B, m) with
A :: {A1, ...,Ak} and C = {01, "'3Cp}

Definition 2.2-6 The entropy of A given C is the number

H<A I C) = -- 2mm.) 2 T1223?” log ”$230)
j=1 J m j

 

i=1
omitting the j-terms when m(CJ-) = 0.
So to get H (A | C) one considers Cj as a measure space with normalized
. . meas
m(-) I m(CJ-) and calculates the entropy of the part1t10n of the set C]. ”re

indUCed by A
and then averages the answer taking into account the size of Cj.

6

Theorem 2-2-1 La (X,B,m) be 3' pmba’bmty Space. If A,C, D are partitions on
X, then:

<1>H<AvCID1=H(A\D>+H(C\A VD).
(2) H(AV C) = H(A) +H(C I A).

<3)H<A1_>. H(A|D).

(4) H(A\/C\ D) _<_ H(A | D)+H(C I D)-

(5) H(A\/ C) s H(A) + 11(0).

(6) If T is measure-preserving, then
H(T—1A I T_1C) = H(A I C), and H(T‘1A): H(A)

Now we extend this conditional entropy to more general Situations. We 13, C =
{A1, A2, ...,} be a countable partition of X into measurable Sets. For each a: E X,

denoted by C (at) the element of C to which :1: belongs. Then th e information function
associated to C is defined to be

IC(:1:) .—_ —1ogm<<(x)> = — ZIogm<A>>< A (x),

Aec
so that [C(17) takes the constant value — log m(A) on the cell A

of C' CIlearly
Hm=Ahmam)

It is useful to consider conditional information and entropy, which t k
- a 9 into _
count information that may already be 1n hand. Let 8‘ be a. Sub‘a'avlgeb ac
ra

can recall that for (25 E L1(X), the conditional expectation E(¢ I OfB- We

8) of -
. . . ¢ glven
an 8-measurable functlon on X wh1ch satlsfies

jFE(<z>I%)dm=/F¢dm

for all F E ‘3; E (¢ I 3X23) represents our expected value for ¢ if We
are give
foreknowledge ‘39. Thus we let m(A I S) = E(XA I S?) and define the mud
1

31's

n the
, . . . tional
information funct1on of a countable part1t1on C glven a o-algebra 8 C B to b

e

W.) ___ .. Z logm(A 1 sum) = — Dogma l emu I <3)
AEC AEC

The conditional entropy of C given 8 is defined by

H“ I <3) 2 £10243?) dm

Lemma 2.2.1 If a and fl are countable measurable partitions of X and g is a sub-
a-algebra of B, then

IaVflIS‘ = [ct/s + lfl!A(a)ve

where A(a) is the a-algebra generated by a.

2.3 Entropy of a Measure-Preserving Transforma-

tion

Definition 2.3.1 Suppose T : X —> X is a measure-

preservin g transformation of the
probability space (X, B, m). If C is a partition of X, then

1 "“1
= 1 —H —l
h(T,C) 1.530 n (yo T g)
is called the entropy of T with respect to C.

This means that if we think of an application of T as a passage f d
' . 0 one a
time, then VIZ; T "C represents the comblned experiment of performi y of
n

g the 011' '
experiment, represented by C, on n consecutive days. The n h (T C) 8111a}

ist

he
information per day that one gets from performing the Original 9): average
P817

meat daily

forever. Now we can give the final stage of the defintion of the entrop
. y Ofa
preserving transformatlon. measum_

Definition 2.3.2 If T : X —> X is a measure-preserving transformatjo
probability space (X, B, m) then h(T) = sup h(T,C), where the Supremum n of the
over all finite partitions C of X, is called the entrOpy of T. 18 taken

If, as above, we think of an application of T as a passage of one day of time, t

h(T) is the maximum average per day obtainable by performing the Same eXp hen
eri

men
daily. t

Theorem 2.3.1 If {an}n_>_»1 iS a sequence of real numbers such that an+p 5 an + ap

for all n, p then
li _"
n—)oo n
exists and equals
. an
Inf —,
n 7?.

Corollary 2 3.1 If T : X ——> X is measure-preserving and a is a finite partition of
n-1 -.- ' t .
Definition 2.3.3 Let T,- be a measure-preservlng transformation of the probability
Space (X. Ci: mi)1i = 1, 2. We say that T1 is conjugate to T2 if there is a measure_
) -—> (Cn'rrh) such that ¢T2 : Tub.

algebra isomorphism 975 : (C2, 7”?
Theorem 2.3.2 Entropy is a CODJUSaCy Invariant and hence an isomorphism

invariant.
Theorem 2.3.3 Suppose A, C are finite partitions of ( X , 3, 7n) and T is a measure-

preserving transformation 0f the probability Space (X, B, m), Then

(1) MT, A) _<_ H(A).
(2) MT, A v C) _<_ h(T, A) + h(T, C).
(3) MT, A) g h(T,C) + H(A).
(4) h(T,T‘1A) = h(T, A).

(5) If k .>. 1. h(T. A) = h(T, v22: T“A).
(6) If T is invertible and k _>_ 1, then h(T, A) = h(T, vf:_k T'A).

Theorem 2.3.4 Let T be a measure-preserving transformation of the probab'ft
1 1 y

Space (X,B,m). Then:

(1) For k > 0, h(T") = kh(T).
(2) If T is invertible then h(T") = IkIh(T),Vk E Z.

2.4 Some Methods for Calculating h(T)

It is difficult to calculate h(T) from its definition because one would need to calculate

h(T, A) for every finite partition A. We consider what conditions on A are needed to

ensure h(T) = h(T, A) The result leads to methods of calculating h(T) for specific
examples of measure-preserving transformations and they also lead to proofs of fur-
ther preperties of MT)

Lemma 2.4.1 Let r _>_ 1 be a fixed integef- For each 6 > 0 there exists 6 > 0 such

that iff = {A1, ...,Ar},n = {C1,...,Cr} are any two partitions of (X,B,m) into 7'
sets with 2::1m(Ai A Ci) < (5, then H“ I 77) + 1“” I 0 < e.

Let C’ be a finite sub—a-algebra 0f 3’ say C = {Ci 5 i = 1, 2, ..., n}, then the non-empty
sets 0f the form BI 0 32m m B'“ where B‘ 2 C‘ or X\Cz', form a finite partition 0f
X. We denote it by 0(0) and we define h(T, C) = h(

T’ C“CV”. If D is another finite
sub-o-algebra, then H(C I D) = H(O(C) I “(Di)-

Lemma 2.4.2 Let (X, B, m) be a probability Space and 30 be an algebra such that
the o-algebra generated by Bo (denoted by 3(Bo)) satisfies 8(BO)=B. Let C be a finite

sub-algebra of 8, Then for every 6 > 0, there exists a finite algebra D DCBO SUCh
that H(’D\C)+H(C\Dl< 0

Lemma 2.4.3 If {An} is an increasing sequence of finite Snb‘algebr
' 38 of B
C is a finite sub-algebra W1th Cc VnAn, then H(CI An) _+ 0 as n ~+ 00 and

Thoerem 2.4.1 (Kolmogorov-Sinai Theorem)

Let T be an invertible measure—preserving transformation of the Dr . ,
fi b. Obablllty SpaCe
a n't su l b °°
(X, 3,7”) and let 3? be 1 e a ge ra. of B such that V";00 Tnél‘223, Then

Lemma 2.4.4 If T is a measure-preserving transformation of the probability S
pace

(X, B, m) and if A is a finite sub-algebra 0f 3 with VEOT"‘A= B then h(T) —
h (221/1)-

10

We shall now calculate the entropy of our examples.
(1) If I : (X,[3,m) '9 (X, 13,771) iS the identity, then h([) = O. This is because
h(I, A) = lim%H(A) = 0- A130, if T” = I for some p 7t 0, then h(T) = 0. In particu-

lar any measure-preseI‘Ving t'1'3Jle01'HlatiOIl of a finite space has zero entropy.

(2) Theorem 2.4.2 Any rotation,T(z) = az, of the unit circle K has zero entrepy.

(3) Theorem 2.4.3 Any rotation Of a compact metric abelian group has entropy
zero.

Definition 2.4.1 Let (X, B, m) be 3 Probability Space. A measure-preserving
transformation T of (X, B, m) iS called ergodic if the only members B of B with
T‘IB = B satisfy m(B) = 0 0f m(B) -__—. 1‘

Corollary 2-4-1 Any ergodic transformation With discrete Spectrum has zero entroPY-
(4) If A is an endomorphism 0f the n—torus Kn: then h(A) = 210g \M where the
summation is over all eigenvalues 0f the matrix [A] With abSOIute value greater than
one.
(5) Theorem 2.4.4 The two-sided {190, ""a Pk—1}-Shift has entr0py __ 2:3 Pi 10g Pi-
Remark: The ‘2-sided (‘i’v %)-shift has entrOpy log 2; the 2-Sided (3} , :1; )-shift has

1
7'5
entropy log 3. Thus these transformations can not be conjugate

2.5 Bogolioubov Theorem

We call the members of M (X) Borel probability measures on X. Each
2: E X
mines a member 6,: of M (X ) defined by 63(A)=1 if :1: E A and 63( A) deters

. . t 0’ otherWiSe
Lemma 2.5.1 Let m, p be two Borel probability measures on the metric s .
pace X. If

fX fdm :: fx fdn,Vf E C(X), then m = y.
We define a map T 1 M(X) ’i M(X) give by (TMXB) = Mfr—1B), we somet.
. 1m
write )1 o T‘1 instead of Ty. We shall have the following. es

Lgmma 2.5.2
[M(Tu) = from/1w 6 cm.

We are interested in those members of M (X) that are invariant measures for T

11

Let M(X, T) = {n E M(XilTp = fl}- This set conSists of all” 5/1/00 making T a
measure-preserving transformation of (X , B, ’u), The follow ,- 11g gives us a method of

constrUCtmg members of M (X, T).

Theorem 2.5.1 (Bogolioubov Theorem) Let T : X —-> X be continuous.
by [in =

If

“”321 IS a sequence in M (X l and we define a new sequence {#nliozi

...1 A ‘ . .
fiizo T1032, then any limit point M Di {pm} is a member of M (X,T).(Such llmlt
points BXiSt by the compactness of M (X ))

Corollary 2.5.1 If T : X ——> X is a continuous map of a compact metric Space

X, then M(X, T) is non-empty.

12

Chapter 3

Topological Entrepy

Adler, Konheim, and McAndreW introduced topological entrOpy as an invariant of

topological conjugacy and also as an analogue of measure theoretical entropy. To each

continuous transformation T : X —+ X of a compact t0pological space a non-negative
real number or 00, denoted by h(T), is assigned. Later Dinaburg and Bowen gave a
new, but equivalent, definition and this definition led to proofs of the result connecting

topological and measure-theoretic entropies. For these materials we recommend Peter

Walters’ book. See [14].

3.1 Definition Using Open Covers

Let X be a compact topologiCal space. We shall be interested in OD en covers of X

which we denote by a, fi,

Definition 3.1.1 If a, 3 are open covers of X their join av ,8 is the 01) cover by all
en

t fthefor AnB h A6Q,BE .Similarl
58 S O m W ere :6 y we can define the jOin vii-1:1 01'

of any finite collection of Open covers of X.

Definition 3.1.2 An open cover 5 is a refinement of an open cover Q wr. t t fl
3 1 en a < ,

if every member of [3 is a subset of a member of a.
Hence a < a V ,8 for any open Covers 0 fl. Also if 3 is a sub

Definition 3.1.3 If a is an open cover of X and T : X —> X is Continuous h
a t en

T—la is the open cover consisting of all sets T’IA where A 6 a.

13

Definiti‘m 3-1-41fa is an Open cover ofX. let We) denote .1... number «sets
in a finite Schover of a With smallest cardinality. We define the entmpy of a by
11(0) =10g N(a).
Remarks:

(1) H (a) _>_ 0

(2) H(a) =0iffN(a)= 1 '1er Ed.

(3)1f a < fl, then H(a) g H(B).

(4) H(C¥Vfl) SH(0)+ Hm)-

(5) If T : X —> X is a continuous map, then H(T‘la) S H(a). If T is also

surjective, then H(T’1C1) : 11(0).

Theorem 3.1.1 If a is an open cover of X and T :‘X ——> X is continuous, then
limnnoo 71EH (Vilgol T"a) exists.
Definition 3.1.5 If a is an open cover of X and T : X —+ X is a continuous map,

then the entropy of T relative to a is given by

11—1
1 .
h, T, : l ._ -1
( a) "£210 H(V7 0)

i=0
Remarks:

(6) h(To) 2 0.

(7) Ifa < ,8, then h(T, a) S h(T, 5)-

(8) h(T, a) S H(a).
Definition 3.1.6 If T : X ——> X is continuous, the topological entropy of T is given
by:

h(T) = sup h(T, a)

where a ranges over all Open covers of X.
RemarkS:

(9) h(T) 2 o.

(10) In the defintion of h(T) one can take the supremum over finite open c
overs

ofX.

14

(11) h(I) = 0 where 1 is the identity map of X.
(12) If Y is a closed SUbset ofX and TY = Y then h(T/Y) _</2(T).

The 119’“ reSUIt Shows that topological entrOpy is an invariant of topological conjugacy.
Theorem 3.1-2 If X1,X2 are compact Spaces and T,- : X,- —-> Xi are continuous for
i :— 1,2, and if ¢ : X1 —> X2 is a continuous map with qle = X2 and ¢T1 = T2¢,
then h(Tl) 2 h(Tg). If a is a homeomorphism, then h(Ti) = has).

In the next section we shall give a definition of h(T) that does not require X to

be compact and we give a definition of h(T) in this more general setting. However,

one result that is false when X is not compact is the following.

Theorem 3.1.3 If T : X —+ X is a homeomorphism of a compact space X , then
h(T) = h(T“1).

3.2 Bowen’s Definition

in this section we give the definition Of tOpOlogical entropy using separating and
spanning sets. This was done by Dinaburg and by Bowen, but Bowen also gave the
definition when the space X is not compact and this Will prove useful later. We
shall give the definition when X is a metric space but the definition can easily be

formulated when X is a uniform Space. See [3]

In this section (X, d) is a metric space, not necessarily compact

The Open ball
centre .7: radius r Will be denoted by B($§7‘) and the closed ball by fo r) Our

definitions Will depend on the metric d on X; we shall see later what, t he dependence
on d is.

Throughout this section T will denote a fixed continuous function, If n is a natural
number, we can define a new metric dn on X by dnfbi y) = maxogign‘i dfTiCT), Tied).
(The notation does not show the dependence on T.) The Open ball

dius r in the metric (in is (ls-=0 T”‘B(T‘:ic; 1').

Definition 3.2.1 Let n be a natural number, 6 > 0 and let K be a compact S b
11 set

15

of X' A Silbset F of X is Said to a (n, g) span K With respect tofjf V27 gKEy E F
with d" (139) S e, i.e.,
n——-l
K C U nT-iB(Tiy;c).
yeF i=0

If n is a na-tural number, 6 > 0 and K iS a compact subset of X let 1",,(6, K) denote

the smallest cardinality of any (n, e) “spanning set for K with respect to T.
Remarkz Clearly m(e, K) < 00 because the compactness of K implies the covering
of K by the open sets 0:01 T“iB(T‘:z:; 6), £13 6 X, has a finite subcover.
Definition 3.2.2 If e > 0 and K is a compact subset of X, let
'r(e, K, T) = limsup l logrn(e, K).

n—ioo 7?.

We write r(e, K, T, d) if we wish to emphasis the metric (1.
Definition 3.2.3 If K is a compact subset of X, let h(T, K) = limHo r(e, K, T). The
topological entropy Of T iS h(T) = SupK h(T, K), where the supremum is taken over
the collection of all compact subsets of X. We sometimes write hd(T) to emphasis
the dependence on (1.
Before giving any interpretations or explanations of this definition we shall give

an equivalent but “dual” definition. This definition will use the idea of separated sets

which is dual to the notation of spanning sets.

Definition 3.2.4 Let n be a natural number, 6 > O and K be a corhpa ct subset of
X. A subset E of K is said to be (12., 6) separated With reSpect to T if it y E E, :1: 7f y,
implies (1,.(33, y) > e, i.e., for a: E E the set 0:01 T_iB(Ti$; 6) contains 1710 other point
of E.

Definition 3.2.5 If n is a natural number, 6 > 0 and K is a compact subset of X
let 3n(e, K) be the largest cardinality 0f any (n, e) separated subset of K with resp e ct,
to T. We Write 3,,(5, K , T) to emphasis T if we need to.

Remark: We have rn(e,K) S 372(5, K) _<_ Tn('§‘aK) and hence 8n(€, K) < 00.

16

Definition 3.2.61fe > 0 and K is a compact subset of X Pat

1
5(5, K, T) ZlimsuPfilogSn(€’K)

fl-‘)OO

We sometimes write 8(6, K, T, d) when we need to emphasis the metric d.
n the Size

Remark: The ideas for the definition come from the work of Kolmogorov 0
an of X

of a metr 1C Space. If (X, p) is a metric space, then a subset F is said to an 45-81)
if Va: 6 X 3y E F with p(:r, y) g e, and a subset E is said to be c-separated if whenever
y, z 6 E, y yé 2, then p(y, z) > e. The e-entropy of (X , p) is then the logarithm of the
minimum number of elements of an 6-Spanning set and the g-capacity is the logarithm
of the maximum number of elements in an c-separated set. So in the definition 3-2-5,
we are considering the metric spaces ( K , (1n) and ”(6’ K) is the e—entropy of (K, d")
and 3,,(6, K) iS the 6-C3Paeiiy Of (Kids) where the e-entropy is the logarithm of the

minimum number of elements Of an e-Spanning set and the e-capacity is the logarithm

of maximum number of elements in an e-separated set. Therefore,

h(T, K) 2 £133 lim sup l(e-entropy Of (K, dn))

n-—)oo Tl

:: £14113 lim sup 1 (ecapacity 0f (K, dn))

71—)00 n

We shall now observe that the definition of h(T) in this section coincides With
that given in Section 3.1 when T is a continuous map of a compact m etrisahie space-
For the moment let us denote by h“ (T) and h*(T,a) the numbers Do curing in the
definition 0f tOpological entropy using open covers. In a metric space ( .X, d) we define
the diameter of a cover to be diam(a) = SUP/tea diam(A), where di am( A) denotes
the diameter of the set A. If 01.7 are open covers of X and diam(Q ) is less than a

Lebesgue number for 7 then ’7 < a. The fOHOWing result is often useful for calculating

h‘(T).

Theorem 3.2.1 Let (X,d) be a compact metric space. If {an}? is a sequence
of open covers of X with diam(0£n) —+ 0, then if h*(T) < oo,1imrHO° h* (T a ) exists
and equals h*(T), and if h“ (T) = 00, then limnnoo h“(T, an) = 00.

The next result gives the basic relationship between the two Ways of d efi .
nmg
tOpological entropy.

17

 

t metric Space

Theorem 3-2-2 Let T : X —> X be a continuous map Of a compaC
(X , d)-

(1) If a is an Open cover of X with Lebesgue number 5, then

n—l

NV We) 3 nus/2. X) s Saw/2X)-
i=0
(2) If 5 > O and 7 is an Open cover with diam(’y) _<_ 6, then
11—1 ,
Tn(€, X) S 811(6) X) S N(V T'J’Y)’ trio Space
ornp . (16
Theorem 3. 2.3 If T: X —+ X is a continuous map Of the C\e ntrop 0,1101
ca
(X’ d)’ the“ h(T)— “‘ ”(T); i..,e the two definitions of topomgl \Tml /
Theorem 3.2.4 0‘ then M
7
(1) If (X, d) is a metric space, T is a continuous map and W X e.
m- hd(T). to \i 0“ ‘
0“ en

i?) Let (X 1-, di), 1 = 1, 2 be a compact metric Space and T‘ IS C @2191“ T“

dz
tine a metric d on X1 x X2 by d(($1,$2),(y1,y2)) ,4 max{d1 ((131, you
haKTt X T2) = ha.(T1)+ hd,(T2).

3.3 Calculation of Topological EntI'Opy

Theorem 3. 2. 1 provided the only method we have given SO far for C a 10111
logical entropy of examples. The followmg 13 an analogue 0f the Ko 01 ting the to
p
theorem and provides a method of calculating tOPOIOgiCal entropy f0 Ogoroks.
. r 80 e Inai
Theorem 3-3-1 Let T : X -+ X be an exPanSlve homeomorphism of th ethilrhl’les
e .
metric Space (X161)-
(1) If a is a generator for T then I). (T) =2 h(T, a).
(2) If 6 is an expansive constant for T then h(T) = Two, T) = 8(50, T) for all 50 < 6/4

Corollary 3 3- 1 All expaDSiVe homeomorphism has finite topOIOEICal entropy

Theorem 3.3.2 The two-Sided Shift on X = “gooey, where Y = {0,1,1 ' H ,k ~ 1},

18

 

 

has to 010 icaI e t
p g n mpy 10% k. :: I13°OOY where

Theorem 3.3-3 Let T ; X _, X be the twosided shift on X

Y={0,1,... , k~1}. Then 110g0(X1)a

(1) If X1 is a Closed subset ofX with TX1 2 X1, then h(TlXI) z limnaw "

°° 6
h that the set {{xn}.—oo

Where 6710(1) iS the number of n—tuples [10, i1, min—ll suc
X 1 [$0 = i0, ..., $n+1 = in+1} is non-empty. b an irreducible
. ' y
(2) Let TA -' XA —> XA denote the topological Markov chaln glven /\ where /\ 15 the
z; 1
k X 1‘5 matrix A whose entries belong to {0,1}- Then h(TA) 0g A is
largest positive eigenvalue of A. 3‘50 when of
T h t (2) holds 6 thect)’
e corresponding one-sided results are true. Par “1th
. (““3
I‘Gducr b] e by arranging the matrix A in lower diagonal block {

Remark: There is a transformation with t0pologiced entrOP‘J
positive real number. haS 'LQYO mpo’
We already know that a rotation T of a compact metric group {act we now
logical entrOpy because there is a metric on G making T an isometIY-
Show any homeomorphism of K has zero entrOpy where K is the unit cirCleo
Theorem 3.3.4 If T : K ——> K is a homeomorphism 0f the 1111 ' .
Corollary 3-3-1 Any homeomorphism of [0,1] has zero t0p01;;i:::le’ the” by? S 0
e

"012%

3.4 The Variational Principle

In this section we describe the basic relationship between topological
. . . entr
measure-theoretic entropy. If T 18 a. continuous map of a Compact metric on), and
S

. Da
h(T) =2 sup{h#(T)],u E M(X, T)}. ' [‘he inequality sup{hu(T)|u E M(X T)}Ce, then
. . ’ S h
was proved by L.W. Goodwyn in 1968. In 1970 E. I. Dinaburg Proved equality (T)

. whe
X has finite covering dimensmn and later in 1970, T. N. T. GOOdman proved e n

(Illality
in the general case. See [3].

We Shall need the follOWing Simple lemma, where we use 53 to denote the bound-

19

 

 

ary of a set B.

Lemma 3.4- 1 Let X be a compact metric space and It 6 M (X )°

(1) If a: E X and 5 > 0 there exists 5 < 6 such that p(68(z;5)) 1"" 0' (A

. m .

(2) If (5 > 0, there is a finite partition «5 = {A1, ' " ,Ak} sueh that dla J
#(aAj) = 0 for each j,

K631“

. 1

riatlona

. f of the V3
We now Collect tOgether some results we will use in the proo

he
3(X) t
° ace and
pr inciple, In this section X will always denote a comPaCt mettle sp
a-aIgebra of Borel subsets.
Remarks;
(1) If 'ui E M(X), 1 S i S n, and pi Z 0,2;1191 = 1’ then
HzglpiuJE) Z ZpiH#i(€)
i=1 q , \ pu’fi
for any finite partition .5 of (X, B(X)). ' e

4 J
T
(2) SUPPOSG q,” are natural numbers and 1 < q < n. F0 {0“}ng

e
. :Ivelh
all” 2 [mg—3)]. Here [1)] denotes the integer part of b > 0' We h
oa(0)2a(1)2~-2a(q—1). 0<
0FiX 0 Si S q—l. Then {0,1,2,...,n — 1} fi “+711 \f— z'/0 <7. < (“fl-’1’ "'
is c — 1}USwhere S = {O,1,...,j — 1,j +a(j)q,j + QC]

79+1 1
. . ”_l , .., fl —- .
Since 3 + a(])q 2 j + [($731) — 1]q : n _ g, we have the card. 7 }
112 . .
mo“ 2‘1‘ (“10’ 012915 at
. . . n- ‘) _
' ForeaCh 0 S] S q— 1, (62(3) -1)<I+J S [(L—qJ-FIJ—FJ l. n\?
. . . _ H - . . T
{J + WW S J S q “ 1’ O S r S “(3) 1} are a dlStht and are 6211 enumbers
than n ”’ (I greater
(3)1fu e M(X, T) and ifp(8A,-) : 0,0 g 2' _<_ n — 1, then
n—l "“1 _ n—l
#(Wfl T4140) = 0 since 3(fl 71—71:) C U TeaAi,
i=0 i=0 i=0

Theorem 3.4.1 (Variational Principle) Let T : X —) X be a Continuous

map of
a compact metric space X. Then

h(T) = Sup{hp(T)I,u e M(X,T)}-

20

3.5 Measures With Maximal Entr0py

The variational principle gives a natural way t0 piCk out some members of M (X , T)-

Definition 3.5.1 Let T : X -—> X be a continuous transformation on a compact
metric space X. A member {1 of M (X, T) is called a measure of maximal entropy for

T if hn(T) = h(T)- And we let Mmax(X , T) denote the collection of all measures

with maximal entropy for T.

Theorem 3.5.1 Let T : X —+ X be a continuous transformation of a compact

metric space. Then

(1) Mmax(X, T) is convex.

. ' m-
(2) If h(T) < 00 the extreme points of Mmax(X,T) are preCISely the ergodic me

bers of Mmax(X, T), a
. dic me '
(3) 1f h(T) < 00 and Mmax(X, T) # (2) then Mmax(X, T) contains an ergo

sure.
(4) If h(T) :: 00 then NImax(X,T) # 0

(5) If the entrOpy map is upper semi-continuous, then Mmax( X , T) is COmPaCt and

non-empty.

Definition 3.5.2 A continuous transformation T : X —+ X of a compact me tri
0 space

X is said to have a unique measure with maximal entrOpy if Mm“ (X T) co ,
’ IISISts of

exactly one member. Such transformations are also called intrinsicially ergo d'
1c.

Remarks:
(1) If T is uniquely ergodic and M (X, T) = {p} then T has a unique measure With

maximal entropy, because the variational principle gives hu(T) = h(T) in this Case
(2) If h(T) = 00 and T has a unique measure with maximal entropy, then T is uniquely

ergodic, because if Mmax(X,T) = {,u} and m E M(X, T), then h%+%(T) 2 00 so

mzu.

21

 

(3) If Mmax(X,T) : {p} then ,u is ergodic. If h(T) = 00 this follows from (2) and if
h(T) < 00 it follows from Theorem 3.5.1.

22

_J. 3h..._

Chapter 4

Pointwise Preimage Entropy

In this chapter we first introduce pointwise preimage entropies hp(T) and hm(T)
which are defined in [5] and [9] After that, we investigate basic properties Of those
. . . . 'th
two invariants, show the existence of forward generator and metric p compatible W1

the topology.

4.1 Definitions of Pointwise Preimage Entropy

Definition 4.1.1 Suppose T : X -—> X and :1: e X. For k=1,2,3,m, the kth preimage
set of a: under T is the subset T‘k(x) of X where T‘k(:z;) = {z E XITI:(Z) :__ 1.} For

N=1,2,..., the Nth branch at a: is denoted by BN(a:,T) C X” and is defin d . th
e m 8

following:

BN($) T) Z {(ZNazN—17 ...,ZO)‘T(Z¢'+1) : Zi,0 S ’t S N ~1al1d 2'0 2

1‘}

To formulate a topological definition, we let 0(X) be the collection of all Open c overs
of this compact metric space X (finite or infinite). Given U E 0(X), let UN be the
open cover of X” by product sets U1 x U2 x x UN, U; E U. For a subset SN C XN
define R(U, N, S N) to be the least cardinality among subcollections of U N which can

cover S N.

Definition 4.1.2 (Pointwise preimage entropies).

23

Let T : X ——> X be a continuous mapping from a compact space X to itself, define

1
hp(T) = sup{ sup [lim sup 7V- 10g NU, N, BN(1‘,T))]}
xeX UEO(X) N—+oo

and

1
hm(T) = sup {lim sup —— log[supN(U, N, BN(x, T))J}
U€0(X) N—roo N 36X
Remark 4.1.1 Continuity of T and compactness of X insure that B N (:17, T) is com-

Pact’ and hence that the numbers N(U.N.BN(2:,T)) are all finite and bounded for
fixed N over :1: E X.

Like the topological entropy, we can show the metric definitions of our invariants
by reinterpreting the numbers N(U, N, SN) in terms of 5-spanning and 6-separated
SQtS. Given any metric space (X, d), we say a subset S c X is 5—separated for some
6 > 0 if distinct points of S are at least e-apartzs 3,5 t E S, 2} (“3, t) Z 6, and say that

. . Let
R C A C X e-spans A if for every a E A, there exists 7‘ E R With dla,"'i< 6
Tl€a 61, Al = min{card(R)|R is c-spans A},
Sléa d» A) = max{card(S)|S is 6-separated A}.

Theorem 4-1-1l9l If (X, d) iS a compact metric space, for any positive integer N let
d” be the metric on X N given by

dN((-rlv "'3 xN), (y1,---,y1v)) : 122§d($iiyi)

Then for T : X —> X continuous, the invariants from definition 4.1.2 Can be cal 1
CU ated
via the following.

. . 1 N
hp(T) — 3161)}:{lg0112njgp l—V— 10g(b(€, d aBN($,T))))},

and

1
hm T =1' limsu —lo ,dN,B ,
( ) 1m{ NaoopN mfg/138k ~(zc T)))}

e—+0
In either formula,
3(6) dN) Bil/($3 T))
can be replaced by 'r(c, d” , BN(:1:, T))

24

 

In topological entropy we define a new metric dn on X by

dn(a:,y)= max d(Ti($)aTi(y))

ogign—l

A subset F ofX is said to (n, e) span K if for all m E K, exist y 6 F with dn (1:, y) _<_ 6
and let rn (e, K) denote the smallest cardinality of any (n, e)-spanning set for K. Simi-
lar definition for (n, e) separated set and 3,.(5, K). We denote N(U) [y to be the smallest

cardinality of subsets in U which covers Y. See [14].

Remark 4.1.2 Let U EO(X), some easy consequences are the following
r(6, d”, BN(.7:, T» = rN(e, T‘N(:c)), 3(6, d”, BN(2:,T)) == sN(e,T‘N(-’Ir))
and N(l/? N) BN(-z'2 T)) : N(V11:,=0 T—nU)lT—N(;c)-

Remark 4.1.3 If T is a homeomorphism, then hp(T) = hm (T) = 0-

Remark 4.1.4l7l and [8] If X is the circle or any closed interval, the“ MU) :=
hm(T) = h(T).

Remark 4.1.5l4l There exists T : X ——>X continuous, X a zero-dimensl0na1 CompaCt
metric space, for which hp (T) = 0 and hm (T) > 0.

Theorem 4.1.2l9] If T1 : X —-) X and T2 : Y -+ Y are topologically Conjugate, then
hp(T1) = hp(T2) and hm(T1) = hm(T2).

Remark 4.1.6 Like topological entropy property, the next one iS triviaL Also, in
section 4.3, we will concern another metric compatible with the tOpOIOgy ofX and

represent its pointwise preimage entrOpy with respect to this metric.

Theorem 4.1.3 If d is another metric on compact set X which defines the s
ame

topology as d, then the pointwise preimage entropy with respect to d are eqna1 to the

pointwise preimage entropy with respect to (i.

In theorem 4.1.2, if T2 is a factor of T1 then h(Tz) S h(Tl). However this inequality

can not hold for pointwise preimage entropy. The easiest example of increase under

 

25

 

 

factors for the pointwise preimage entrOpies is obtained via inverse limits. Recall that

the inverse limit of the map f : X —+ X is the shift a, defined on the sequence space

2 = {{xi :0 I f(:13i) =1 $1-1,7; = 1,2,...}

I
by
0f(330,1131,m)=(f($0), f(x1),..)=(f(:r0),$o,$1,-~)

The product topology on E] C X N makes 2} compact and a, a homeomorphism.

Furthermore, if f is surjective, then it is a factor of its inverse limit via the PYOJeCtlon
<p({mi :0) = 330.

By Remark 4.1.3, we have
hp(0’f) = hm(0'f) = 0.

Thus, any map f with hp( f) = hm( f ) > 0 gives an example showing

and
Remark 4.1.7 There exist maps f : X —> X, g : Y —+ Y with f a factor 0i 9;

hm(f) = hp(f) > hm(g) = hp(g)-

An easy example is the standard expanding map of the circle:set .51 2: R/Z and
define f(x+Z)=2x+Z. It is easy to check that

h2:2(f) = hm(f) =1032-

We turn now to Cartesian products and additivity. Subadditivity of all t
. W0 in-
variants IS relatively easy to prove:

Lemma 4.1.1 For any continuous maps Ti : X, ——> Xi,z' = 1, 2, we have
ha(Tl X T2) S ha(T1) + ha(T2)

where a = p or m.
Topological entrOpy is multiplicative under iterates and we can show that the same

is true for pointwise preimage entrOpy.

26

 

Lemma 4.1.2 Suppose T : X -+ X is continuous, where X .
15 a Compact metric

space. Then for every k E N, we have
ha(T") = k - ha(T)

where 0! = p or m.

4.2 Forward Generator

After finishing the first version of this section, we found that D.Fiebi g, U.Fiebig and
Z. Nitecki used the tool of graph theory to get, Similar but better resultS- See [4].

Definition 4.2.1 Let X be a. Compact metric Space and T : X ..y X a continu'
ous function. A finite open COVer a of X is a forwar d generator {or T if for

sequence (Any? of members of a the set {1:020 TWA" Contai t most one
113 a

. . : X —> X ' . tmettic
Lemma 4 2 1 Let T be a continuous funCtlon with a Compac
space (X, (1). Let a be a forward generator for T. Th
that each set in Vnzo T ‘"a has diameter less than c.
PROOF
Suppose the theorem does not hold. Then 36 > 0 such that Vj > 0 3
, . . - . ’ ‘2'
e and 3A“ 6 a,0§1 SJ Wlth 333', yj G flLO TflAfi- There is a Subs 0339', 072.
Q J"
natural numbers such that :L‘jk —> a: and yjk -—> y since X is compact Hence { .
, . _ . W ‘7‘} Of
Con81del‘ the Sets Ajk,0- Infinitely many of them coincide smce Q is fl
. _- . nit .
Jahyjk 6 A0, say, for infinitely many I»: and hence x, y e A0. Similarly f0:
) ea

. . , . _ ch 77.
infinitely many AM," comelde and we obtain An E a Wlth 13,31 6 TWA". Thus

(X)
117,21} 6 (WT—11A",
0

contradicting the fact that a is a forward generator. 0

Definition 4.2.2 Let T be a continuous function of a compact metric space (X, d)

27

 

to itself is said to be forward expansive if 36>0 and X76 y E X, then 312 E N with

d(T":1:,T"y)> 6. We call 6 a forward expansive constant.

Lemma 4.2.2 Let T be a continuous function of a compact metric space (X, (1)
to itself. Then T is forward expansive iff T has a forward generator.

PROOF

Let 6 be a forward expansive constant for T and let a be a finite cover by open balls of
radius 6/2. Suppose that :c,y E 08° T‘".4n where An E 0. Then d(T“(:r),T"(y)) _<_
6 for all n E NU[0] so, by assumption :1: = y. Then a is a forward generator.
Conversely, suppose a is a forward generator. Let 6 be a Lebesgue number for a,
If d(T"(x),T"(y)) g 6 for all n ENU[O], then for all n EN exists An E a with
T"(x),T"(y) E An and so, :r,y E 03° T‘"A,,.

Since this intersection contains at most one point we have a: = y. Hence T is forward

expansive. 0
Example: {{1, 2, ...m}N, a} where a is left-shift.

As section 4.1,we let 0(X) be the collection of all covers on X. Let U E 0(X)

and x EX, we denote
1
hp(T, U) = sup{lim sup — log NU, N, BN(:1:,T))},
:cEX Ill—+00 N
and

I
hm(T, U) = lim sup N log{sup MU, N, BN(:1:,T))}.

N—mo IEX
Let Y QX, N(U)|y be the smallest cardinality of subsets in U which covers Y. For

any fixed a: in X, we have N(U, N, BN(:r,T)) = m(vfzor-nU)|T-~(,,.

Theorem 4.2.1 Let T : X —> X be a forward expansive continuous function of

the compact metric space (X,d). If a is a forward generator for T, then

hp(T) = hp(T,a) and hm(T) = hm(T,a).

28

...“:

PROOF
Since a is a. forward generator, for any U e 0( X)

SllCh that U < vrllV=0 T‘"a,

1 we can 0110056 /V large enough

I: k N
10s N( V T‘"U)lT——k(n) 5 log N(\/ T-n V T-"a)1T_.(,,, for any 1:.

11:0 71:0 11:0
Then

n 1 k N
11111 Supfilog N[V T UllT‘k($)<1imsuP-10g R(V T-n(\/ T "at“ l‘T my

k—>
fl: 0 11:0 11:0

k+N
= lircrim sup E log N (V T‘na)
n=0
S 1im sup % 10g NIT/NT”,

k—mo

“Wm

CY) |T_(k+N)(a:)
n=o

 

. k +
: 11m sup 72 I k+N T/(Hmtxl

,Hoo k k+N10gN( V T’“a)\

k N "’0
Slimsup +

lim 1
’HOO ’9 Ic—lsup m Ioglih RS],

1"“ 0L)\T-U=+N 3 (I)

 

1' 1 lc+N
= 1m sup logN _

 

So we can get hp(T, U) S hp(T, a) for any fixed :15 in X this ,
, Imp] ‘

Q
am =sup{ ”to h ,.(T H» < suphp (T a) S

Then MT) : h,,(T, a)-

Similarly. since U < szoTwaa NVLOUHT—ke) s NVLOVLOTWQMT-
k(:t),

hm (T, U) =1imsupk 10g[SUpN(VT "UHT— we)

1:: 0
k N
< lim sup k110g[suP NV T "(VT—n 0)] l—IT h(x),
Ic~+oo 11:0 73:0

BY asilhilar calculation, we can get hm(T, U) S hm(T,Cr). Finally we get hm(T) =
hmfliQ)’ O

29

 

Theorem 4 2. 2i4 Le] t T X ——)X be forward expansive with é”) Tb
\\~ /0g/I 5'”

there is 2: e X With card(T "23) 2 A" for all n and in part1 cu/a,

MT) = mm = h(T).

4.3 Metric ,0 Compatible with the TOpology

Lemma 4.3.1 Let T : X —+ X be a continuous map of a compact metric Space (X, 61)-

(1) If a is an open cover of X With Lebesgue number 5, then for any £6 E X,

N(v::olT-ia1T—nm)s rn(5/2,T‘"(x)) s saw/arm»

(2) If 6 > O and 7 is an open cover With diam (’7) < 6, then for any 2: Ex,

me, T“"(x)) < sue T "<$>><N<V?=‘l —o 7’ Wm).

PROOF (1)1t’s abvious that Tn(e,T‘" (:1: )) < Sn“ T‘n($))
Consider any T‘" (as) of a: and let F be a (n, 6/2) Spanning Set for T4101;
rnt5/2,T“”(a:)). Then

oi cat dummy

T‘"(:I:) Q U {—1 T—‘B(T"x;5/2)

xEF £20
and since for each 2 B(T’$ 5/2) IS a SUbset 0f a member Oka

-T— alT .m) _<_rn(5/2T "(20) Q We have WV

(2) Let E be a (n e) separated set of cardinality 8,, (e, T "($)) fOr T\
(.2:
). NO Them

"My

ber of the cover V":01T ’YlT "(33) can contain two elements of E so

WV; 01T_z’7l’1‘~n(x))- O

8'1(€,2»\

Lemma 4 3. 2 Let T: X -) X be a forward expansive continuous function fOr
compact metric space (X, d) with forward eXpansive constant 6. Then

hp(T)~ :-_ sup{lim supn - 1log Tn(5oa T‘"($))}

36X n—)OO

1
:: sup{lim supn — 108 Sumo, T "(17)”

xEX 11—100

30

 

and

1
hm T :1 — ~72
( ) 1:2:an Egg/i3 rn(6o,T (2))

. l
= 11m sup — 10g sup 3,1050, TWO?»
n—>oo 72 sex

PROOF

We can let 6° < 6/4- For all x E X, choose $13$2anw$k such that T403) Q
Uik=1 30%; g — 260). This cover Oz 2

. {B (311-; e/ 2M1 S ’i S k} is a forward generator
Wlth the Lebesgue number 260. So by Lemma, 4.3.1

- 1
hm T,a Sllmsu —1 _
< ) Moopn Ogi‘é}? ""0501 "(m 5 MT)
Similar calculation for hp(T)- <>

Lemma 4.3.3 Let T be a forward expansive Continuous function from c pact metric
space (X, (1) to itself with forward expansive COHStant 6. Then for a“ e7 (LEN > 0:
such that d(T‘x,T‘y) _<_ e for all z' with 0 g i S N, this implies (1(33 y) g 6'

PROOF ’

We may assume there exists 6 > 0. For all N = 1, 2 3

fl. 3 We C
s-t- d<T WWW s :20 s n _<_ N and d(xn,yn) 2 6. Q, a” Emmy" EX
3”" -+ y' The“ WW) > 6’ b‘” d(T‘x,T‘y) s e for all 2' = 1 0%

2’ 3) - .
the forward eXpansive property of T. 0 ‘

Q
2"”. ‘7‘ .Z' and
1'
s contradicts

Lemma 4.3.4 Let X be a compact metric space and T : X ~> X

. , a for“,

pansive Wlth forward expanswe constant 6. Then for 0 < e < 8/2 and 6 > a
. . - . h

ex1sts 05,5 such that for all posrtlve integer n and all :1: in X, we have ere

3n (5, T‘" (312)) s Cé,esn(€’ T‘"(z)).
PROOF

For 0 < 6 < e/ 2 and O < 6, by Lemma 4.3 and uniform continuity of T on X there

exists a. positive integer N and a > 0 such that

31

if dN(:r, y) g 26, then d(a:, y) S (5
and if d(:z:, y) S a, then dN(:z:, y) g 6

Now fix a: and let n be big enough, assume E is a maximal (n, 6)-separated set of
T‘"(:I:) and F is a maximal (n, e)-separated set of T‘"(a:), then for :t’: E E there
is a z(:1:) E F such that d,,(a':,z(a’:)) < e. Let E; = {2'3 6 E : z(a’:) = 2}, then
card(E) S Ezep card(Ez). But if 3:,y E E2, then dn(x,y) S 26 by definition of E2,
hence

d(Ti(:r),Ti(y)) S 6 for 2' : 0, l, 2, ..., n — N.

Since any E E, dn(:r, y) > 6 and if d(T"‘N+1(z),T"‘N+1(y)) g a, this implies that
dN(T"‘N+1(:1:),T""N+1(y)) g (5, then dn(:c,y) S 6. This is a contradiction. So
d(T"‘N+1(2:),T"“N+1(y)) > a and T”"N+1(:r),T""+1(y) E X, X is compact. This
implies that card(Ez) is bounded by some constant 06‘. Therefore, card(Ez) g
06,5card(F). 0

Remark: We can show Lemma 4.3.2 from Lemma 4.3.4.
During the remainder of this section we will assume that T is a forward expansive

continuous function of a compact metric space (X, d) onto itself with forward expan-

sive constant 6 >0.

Now for any integer n 2 0, we define:
Wu 2 {(cc,y) E X x X : d(Tix,Tiy) g e for 0 g 2' S 72}

It’s obvious that ”:0 W}, = A where A = {(1:, :r) : :1: E X}.

Take 5 small enough such that 36 _<_ 8. Choose N from the above Lemma 4.3 with

respect to e. We define Vn = WnN for n=0,1,2,3,...

and (1:, y) E Vn+10Vn+10Vn+1 means there exists u, v E X st (:13, u), (u, v) and (v, y) E

Vn+1-

Lemma 4.3.5 The sequence Vn is a nested sequence of symmetric neighborhoods

of A whose intersection is A and such that VH1 o V,,+1 o Vn+1§ Vn for all n 2 O.

32

 

PROOF

Let (11:,y) E Vn+10 Vn+1 o Vn+1, then exists 22,2) 6 X st

(2:, 22), (22,22) and (22,31) 6 Vn+1

i.e. d(T‘:r,T‘u) 3 8,0 3 2 S (n +1)N,
d(T‘u,T‘v) g 8,0 g 2' g (n +1)N
and d(Tiv,T‘y) S 6,0 3 2' S (n +1)N

By Lemma 4.3.3,
d(Ti:r, Tiu) S e,d(Tiu,Tiv) S 6 and d(Tiv,Tiy) S e for 0 5 2 S nN.
The triangle inequality can imply
d(Tia:,Tiy) S 36 S e for 0 g 2 g nN.

This implies that (2:,y) 6 Va. 0

Metrization Lemma 4.3.6[6] Let V, be a sequence of symmetric neighborhoods
of the diagonal, A , of Xx X with VI) = X x X such that Vn+1 o Vn+1 o Vn+1 C Vn for
each n and fl? V, = A. Then there is a metric D compatible with the topology of

such that the following condition holds for 112 1,
Vn C {(II:,y) : D(x,y) < 1/2"} C Vn_1

We define Nd(A; c) = {:r; d(:r, A) < e} where d is a metric on A.

There following consequence comes from Lemma 4.3.5 and Lemma 4.3.6

Lemma 4.3.7 There is a metric p compatible with the topology of X such that
Np(A;1/2"+1) g V" <_: N,,(A;1/2")

for all positive integer 72.

Lemma 4.3.8 There is a metric p compatible with the t0pology of X and there
is A, 0 < /\ < 1, such that

Np(A; AM”) 2 W... C; Np(A;A'""”)

33

 

for all positive integers m.
PROOF

Consider any positive integer m = nN + j, 0 S j < N, it is easy to see that

Vn+l = W(n+1)N = WnN+N g WnN+j : Wm g VVnN = Vn-

Therefore Vn+1 g Wm g V". From Lemma 4.3 we can get N(A ;1/2"+2) g Vn+1 and

l

Vn Q N(A;1/2"). Now we let A = (5-)?

Np(A; Am+2N) g NP(A;Am+2N-‘j) : NP(A; ,\("+2)N)
: p(A; (%)n+2) g Vn+1 g Wm g V12

2 Npm; 3.1;) = Npm; (amt)

=Np(A; W) = Npm; ,...-.) 9 NM: W”)

This finished the proof of the Lemma. 0

Theorem 4.3.1 Assume T is a forward expansive continuous function of a com-
pact metric space (X, (1) onto itself with forward expansive constant e > 0. Then
there is a metric p compatible with the topology of X and there is A, 0 < /\ < 1, such

that
1
hp(T) : sup{lim sup E log 7‘1(/\k, T"k(:r))}

xEX [ft—’00
and

1
hm(T) = lim sup 7 log{sup r1(/\",T_k(:1:))}

lc—roo IEX

with respect to this metric p.

PROOF

For any .27 EX we consider T‘k(2:). Let E be a (k,e)-spanning set of T""(:r) with
minimum cardinality. For any y E T‘k(:1:), there exists 2 E E s.t. d(T‘y,T‘z) S e
for 0 S 2' < k. So (y,z) E Wk_1. From Lemma 4.8 we can find a metric p on X
and A,0 < A < 1, such that (2:, y) E NP(A;Ak'1’N). This means that there exists
an F which is (1,/\"‘1"N)-spanning set with metric p and 11(F) S ME). Therefore
r1(x\"‘1‘N,T"‘(x)) S rk(e,T"‘(:c)).

On the other hand consider F to be a (1,/\"‘1+2N)-spanning set of T‘k (1:) with respect

34

to this metric p and with minimum cardinality. Thus for any 2; E T‘k(:c), there exists
26 F such that (y, z) E Np(A; A"‘1+2N). So (2:, z) E Wk_1 by Lemma 4.8. This means
d(T‘y, T 1'.Z) S e, 0 S 2 < k. and this implies that we can find E which is (k, e)-spanning
set ofT’k(.2:) with card(E) S card(F). Therefore rk(e, T‘k(:c)) S r1(x\"’1+2", T‘k(:1:)).

Since p is fixed, let 6 be small enough and using Lemma 4.2, we have

hp(T) = sup{lim sup % log m(Ak, T“k(:1:))}

xEX k—mo
and
hm(T) 2 lim sup 1 log sup r1(Ak,T'k(:1:)).
k—mo xEX
O

35

Chapter 5

Modified Pointwise Preimage

entropy

. . all it upper
In this chapter we modify the original pointwise preimage entropy and c

. . . - “$109), an

Preimage entropy. Then we show the relationship between cond1t10n3\ e h the
. O S OW

upper preimage entropy. Finally, we follow the S-M-B method and use W t

asymptotic behavior of metric preimage entropy and ergodic decompSililon'

5.1 New Definitions

We continue to consider a continuous self-map T of the compact met .

IC 8
Given a subset K C X, a (5 > 0, and an positive integer n, we set Pace (Ax, 0').

7.0175, K) : THUS, K, T) :2 max{card(E) : E Q K,E is (71,6) \ Sep
{Hated}-
Definition 5.1.1(Upper Preimage Entropy)

1
htop(T l g“) = limlim sup — log sup r(n, 6,T"‘:r)

1 "-1
— SUP lim sup - log sup N T‘i _
a Open Cover 71—900 77; kZO,-'17EX (g) a)iT k3

Remark 5.1.1 hp (T) S hm(T) S htop(T | 5") S h(T)
Example: Consider S; {1,2}N —> {1, 2}” and T: {1,2}Z —) {1, 2}Z where S and T

36

 

are left-shifts. Then we can get

 

M5 X T) = hm(5' X T) = hiop(S x T I F) = log2 < h(S x T) = 210g2

Next, let 6 denote the point partition of X which we also identify with the 0-

algebra B of Borel measurable sets.

For n > 0, we set
5‘" = T-"g
- . ' ' , let
Given a finite partition 01, let an 2 VzlgolT‘Qy, For a T-invarlant pI'Obablhty ’1’
H11 (0" l é—k)

' ~k call this the
denote the conditional entropy of a" given the a-algebra T B- We

entropy of a" given the preimage partition 5"“.

. ' the
- . . . easlng 111
Note that, since H,,(- I ) is mcreasmg 1n the first variable and deer

second variable, we have n 2 m,l 2 12 implies
HM l 5') 2 Hue“ I 6"“)-
Set
Hp(a" |g‘) : Hp(oe" | 5“”) 2:11:13 HAG" l 64:) ‘2 [£1220 H,,(an ‘64:)

Lemma 5.1.1 The function an = H,,(oz" | f‘) is subadditive.
PROOF

We need to show
an+m S an + am.

We have
an+m : £320 H“(an+m i €_k)
= klim HAG" V Twam l g—k)
._—. um ulna" | U) + HAT—"01’" l a" v m)
Ic—mo
g klim H,.(a" I 6"“) + gig, Hu(T‘"a’" I 6"“)
= klim Hum” | g-k) + klggoHuam I 5"“)

Zan+am

37

 

0
Definition 5.1.2(Metric Preimage Entropy)

1
hu(T I 510) = M(a I 6—) = 111320 EH,,(o" | g‘) = "13:0 %H#(a" 15—)
and
hu(T l €_) 2 Slip 12,,(05 l g.) = SUP h#(T l {101)

. ' 1].“...
Lemma 5.1.2 Metric preimage entrOpy hu(T | 6-) is a measure-theoretic co J

' ° . ° 'u ac
gacy invariant and upper preimage entropy htop(T l f) 15 a topological con) g y

invariant.

PROOF

. ve the topo—
It is easy to show the measure-theoretic conjugacy invariant. Here we pro X a X-
T' 1 . 1
logical conjugacy invariant. First we let X1, X2 be compact spaces and . 2 With
- map
be continuous for 2' = 1,2 and 45 1 X1 —> X2 be a homeomorphlsm have
. g x) We
¢T1 = T2¢. First we let a be an Open cover of X2- Then If (M?!)
for k 2 0, N(¢“1a)lTl—k(y) = N(QHTflm'

Hence,

12—1

1
h T ‘,a =limsu —log su N T” _
..,.( 2‘6 > ,.-..P. ,._,.P;. V 2 any. kt)
n~1
= lim sup-l- 10g Slip N(Cb—l V T2“a)/
n—mo n k20,y€X1 i=0 T}\lc(y)
1 12—1
2 lim sup — lo su N ”T”
n——)oo n ngOyEX; (>__/0¢ 2 aNTerI)
1 n—l '
=1imsup—lo su R T" *0;
n—>oo n gk20,yIE)X1 (i\:/() 1 ¢ )lTl‘kQI)

= htop (T1 IF, 45—10:)

Hence ht0p(T2 | 5‘) _<_ hm,,(T1 | g“). If gb is a homeomorphism then ¢-1T2 2 (pm—1 SO
by the above, htop(T1 '6‘.) S htap(T2 '€—). 0
Lemma 5.1.3 Let g and 77 be two finite partitions of X, then

hMC l E‘) S hum l E‘) + HAG l n)

38

 

 

PROOF

H,,(V'T"C|€ P) <H(VT“<VVT"77)|€ )

2:0

H(VT nlé k)+H.. (VT‘iCI VT"r/V€ )

i=0
<H.(VT nltPl+H (VT"'C|VT 2)
i=0 i=0

Let k -> 00, and H).(V:'_0' T_'C l Vii—:01 T477) S 22 ° Hu(< l TI)

This implies that H.(<o I 6 ) S H.073 151+ 72 . Hu(( | n).

Divide by n and let 72 go to infinity, then mg I g-) g h..(n \ E”) + HP“ ‘ 7” 0
Lemma 5.1.4 For any fixed k,

h,,(T l £',a) = h,,(T | g-,\/T-Pa).

izo

PROOF

k
MT l «5‘.VT"'a) = lgn lacuna") lé’)

i=0

gigg- u(\/'T“(VT 0016 )

 

i=0 i=0

k+n— 1
= lim —H,.( V T"a) )lé)

i=0
k+n —1 1 1....-.
= lim Hp( T\i
n—+oo n k-l-TL-l iZVO Q/€~)

: h#(T i E—va)

0
Lemma 5.1.5 If {An} is an increasing sequence of finite partitions of X and C
Is a,

partition with C S V20 A,, then HH(C | A") —> 0 as n—> 00.

Let C be a finite sub—a—algebra of 3, say C = {Ci 3 1' = 1, 2, "'2n}, then the non-empty
Sets of the form Bl n B2...r1 Bm Where B.- = 02' 01‘ X\Ci, form a finite partition of X.
We denote it by (1(0) and we define hp(T | 620) = M(T l €-,a(0)).

39

 

 

Theorem 5.1.1 (KolmOgorov-Sinai)
Let T be a m.p.t. of (X, 3,22) and if? be a finite sub-a-algebra s.t. Vf=0T’"(a($i))—‘=Ba
then
h.(T I r) = h.(T I613?)

PROOF Let C be any partition, we show that h,,(T I '5”, C) S M(T I g", a(§R))
For 12 Z 1, by Lemma 5.1.3 and Lemma 5.1.4,

hu(T | £1 C) _<_. h..(T l 6‘, vg'ZOT-P'aaln) + HA0 I vysoT‘pGo)
= MT l t“. a(§R)) + H,,(C' I vaOT"a(§R))
Let An = V?=0T“a(A) be Lemma 5.1.5, H,,(C I An) ——> O as n --> 00- 0

Lemma 5.1.6 Let (X, B,p) be a probability space. If Bo is a sub-algebra
3(Bo)=B then for m.p.t. T : X ——> X we have hp(T I 6‘) = sup hpr I g 2

of 8 With
A) where

the supremum is taken over all finite sub—algebras A of 30 -

PROOF

Let c > 0. Let C _C_B be finite. Then there exists a finite 1) £30 such tb
D) < 6.

Thus

at. HILXC \

h#(T I 6-20) S h#(T I 6—21)) + H#(C I D)
_<_ hu(T I €_,D) +5

Therefore h,,(T I EZC) S C + sup{h“(T I §_,D) 3 D CBo,D fin'
Ite

h,,(T I E") S sup{h,,(T I 5“,D) : D CBO,D is finite}. The Opposite _
1

obvious. O

} and thus

”equality is

Lemma 5.1.7 Let (X , 3.11) be a probability Space and let {An}?0 be an inc
I"Basin

sequence of finite sub-algebras of B such that V211 An :8. If T : X _+ X is m p t8

then ' "

hu(T I 6—) = "lgghAT I €_,An)-

40

 

Lemma 5.1.8 Let a,- be a finite partition and S}.- be a sub-a-algebra of (X, 8,, mi),
for 2' = 1,2, then

Em... | 8) == E(XA I%1)-E(XB | 32),”,
where S? = (351 x (352, A E 021, B 6 (12, 2‘31 is a sub-o-algebra of I31 and 32 is a sub—a-
algebra of 82.
PROOF

We have 109cm?!) 2 " ZAXBEO 10g E(XAXB I %)XAXB($3 y) and let # : on X m2,

' E
SlnCe fFl E(XA I %I)dm1 = fFl XAdml, fF2 E(XB I (32)de :2 sz XBde’ for F1
81) F2 6 82.

Then

ffleFz E(XA><B I $)d/L : ffFlXFz XAdeH

= ..(A (1le BnF.) = m1(AFlF1) - m2(Bo F2)
And

IF, E(XA I sodm. .sz E(XB I (392)de = m1(A 0F.) .m2(B 0 Fa)
: ffleF2(E(XA I (31) ' E(X3 I C332))dp.

This implies that E(XAxB I 3) 2 E001 I (31) 'E(XB I 82) 21.6. O

Theorem 5.1.2 Let (X1,Bl,m1) and (X2,B2,m2) be Probability Sp

C:
X1 —-> X1,T2 : X2 —-> X2 be m.p.t. Then es and let 7i :
h,.(T1>< T2 I §_) 2" hm1(Tl I 6—) + hm2(T2 I {_I

where p = m1 >< m2.

PROOF

By Lemma 5.1.6,

Let f0 denote the algebra of finite unions of measurable rectangles. Then [3(fo)

=B1X82.
h,,(T1 x T2 I g") : sup{h,,(T1 x T2 I £‘,C) : C C.7-'o,C finite},

Hence

But ifC is finite and C CFO, then C C 01 X 02 for some finite Ol1 C812 012 C32.

41

(T1 X T2 I 6—) = sup{hll(Tl X T2 I €_aal X 0’2) :01 C81,02 C82, (11,02 finite}

Let a = a(a1 x 02) = 0(01) X 0(02),

ff] aI§— k(a:,)duy -Z fflogEOchBIg- k)XAxB($ 30d”

AxBEa

— - Z / [(‘OgEI IXA Ié k)+10gE(XBIE‘kDXAxBISUI-y) dmldmz
AxBEa

= — (if/1031“ (XA I5 k)XAxB($ y)dm1dm2
AxBEa

+//10gE(XB Ié—k)XAxB(x,y))dm1dm2)

—— Z logE(XA If‘k)XAdm1

1460401)
_ Z IOgEIXB I E—k)XB dmz
B€a(a2)
This implies
Hm I t") = Hamel) | £4“) + Hm<a<a2> I 6"“)-

Then
Hu(T1 X T2 I 5 a): m (T I 5 ,Ot(011)) +Hm2 (T I §‘,a(QQII.
So that
hu(T1x T2 I 5’) = hm1(T1I§")+ hm2(T2 I f-)-
0

Theorem 5.1.3 Let T,- ; X. —+ X,,z' : 1,2, be a continuous map 0n th
e

metric space X,_ Then hmp(T1 x T2 I 5*) = ht0p(T1 I 6") + htop(T2 I f‘). Compact

PROOF
Let d,- be the metric on X, We use the metric d((:1:1, $2), (311, 312)) = max(d1(x1,y1) d2“:
1 2, y2))

On X1 X X2-
If F is an (n (3)-Spanning set for T‘kxi E Xi then F 1 X F2 is an (n, e)- Spanning set

for T141331) X T241162). Hence
T(n1€a (T1 X T2)—k(xlix2)) _<_ T(n’€’T—n($1)) . T(TL,€,T-k($2))

42

 

which implies

T(n7 6) (Tl X T2)-k($1) $2)) S 10g sup T(n1 £3 Tl—k(xl))

10g sup
k20,xixx2€X1XX2 kZOJIEXi
—k
+log sup r(n,6,T2 (1132))-
1:20,:L‘26X2

Therefore
hd(Tl X T2I€_) S hd1(T1I€—aX1) + hd2(T2I€_,X2)‘

Now we show the other inequality. - 1
. . ' ' al final) 6
For all Tl-invariant measure [11 and Tg-mvariant measure #2, by varlatlon p

(Theorem 5.2.1) we have
hdl(T1I£—3X1)_>. hu1(T1 I 6—) and hd2(T2 I €_,X2) Z hp2(T2 ‘6 )

Then
hd,(T1 I £1 X1) + we I 52X» 2 h... (T1 I 6‘) + h,,,(T2 I a”)
= hmxp2(Tl >< T2 I g—)

This implies
hd1(Tl I €_)X1)+ hd2(T2 I €—,X2) 2 sup h/11x#2(T1 x T2 I {-7
mxm
: hd(T1 X T2 If‘)

So we can get the equality. 0

Theorem 5.1.4 Let T be a measure-preserving transformation 0f the prob . .
space (X, 8,”). Then the map it —~) h,,(oz,T) is affine where a is any finite paiélhty
of X. Hence, so is the map H —+ h,,(T I 5‘). ""011
PROOF(See [3].)

For any integer n, constant 0 < /\ < 1 and measures [1, pl, #2, with

I“ = A#1+(1 _ ”#2,

we have
0 S Hu(an)) — AIIM1((1n)-_(1 "' A)Hu2(an) S 10g2

43

 

Hence,
(Ma) = MIMIC!) + (1 - AWAGO-
Now, We proceed to h,‘ (T, €_)-

Fix a finite partition (1 and an increasing sequence ,81 < 52 < converging to 5

ash—+00.

Then , for a positive integer n, we have

HI,(a I 6—) = lim lingo H,,(oz I T‘mfli)

14m m

Next, consider the finite partition 0, fl. For any measure Ha We have

Hu(a I 5) = H#(0 V ,3) " Hum)-
Using the finite partitions a" and T‘mfi,, we have
.1)
n - . <log‘2 (5
0 3 Hum" v T-ma» — AHm(a" VT‘mfi,) _ (1 .. A)H,,2 (a v T “(3.) ,—

and
_ ._ a 2 (5.2)
0 Z —IHu(T mfii) "— )‘Hp1(T mfii) — (1 ‘_ A)HH2(T 771,80] Z -— 10%

The second term of (5.2) is non-positive, so adding it to the second term Of (5'1) does

not increase the latter’s value, so

HAO!" I T—mfli) — AHMUI" I T—mfii) — (1— ”Hm (an I T_m;6-)
z SIOgQ
Similarly, adding the second term of (5.1) to that of (5.2) does not degre

389 the
value, so latter,s

__ log2 g H,,(an I T‘mfl,) — AH“, (an I T“’"fl,~) — (1 — /\)H#2(Qn I T‘mfia
Putting these two inequalities together gives
— log? 3 Hm“ IT—mfi.) —— AHMa" l rm.) — (1 - A)Hm(a" I T—mfi.) _<_ log2
Letting z' ——+ 00 and then m —-> 00 gives
~Iog2 3 HM I 6‘) — AHmIa" I E") — (1 — A)H...Ia" I g-I _<_ Iogz

44

 

Now dividing by n and letting n ——> 00 gives that
“(a I 6‘) = My. (a I 6‘) + (1 — A>ha<a In

as required. 0

5.2 Variational Printiple

Lemma 5 2 1 For each positive integer 7', hu(Tr I '5 ) — 7‘ h (T I 5 )

PROOF
First Show that h (T'r I {TVZIi TflIA) = 7‘ ' h). (T I 5‘, A) for any finite partition A

We have
nr— 1
_ — 1' __ H“( ——iA T—rn— —k
7‘ M(Tlé ,A)== "31,30”ng (V T I (6))

r— l

_- lim -supH,, ("\/ T’rj(\/T iA) )IT—m k(€)))

n
fimnk— 3:0 i=0

= h,,(:rr I g: \/1 T“A)
i=0

Thus
7‘ h,(T I g )-—- r SUP/l h#(T I {214) = sum M(T’ I EZVZJ T"A)

S SUPc hu(TT I g c) :: hu(T' I 5‘) where c is any finite partition,

On the other hand,
(7" I 51/1) 3 MT" I 52%;; T“A) -—- r - MT” I §-,A)

This implies that MATT I 5’) Z T ‘ hp(T I 5 ) 0

Definition 5.2.1
T _ su 1l T k a:
ht0p( If ) Ilnglirtn P— ngziufs r(n 6 ( ))

limsup—log SUP N(rVT_iflIT-”‘(I))

sup
k>0, JIEX
i=0

open COVCI‘ I6 n—yoo n

_—
.—

45

where r(n,e,T’k$) is the minimal cardinality of (n, 6) spam 11ng set in 7, 7 j b
' .2' or t e
. . - - —k .
max cardmahty of (HIE) separating set In T (33) and N(fi) ’ Twin) 13 the minimal car-

dinality of subcover of fl which can cover T‘k(:1:).

Lemma 5.2.2 ht0p(Tm I ,5) = m ° htop(T I g‘) for all positive integer m.

PROOF

Here we consider the spanning Set-

Since for a: e X, r(n,e,T"‘,T"kml1‘)) _<_ r(mn, e, T, T‘k(a:)) for all k 2 1

We have i log supkzmex 7‘(n 7 €,Tm,T—km (513)) S % log supkzoflex 'r(mn, 6, T, T_k(~73))

Therefore ht0,,(Tm I F) S 777’ 'htOPIT I 5-)
Since T is uniformly continuous, Ve > 0,36 > 0 such that d(:r,y) < 6 implies

max05,3m_1d(Tix,T‘y) < 6 for all 21:,y in X.

So an (n.6, T—km$)-spanning set W.r.t. Tm is also an (nm,e)-spanning set for T“":1:

w_1°.t. T.
Hence r(n, 6,Tm,T“°ma:) Z 'r(mn, e, T, T’kcc), so

m _k 1
————— log sup TURN, 6, T, T :1: g — 10 su r 771 ’km /
m'n. kZO,r€X ( )) n g kZOp’tiX (n, 67 T a T (1.))

Therefore, m - hmpIT I 6’) S htop(Tm I 5“). 0
Lemma 5.2.3 For all 6 there exists a: and k > 0 such that E1, is an (71,6) separated

SBL in T—k($) With card(En) = supk20,xex T(n, C, T_k($))

PROOF
Because r(n, e, T'k(:r)) g r(n, 6, X), it’s a finite bounded positive integer for all :1: in

X and positive integer k. We can easily find such En. 0
Lemma 5.2.4 Let E C T_"(£L‘), then E C T_"—k(Tk.’13) for all positive integer k.

Remark 5.2.1 [See [14], Remark 8.2.2]Assume 1 < q < n, for 0 g j 3 q — 1,

put a(j) == [917—3)],where [b] denotes the integer part of b. We have :

(1) FixOSqu—l. Then
{0,1321-.-,n—1}= {j+rq+zIOSTSa(j)—1,OSZ'Sq_1}US

46

where S : {0,1, ...,j " 1,j + a(j)qij +a(j)q + 1: ...,n _ 1} and the Cardjljaljty ofS’

is at most 2q.
(2) The numbers {j + rqIO S j _<_ (I “ 1, 0 S T S 0(3) " 1} are all distinct and are all

no greater than n — q.

Theorem 5.2.1 Assume p]- ——> )1 With Mad) 2 0, then

(1’1 . 9‘1 _
_lim H,j(\/ TTQ I 6"“) = Hu(\/ T"a | 6"“)
]-+oo

£50 i=0

for all finite partition 01.

Lemma 5.2.5 Assume [13' ——> I‘ With M30) = 0, then

j-l q_1 . q—l -i __
lim lim Hfln(\/ T"a I 5"“) = 3530an Tea I 5*) = HuIV T a IE >

n—mo k—mo i=0 i=0 i=0
PROOF
By Lemma 5.3.6 we know that
0—1 q__1
“m lim H#n(v IMO I {_k) = “In H "(VT—'0 I lim (k)
“rt—>00 k—ioo i=0 n—>oo i=0 [It—>00
q-l .
: HuIX/(f—‘a' .‘iaé‘ki
q—l
= £1.12. Hu(_\_/0T“'a I 5"“)
q—l
= ”(VT—'01 I 5")
i=0
0

Measurable decompositions are necessary to show the variational principle. See [12]_
Let C be an arbitrary decomposition of the Lebesgue space X and X I; be the factor

space. Let the factor map 71' I X ‘9 X IC be 7T(J,‘) = C where IE E C E C. Then

M(A) = [meow (7 C) Kim/1

Where MC is the conditional measure on C.

Lemma 5.2.6 Let a be a partition of (X, 8,“), consider the factor map 7r,c : X ——+

47

X IT-kg and pm), is the conditional measure of ,u on T""(:r;) - The”

n—1 n—1
Hu(\/ T—‘a I 6—16) 3 / ~ H”’I*(V T_'aIT~k($)) d717,,”

Lemma 5.2.7: Let 77 = (30,31....,Bk} be a partition of X such that ,8 =
{Bo U 31,...,Bo U Bk} is an open cover of X. Then

N(n\-/ TAMI)» S N(n\_/ T“fi)IY . 2"

{:0
for any subset Y of X.
PROOF:
Consider the subcover 3 0f 16 With cardinality N(V?;01 T_iB)IYI
let A.- = (Bo U Bio) n (fr—130 UT‘IB,,) n n (T-("-1)Bo uT’In'llBin_,) 65

N ow we decompose A, into the partition
{’11 :3 {Bio flT—lBji fl fl T_(n-1)Bjn_lijk 1‘ 0 or ik,0 _<_ k _<_ n *1}

Then I‘M/A1,) 2’ 2" and we have

n—1 n—1

{\l T“n n Y: V T‘in n Y aé as} g (1,6,.4,
i=0

i=0

This implies that ”—1 1
N(V T‘in)Iy s N(V T"fi)IY . 2n
i=0 i=0
0

Now we are ready to show the relation between upper preimage entrOpy and m etric
preimage entropy. Here, the main technique used is the construction made by M.
Misiurewicz.
Theorem 5.2.2 (Variational Principle)
Let T : X —+ X be a continuous map of a compact metric space X, then
htop(T | 6“) = 811p (MT I 6')
uEM(X,T)

PROOF
Let p 6 M(X,T). We Show that hu(T I 6—) S hton(T I €_). Let C = {A1,---,Ak}

48

 

be a finite partition of X. Choose 5 > 0 so that e < ——"',c 1013:. Then W6. can choose

compact sets Bj C Aj. 1 .<_. j _<_ (C, With #(Aj \ Bj) < 6 and ‘8: H15}, =¢jfz'7éj, Let
k

77 = {BO,B,, ...,Bk} where Bo = X\U,-=1 Bj- We have ”(130) < Ice, and

k Bin/13' Bi Aj
H..(<In>=—ZnIBt->Zfl;(§5‘)‘°gfl( I; )

 

z: j=l
k
”(Boo/11')1 M(BoflAj) . . . p(B,-flA,-)
:_ B —————/"0g srnce1f2#0,———————=OOI‘1
M 0) 32:; M30) Bo “(8.)

S #(Bo) logk

< kelogk <1

SO we have H,,(C I 77) < 1.
Then fl = {Bo U 31,..., Bo U Bk} is an open Cover of X. We have if n, k 2 1,

Hug; k(V?:01 T—inIT-k(x)) S 108N(V?;.‘T"an—k(,,) where #3,). is the conditional mea—
sure Of “ on T4: (33) and N(Vilz—ol T—inIT"“(2)) denotes the number of nonempty set in

the partition V22] T"n under T‘k(a:). Let me : X ——> X IT-I.E be the facotr map, by

Lemma 5.2.6 and Lemma 5.2.7

n—1

n—1
H,(\/ cram-k): / . H..,.(VT"nIT—n.)>d7rk.u
11:0 XIT- 5 i=0
n—l
S 10g(sup N( V T—in)IT—k(x))
16X i=0
n—l
S 10 su N T‘i -., -2"
flag. I_\_/ mIT I) )

Let k go to infinity, divide by n and n approach to infinity, therefore
hm I é“) _<_ hiop(T | £15) +10s2 S hmp(T I E“) +10g2

So by Lemma 5.1.3

hu(C | 5‘) S hu(77 I 5') + HMC I 77)
S htop(T I 6—) + log2 +1

This gives hpIT I 5’) .<_ htop(T I 5‘) + log2 + 1 for all ,u e M(X, T).
This inequality hows for T" SO " ' MT | 6‘) S n - hwp(T. 6‘) +10g 2 + 1. We divide

49

 

by n and let n approach to infinity. Hence hfl(T I 5‘) g htop (37¢)-

Now we show the other inequality.
Let c > 0 be given We should find some invariant measure ,u such that

1
_>_1imSUp—lo su rnHeTkx
new) ., g..xi’>. ( ())

Let 8(6, X, T) be the right Side Of this inequality.
As in Lemma 4 5 4 let En,k(:r:) be SUCh an (n, 6) separated set for T (:12) of maxrmal

cardinality 3,, ,.(6 X) Let O'n,k.:c G M (X ) be the atomic measure concentrated uni-
6 . LBt [1111,]: E M(X)

formly on the points of E. I. (I), i.e. any... 2 m ZyeEn,k($) y
be defined by p" k —— - "2:131 On ,1... 0 T4. Since M (X ) is compact we can choose a

subsequences {nj, kj} of natural numbers such that

1
lim ilog 8.1,ch (6 X)— — limsup —log sup r(n, e, T k($))
n—mo k_>_0, :rEX

j——)oo ”j
and {#11 k } converges in M (X) to some n E 11/! (X ) We know that It IS an invariant

measure.
Now we choose a partition (1 = {A1,A2, ...,A.} of X so that diam(A,~) < e and
MBA) _— O for 1 < i S k- Since no member 0f V?:01T_ia can contain more than one

member of En,k($), then as in Remark 5.2.1

1Ogsnk(€ X)—_H0'n,.kx(v T—ia)

= H..,...<V T—ia I 5*)
i=0

n—l

< Han k I V T‘ioz I {_k“m) for all positive integer in
i==01
=H...,(V T—‘a I 6)
i=0
a(J’)-1 q-1
s E H..- (T-IWIIV T-‘a I 5*) + 2: H.-. (T400
r=0 i=0 (ES
a(j)-1 q-l .
S E: Han.k'xoT"('¢l+j)(V T-‘Ia I C”) + 2q10g(l)
r=0 i=0

50

Sum this inequality over «7 from 0 to q _ 1 and by Remark 5‘21 til en

n—l 0—1
qlog ancIé, X) S ZHan,k..oT-P(V Tfla I 5-) ‘I‘ 27210g(/).
i=0

10:0

If we divide by n and use Remark 5-2-1 and the concavity 0f —.’L' Ing we can get

q—l
_,- _ 2 2
fiIog..,.(.,x> .<. H..,.(VT a I€ )+ 73—10%)

i=0

(5.3)

Since the members of V3; T40! have boundaries of n-measure zero, by Lemma 5.2.5

we can claim that

9‘1 q_1
_lim H .1. ,..,(V T“‘a | 5‘) = H.(\/T—"a I 5‘)
.raoo i=0 .20
Therefore replacing (n, [C ) by (nj, [6,) in (5.3) and letting j go to infinity we have
0-1 .
qs(e. x. T) s H.(V T“'a I 5‘) -—— Was“ I 5)-
i=0

where 8(6, X,T) = llmjaoo ”flog snj,kj(c,X). We can divide by q and let q go to
infinity to get

8(6.X, T) S M(T I €30) S (MIT I 6‘)

5.3 Preimage S-M-B Theorem

For each finite partition a of X, let 8(a) be the U-algebra generated by 0.
Definition 5.3.1 We define

(1) 0": : i:iT’ka

(2) mammal, v we» =..f8(1im._...Iaa v We» = mamas v T"‘(€))

Now let (X,B,n) be a probability space, {(Bn)} be a sequence of sub—a-algebra of
B and {Xn} be a sequence of random variables. Then {(Xn, B") : n ::= 1,2, ...} is a

martingale if

(1) B7; C Bn+1

51

(2) Xn is measurable w-r.t. Bn
(3) EIIXnI] < oo
(4) EIXn+1/Bn] = X. a.e.

Theorem 5.3.1I1] Every L1 b0unded martingale converges a.e.

{(X_,,,B_n) ; n = 1,2,3,...} iS a reversed martingale if (1),(2),(3) and (4) hold

for n 2 1.
Remark 5.3.1: For a reversed martingale, limnfioo X _,, = X exists and is integrable.

Lemma 5.3.1II1],Theorem 35-9] As above, we have for all A 6 oz

lim E(XA I 0‘71l V T_k(§)) = E(XA I klingoM? V T’k(€))) a.e.

k—ioo

Lemma 5.3.2 Let 9,, == limit—+00 IaIa'lth-k(§) for all n=1,2,3,.. and g“ — supnzlgm

then for each A Z 0 and each A 6 a, we have
uh: E A:g*(:1:) > A} S e")

PROOF
For each A E a, and n=1,2,... Consider

9A : £320 IAIagvT-Hg) = — .132. logE(XA I a? V T”‘(€))

= — log E(XA I [33250711 V T—k(€)))

This shows that 91’] exists and consider

Bf? = {:13 = 91%), -..,/H.) _<_ A,g::(x) > A}

Since BIA eB(1im,.....(aI v T"‘(€)))
MB? 0 A) =/ XA (#1
BIA

._= LA E(XA “132m: VT_k(f))) d#

=/ 8’9? dnge’AMBlA)
3:"

52

 

Therefore
Ha: e A = 9*(rc) > A} = ZHIB: n A) s .—»\ 2 3.5;) S.-.
k=l k:1

0

Lemma 5.3.3 [[10],Corollary 6-2-2l 9* 6 L1.

Lemma 5.3.4 hm.-- 106...-..) exists.
PROOF '
a... 1.3.-..) = —— 22...... 10sE(XA I T"°(€))XA,
And B(T"° (5)) 33(T’(k+1)(€)) for all k 2 1
We also have E(E(XA I T'_k+1(§)) IT‘k(§)) = E(XA I T—k(§))
And E(E(XA I T""c (é) )) < 00 for all positive integer k.

by reversed martingale theorem, the limit exists. 0

Lemma 5.3-5 Let 9.. == llmk—m IaIa'l‘VT‘k(£)i

then

= lim 9n = lim lim IQIQWTHIEI exists a.e in L1.
"400 n—>oo k—>oo

PROOF
Since B (011 V T’k(€)) CBW?“ V T_k(€)) for all k 2 1.

then B(limk_,oo(a? V T_k(€))) CB (“Ink—onO/f+1 V T_k(§)))

And
9" = .152. IOIOI‘VT-Hé)
= aIlimk_.OO(a?vT-k(§)) by Lemma 4.6.1
= - 2 log EIXA I grew? v T-kmnx.
Aea
with

E(E(XA I 1:15:10 a?“ V T4(5) Ilium... a?VT—k(£)) = E(XA I 1.112,]. a? V T—k(§))

53

AIso‘ EKE (X A \ Emit—~00

1‘ v T—’c
a 1. (5)» < 00 for all n by Leflflm 533
a . . .
By Martingale Comet

gel’lce Theorem, 2 ' °
9 hm.H00 9., exists a.e. in D. o

Lemma 5.3.6 We let 60° =

V°°= B i .

mg n .E {"3 f {811} is an increaSlng sequence of SUb‘U"
: l -

algebras oi X and let Boo n "} 1s a decreasing sequence, and a is a finite
partition, then

n——)OO

‘ 3

1"“ Hem" n) = H..(a I 3...).
PROOF

We show the decreasing case an

“10], proposition 5.2.11]
Let A e 01, because E(E(XA IBn—i) I3...) = E' (XA I3”),

. . ' . uenCe-
d a Similar (llSCllssi0n for the increasmg seq

by reversed martingale theorem and Billingsley [1], Theor em 35.9,
mm... E(XA I8.) = E(XA I3...)- . b
B )-E(X IB - Smash. Y
And IaIBn 7‘ —:AEalOgE(XA I " A '1) IS a bounded continuofi
the bounded convergence theorem, we can get
limnaoo HIC! I B”) :2: limit—+00 I IaIBn d”
: Ilimn-aoo IaIBn d” z: H(O.' I 800)

0

Lemma 5.3.7 Let a be a finite partition, then

n—l
" z: _i :2 1 ’ —
Imam ) MITIE 0‘) .$£L%H#(QI VT ’avre.

(:1

11-1 (6))
: Hui“ I lim lim V T—laVT—k(€))
(-1

"-900 k—+oo

PROOF

' I'm 00 H a V T“a V V T”(j‘1)aIT—k(€) 21.1 .
Slnce 1 k4 LA mkflOo Huia I Vi: T“'Q \/

54

 

 

T—klgl) + limk—mo H11 (Vi—:11 T_la I T—k(€))a
Then
lim,HOG HAO! l Vi: T40 V T—k (§))
: 1fink—mo Hu(a V Tnla V V T—(j‘lla ‘ T_k(€)) - [fink—>00 Hp( ljrjllT—la l
T”‘(€))
= 11mm... H,.(v{;3 T401 | 714(6)) ~ lime... Hflwgg T"a \ T‘“‘“l(€)l

We can get

$2211me H..(a \ Vii-i T40 V T "‘(€))
= nmk... HAVE}? T40 ‘ T“‘<€>> - “mm Hm | Two)

By Cesaro theorem and Lemma 5.3.6,

n—aoo k—mo

11—1
“(0 l g) = lim lim Hu(a | V T40. v T—k(€))
1:1

-n-—+oo k—>oo

11—1
2 Hu(a| lim lim V T"a v T"‘(€))
1:1 .

Now we are ready to Show the following theorem Which is Simil t Sh
at 0 an
McMillan-Breiman theory. "011-
Theorem 5.3.2 (preimage S-M-B theorem)

Let T : X -+X be an ergodic m.p.t. on the probability space (X, B a)
3 and a a fi .
111

partition of X. t9
Then
. . 1 1
31.12.23; 571' VI‘=oT-‘aIT-k(o($) = MT I 51a) a.e.
pROOF

Let g" = limkqoo IalV?=1 T-l(a)\/T‘k(§) fOI' ”21,2,3,...

55

Then we can find that

EEOIOS‘T—km =k1§2° IT—Iov...vT—na|T k(£)

+133qu Q‘lT ‘av. .VT-"aVT *(E)
: [03

kl—i—{go 1—|T (k—1)(€) 0T

+ lirn I

k—>oo QlT 10V" vT- navT— k(€)

...—
..—-

g:+gn~loT+u +gloT‘n—l+go OT”

:2: 2972— —-s 0 T where 90 — lim oo-IOIT "(0
3:0

. 1
Let .9 = limn—mo limk—mo IalV?;11T"aVT"‘(€) 83’ Lemma 5.3.5, g exists a e 1n L ‘
Then we can write

 

       

1 1.111 I T "(é)’
l n __
7l+lk—>oo 00'

_30

       

1 s
3 x T
T +n+1 2(gn—s’g)o

By Birkhoff Ergodic Theorem, PTOPOSitiOI1 5-3-6 and 5.3.7

n

‘ f
0T3: gdp
Alyson—+129 X

320

 

= [3520232010 alaWT- wad/u

: hu(avT l 6—) a e

Therefore, we must show the following to prove the Theorem

            

 

1

lim 3: lgn—s ‘_ gl 0 T8 = 0 a.e.

n—mo’n Tl+ 1__0 (5.4)
For each N=1,2,3,...,let G'N 2':- SUpsZN lg, —- gl, Then

n—N n
/Zl9n— s—g|oT"_ gloTs+ j: lgn‘s‘glOTs
3:0 8:013=n-N+1
n
3
— <n+1IZ:GNo OT + n+1 Z lQn—s—9|0T’.
s=n-N+1

56

 

 

We fix N and let 7" go to infinity. Since lgn—s * 9| _<_ 9* +9 6 £1, thesecond term
above tends t

0 0 a.e. Slmflarly, 0'” S g" + g E Li’so we may apply BirldzoffEI‘gOdic
Theorem to the first term:

n—N

1 S
/ lgn- *90T5<1 0N T
3:22.... 23 I .32.; §~

X

0
at (5-4)'
by the dominated convergence theOI‘em and that G N __, O a.e. then we can g

 

Lemma 5.3.8 12,.(T I 6‘) 3 MT)-
PIROOF 1 ( \Vfl’11T’ia)
- - - H (alVIL‘l T“aVT—k(€)) : Hu“ 5
n finite artition a, u
For a y p (a I vn:11T_ia)
afld h" (T, a) = limnaoo Hp.

By Lemma 5.3.7, h,,(T l 5’) S hp(T)- <>

. o n—
5 4 Ergodic Decomposition of MQtriC Prelmage E

tropy

Lemma 5.4.1:([14], Lemma 4.15) Let r 2 1 be a fixed integer F

. I‘ each
exists 6 > 0 such that if C ={A1,...,Ar},17 = {01, .. MC} are an 6 > 0 there

X into 1' sets with EiziM(A AC.) < 5 then Hp(< l 77) + H #07 I C)< Wopartitions 0f
Lemma 5. 4. 2 (cf. [14L Theorem 8. 3) Let T: X fix be a Conti
111

“Cu
compact metric space. Let (Cali: 1be a sequence of partitions SUch t at 8 map Of

mama.
Then

J“ 0.
MT ‘ 5') ._. .332.”u<T I {14.)

Proof:

Let 6 > 0. Choose a finite partition C = {Ah/12’ A } Such that h (T l g C

h,(T\§-)——eifh..(T|§‘)<oo, orh “W >1/6ifh (Tlé‘ )_ )>

00' ChOOSe
5 > 0 to correspond to e and r in Lemma 5.4.1. Choose compact Sets K C A

ii 1 < i <
T With “(‘4' \ K’) < 6/” + 1) Let 6 _ int?! (“Kb K j) and choose 11 Wlth diam(€n )

57

 

< 6 /2.

POT 1 S. i < 7” let E?) be the union Of all the elements of C}. that intersect If} and
let ES) be the union Of the remaining elements of Cu. Since diam(Cn) < 51/2 9361’
CE G; can intersect at most One Ki. Then (5 = {129), ..., Erir)} is so that C; 5 Cu and
#(ES)AA,-) < 6' By Lemma 1 We have HpiC l Cn) < 6. Therefore ifn is such that
diam(Cn) < 5/2, then

hu(T [6-, C) S h#(T I 61¢) + 6 by Lemma 5.2.3
S Mme—.e.) +6

(T \ 5’)

Then diam(Cn) < 5/2 implies hu(T "5—1Cn) > h,‘(T l f“) -— 26 if hp (T \ 51%)

_ . h
hP(T, C") > (1/6)-—€ ifh,‘(T I E ) = 00. Therefore we Show that hmnaw ”
exists and equals h# (T l 5’)-

<7

Ergodic Decomposition of Invariant Measures

W“ T ‘ X —> X be a measurable map' we define EMT) as the set of POints

a: E X such that, for every continuous f : X -——> R, the limit

~

n—l
f =1im lZf(Tj(x))
1:0

n—mo'n.

exists. Further, let C0 (X ) be the space of continuous functions

with the norm llfll = Supxex Hf($)ll- For 5’3 e 20 (T) we define L X 5‘ R endows
”—1 ~73 . CO(X) d
, 1 . s R b
L.(f) = 331;; 2f (T3010) 3,
i=0

Then Lgc is a positive linear functional and L3( f ) = 1, so that by Riesz’

theorem there exists a unique probability measure [it on X SUch th t
a

[X fdu. = Lz(f)

We define EAT) as the set 0f 35 E EMT) Such that
define 22(T) to be the set of a: e 21(T)

entatio11

Ma: is T-invariant. Then we
f . .

or wh1ch pt IS ergodic, and 2(T) to be
58

 

 

 

the set of a: 6 22C?) for Which 17 belongs to the support of #1. Now let/1 be an

invariant measure- Then every integrable function f is [tr-integrable for [1-311110%

every 6 ZCI‘) and
x [((Lfdflxmiiz/defl

Theorem 5.4.1: Ergo dic Decomposition of measure-theoretic entropy
' 6
Let (X,T) be a compact dynarnICaI system and a a finite partition. Let ,u
‘ are
M(X’T) and {11.3122 6 E} with ”(E) = 1- Then a: —> h#z(Taa) and :13 z) MAT)

measurable functions with

hu(T, C!) = A h”: (T) a) dfl
arid

h#(T) = All”; (T) dp

r[‘heorem 5.4.2: Ergodic Decomposition of Metric Preimage Entropy L t e
Let (X,T) be a. compact dynamical system and a a finite pattmm' e 11

Mom ...“... - . e E} withuw) =1- Then a: .., t. (T 1 :30) am .4

hpxtT \ E’) are measurable functions with

T - = ‘
m IE ,a) thufllé .045.
and

h (T g- =/ h I T -
Proof:
By Lemma 5.3.7, we have

MT I €10) = We ‘ 2:12.332. 0?” V The)

— lim lim ,.-
X n—mo k—+oo Ialo‘l 1VT_"(€) d.”

= lim 1'
[((L n—>oo 1:32) lulu?‘1vT-'=(o due) din

: / hu:(T l €_,a) du
X

59

 

BY Lemma 5.4.2, we choose a sequence 0f finite partitionS (Cm) SucIJ that (flam{(m) ’7‘

0. Then
puma wiring" (TIE Cm)
a: 4:13; 31,"; 33g}! (cm I Km)?” v TWO)
: lingo HM lim lim((m](§m)'1"1vT"k(§))
: nil/1P ( an-l—fgo 13320 IleKm)" “”4“” d”
. x du
3 #13100 :/ nllrrgoklggolquCm )" ‘vT *(é’dfi)
:: 121100 hi4: (T l 6-, Cm)d/1'
m x
2: / Iim MT l sacrum
Xm—mo
z] h#I(T|€‘)du
X
o

60

 

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62

   

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