On single-matrix graph-directed iterated function systems ✩

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Journal of Mathematical Analysis and Applications

www.elsevier.com/locate/jmaa

On single-matrix graph-directed iterated function systems ^✩

Luo Jun

^a

, Ya-Min Yang

^b

^{, ∗}

aMathematics Department, The Chinese University of Hong Kong, Hong Kong bCollege of Sciences, Huazhong Agriculture University, Wuhan, China

a r t i c l e i n f o a b s t r a c t

Article history:

Received 20 March 2009 Available online 21 July 2010 Submitted by C.E. Wayne Keywords:

Graph IFS Weak norm Open set condition

We study graph-directed function systems where each contraction in the system has the form fe(x)=^A−1(x+^de), where A is an expanding matrix. We show that a certain discreteness implies the open set condition, and the latter implies the strong open set condition. Hausdorff measures and dimensions (w.r.t. a weak norm) of the invariant sets are investigated. The stationary Markov measures of the system are proved to be translation invariant.

©2010 Elsevier Inc. All rights reserved.

1. Introduction

1.1. Graph IFS. Let

(

V

, Γ )

be a directed graph with vertex setV

= {

¹

, . . . ,

N

}

and edge set

Γ

. We call

{

^fe

;

^e

∈ Γ }

, a collec- tion of contractions f_e

:

R^d

→

R^{d, a}graph-directed iterated function system(graph IFS).

Let

Γ

i j be the set of edges from vertexi to j, then there are unique non-empty compact sets

{

^Ei

}

^N_i=1satisfying [27]

E_i

=

N

j=¹

e∈Γi j

fe

(

E_j

),

1

ⁱ

^N

.

(1.1)

We call

(

E₁

, . . . ,

E_N

)

theinvariant setsof the graph IFS.

The graph IFS is said to satisfy theopen set condition(OSC), if there exist open setsU1

, . . . ,

UN such that

N

j=¹

e∈Γi j

f_e

(

U_j

) ⊂

^Ui

,

1

ⁱ

^N

,

and the left-hand side are non-overlapping unions [14,27]. In addition, if Ui

∩

^Ei

= ∅

^{for all 1iN}, then we say the graph IFS satisfies thestrong open set condition(SOSC) [31].

Let us define M

= (

mi j

)

1ⁱ,j^N to be the associated matrix of

(

V

, Γ )

, that is,mi j

=

^#

Γ

ji counts the number of edges from j toi. We say

(

V

, Γ )

isprimitiveifMis a primitive matrix, i.e.,Mⁿ is a positive matrix for largen. From now on, we will always assume the graph

(

V

, Γ )

in consideration is primitive.

✩ The authors are supported by CNSF 10631040.

*

Corresponding author.

E-mail addresses:yangym09@mail.hzau.edu.cn,yangym05@mails.tsinghua.edu.cn(Y.-M. Yang).

0022-247X/$ – see front matter ©2010 Elsevier Inc. All rights reserved.

doi:10.1016/j.jmaa.2010.07.001

1.2. Single-matrix IFS. If a graph

(

V

, Γ )

contains only one vertex, then the graph IFS

{

^fe

;

^e

∈ Γ }

simplifies to aniterated function system

{

^fj

}

^N_j=1 ([14]). If all f_jhave the form

fj

(

x

) =

A⁻¹

(

x

+

dj

),

1

j

N

,

(1.2)

where A is ad

×

^dexpanding matrix anddj

∈

R^d, then we call

{

^fj

}

^N_j=1 ^asingle-matrix IFS. (A matrix isexpandingif all its eigenvalues have moduli larger than 1.) Let us denote byq

= |

^detA

|

^.

Some special cases of system (1.2) define number systems. The study of such number systems goes back as early as 1970’s [18,28,15]. See also a recent survey [6].

Another special case of (1.2) is the so-calledself-affine tiling system, when N

=

^q

:= |

^detA

|

and the OSC holds. The self- affine tiling system has been studied by many authors [4,16,17,11,22–24,3].

In the above studies, we generally concern the following questions:

(Q1) When does the system satisfy OSC?

(Q2) If the OSC holds, how to compute the Hausdorff dimension and Hausdorff measure of E, the invariant set of(1.2)?

(Q3) Does OSC imply SOSC?

•

^{If A}is a similitude, these questions have satisfactory answers.

LetD

= {

^d1

, . . . ,

dN

}

be the set of translations in (1.2). Define Dn

=

^An−1D

+ · · · +

^AD

+

D

,

n

¹

.

Let dimE denote the Hausdorff dimension ofE, and letH^s

(

E

)

be thes-dimensional Hausdorff measure ofE. A setGis said to ber-uniformly discreteif

|

^x

−

^y

| >

rfor anyx

,

y

∈

^{G. Then}

(i) OSC holds if and only if#Dn

=

^NnandDnis r-uniformly discrete for some r

>

0independent of n.

(ii) If OSC holds, then s

=

^dimE

=

^dlog_logq^{N and0}

<

H^s

(

E

) < +∞

^. (iii) OSC implies SOSC.

Especially in the self-affine tiling system case, many deep results on (Q1) and (Q2) have been obtained by Fourier transformation method [16,11,24]. For (Q3), Schief [31] gives a positive answer for general self-similar IFS.

•

^{In case of A}is not a similitude andNdoes not equal to

|

^detA

|

, it is much more complicated. Actually, in this case the second assertion does not hold. McMullen’s carpets provide counter-examples [26].

To overcome the difficulty that A is not similitude, Lemarié-Rieusset [20] introduce a weak norm

ω

of R^{d such that}

ω (

Ax

) =

^q1/d

ω (

x

)

. Under the weak norm, Ais a ‘similitude’. He and Lau [12] introduce Hausdorff dimension and Hausdorff measure w.r.t. the weak norm, which will be denoted by dimω andH^sw respectively. [12] proved that the above results still hold except the second assertion is replaced by

(ii) If OSC holds, then s

=

^dimωE

=

^d_logq^{logN and0}

<

H^s_ω

(

E

) < +∞

^.

1.3. Single-matrix graph IFS. In this paper, we investigate the graph IFS

{

^fe

;

^e

∈ Γ }

with the form

fe

(

x

) =

A⁻¹

(

x

+

de

),

(1.3)

where A is a d

×

^d expanding matrix andde

∈

R^d. The paper is motivated by the questions posed by Professor S. Ito in a conference in Beijing in 2006. To state the questions, we need some notations.

Denote by

Γ

_{i j}ⁿthe paths from vertex ito vertex j with lengthn. For I

=

^e1

· · ·

^en

∈ Γ

_{i j}ⁿ, set fI

(

x

) :=

^fe1

◦

^fe2

◦ · · · ◦

^fen

(

x

)

and define

d_I

:=

^An−1de1

+

^An−2de2

+ · · · +

^Aden−1

+

^den

,

then f_I

(

x

)

has the form: f_I

(

x

) =

^A−n

(

x

+

^dI

)

. Set Dⁿ_{i j}

:=

d_I

;

^I

∈ Γ

_{i j}ⁿ

.

(1.4)

(P1) Does r-uniformly discreteness ofDⁿ_{i j}^{imply OSC?}

(P2) Does OSC imply SOSC?

(P3) Let

μ

ibe the stationary Markov measures on E_i, are

μ

i‘translation invariant’ on E_i?

Stationary Markov measures will be introduced in Section 2. For a measure

μ

supported by a set E, we say

μ

is translation invarianton E if for anyB₁

,

B₂

⊂

^{E andB}1

=

^B2

+

x, it holds that

μ (

B₁

) = μ (

B₂

)

.

1.4. Main results. In this paper, we generalize the results of [12] to single-matrix graph IFS. These results are worth to be documented, since graph IFS are frequently encountered in practice, for example, in the study of IFS with overlap structures

[30,7,13], in self-similar tiling theory [33,34,5], in Rauzy geometry [29,2,32,19], etc. Our results are in great general form and contain many previous results as special case. Also, they give satisfactory answers to the questions of Professor Ito.

Theorem 1.1.For graph IFS(1.3), the following are equivalent:

(i) OSC.

(ii) #Dⁿ_{i j}

=

^#

Γ

_{i j}ⁿand there is an r

>

0such thatDⁿ_{i j}is r-uniformly discrete for all1ⁱ

,

j^{N and n1.}

(iii) SOSC.

In Section 6, we show by examples that OSC does not imply SOSC if the system is not a single-matrix system.

Theorem 1.2.For graph IFS(1.3), let

λ

be the maximal eigenvalue of M, the associate matrix of

(

V

, Γ )

. If OSC holds, then for any 1^iN,

(i) s

=

^dimωE_i

=

^dlog

λ/

logq.

(ii) 0

<

Hω^s

(

Ei

) < +∞

^.

(iii) The right-hand side of(1.1)is a disjoint union in sense of the measureH^sw.

Remark 1.3. For Theorem 1.1 and Theorem 1.2, the case that A is a similitude has been studied by Li [21]. The case that (1.2) is a tiling system has been settled by Lagarias and Wang [25].

The next theorem answers question (P3). The translation invariance of a measure on a fractal set has never been con- sidered before. Theorem 1.4 seems to be the first result of this type. Moreover, although the result seems very nature, it is hard to prove without using the weak norm technique.

Theorem 1.4.For1ⁱN, the stationary Markov measure

μ

iis equal to a⁻_i¹H^sω

|

E_i, where ai

=

H^sω

(

Ei

)

. Consequently,

μ

i is translation invariant.

1.5. Applications. Recently, Furukado, Ito and Rao [9] apply the above results to the study of atomic surfaces of hyperbolic substitutions and obtain some interesting results. According to a substitution

σ

, [9] constructs a single-matrix graph IFS.

They define a fractal domain-exchange transformation

Φ

on E

=

N

i=¹Ei, the union of the invariant sets.

Φ

preserve the stationary measure

μ

by our results. [9] shows that

(

E

, Φ, μ )

is (measure theoretically) isomorphic to the substitution dynamical systemdefined by

σ

, i.e., there exists a measure-preserving bijection between two systems except a measure zero set.

Akiyama and Loridant [1] apply our results to study the parametrization of boundaries of self-affine tiles.

The paper is organized as follows: In Section 2, we recall some known results on Markov measures. In Section 3, we give a brief introduction to weak norm. Theorem 1.1 is proved in Section 4, Theorem 1.2 and Theorem 1.4 are proved in Section 5. In Section 6 we give some remarks on SOSC.

2. SOSC and Markov measures

In this section, we consider general graph IFS (1.1).

2.1. Markov measures. LetM(R^d

)

denote the collection of probability Borel measures onR^dwith bounded support.

Let p

: Γ → (

0

,

1

]

be a function satisfying

N

j=1

e∈Γi j

pe

=

¹

,

1

ⁱ

^N

.

We shall call paprobability weighton graph

(

V

, Γ )

.

It is well known that there is a unique vector

( μ

1

, . . . , μ

N

) ∈ (

M

(

R^d

))

^N satisfying the equations

μ

_i

₌

N

j=1

e∈Γi j

pe

· μ

_j

_◦

f_e⁻¹

,

1

ⁱ

^N

,

(2.1)

we call

μ

1

, . . . , μ

N theMarkov measuresdetermined by the weights

{

^pe

;

^e

∈ Γ }

^. Remark 2.1. For an IFS, Eq. (2.1) simplifies to

μ =

_N

i=1p_i

· μ ◦

^f_i⁻¹

.

The measure

μ

is called a self-similar measurewhen the mappings fiare similitudes [14].

2.2. Measures on symbolic space and their projections. Let

Σ

_i^∗

=

N j=1

k1

Γ

_{i j}^k denote the collection of all finite paths with initial state (or vertex) i and

Σ

^∗

=

_N

i=¹

Σ

_i^∗. Denote

Σ

_i^N to be the collection of infinite paths with initial state i, denote

Σ =

N

i=¹

Σ

_i^N be the set of all infinite paths.

For I

=

^e1e₂

· · ·

^ek

∈ Γ

_{i j}^k, we define b

(

I

) =

^{i be the} initial state of I, and t

(

I

) =

^{j be the} terminate state of I. Denote EI

:=

^fI

(

Ej

)

where j

=

^t

(

I

)

.

Denote

[

^I

] := {

^e1e2

· · ·

^ek

· · · ∈ Σ :

^e1e2

· · ·

^ek

=

^I

}

, and call it a cylinder of

Σ

. GivenI

,

J

∈ Σ

^∗ witht

(

I

) =

^b

(

J

)

, denote by I J the concatenation of Iand J.

LetS be the shift operator on

Σ

whereS

(

e1e2e3

· · · ) =

^e2e3

· · ·

^. LetPi be the probability measure on

Σ

_i^N satisfying the relations

P

i

[

^e1

· · ·

^en

] =

^pe₁

· · ·

^pe_n

,

e1

· · ·

^en

∈ Σ

_i^∗

.

(2.2) According to formula (1.1), it is seen that

{

^fe1···^en

(

Et(en)

) }

n¹ is a decreasing sequence of compact sets and their inter- section is a single point in E_b(e1). Define a projection

π _: (Σ

₁^N

, . . . , Σ

_N^N

) → (

R^d

, . . . ,

R^d

),

where

π

i

: Σ

_i^N

→

R^{d is defined} by

π

i

(

e₁e₂

· · ·

^en

· · ·)

=

n¹

f_e1···en

(

E_t(en)

).

Then fore

∈ Γ

i j and J

∈ Σ

^∗_j,

f_e⁻¹

◦ π

i

(

e J

) = π

j

◦

^S

(

e J

).

(2.3)

Set

μ

i

=

Pi

◦ π

_i⁻¹

,

then

μ

iis a probability measure supported by E_i.

Proposition 2.2.The projection measures

( μ

1

, . . . , μ

N

)

satisfy Eq.(2.1)and thus are Markov measures on E_i. Proof. Pick any Borel set AinR^d. For any edgee

∈ Γ

i j, set

H_e

= π

⁻_j¹

◦

^fe⁻¹

(

A

).

Then He

⊂ Σ

^N_j and thuse J

∈ Σ

_i^N for any J

∈

^He. Denote e He

:= {

^{e J: J}

∈

^He

}

^{, then}Pi

(

e He

) =

^pePj

(

He

)

by the definition ofPi’s.

First we show that

π

_i⁻¹

(

A

) =

N

j=¹

e∈Γi j

e H_e

.

(2.4)

Suppose I

∈ π

_i⁻¹

(

A

)

and the initial edge ofIise

∈ Γ

i j, then I

∈ π

_i⁻¹

(

A

) ⇐⇒

^f_e⁻¹

◦ π

i

(

I

) ∈

^f_e⁻¹

(

A

)

⇐⇒ π

j

◦

S

(

I

) ∈

f_e⁻¹

(

A

)

by formula

(

2

.

3

)

⇐⇒

^S

(

I

) ∈ π

⁻_j¹

◦

^fe⁻¹

(

A

) =

^He

⇐⇒

^I

∈

^{e H}e

.

So formula (2.4) holds. Applying Pito both sides of (2.4), we obtain

μ

_i

(

A

) =

N

j=¹

e∈Γi j

P

i

(

e He

) =

N

j=¹

e∈Γi j

pe

◦ P

j

(

He

)

=

N

j=¹

e∈Γi j

pe

◦ P

j

◦ π

⁻_j ¹

_◦

f_e⁻¹

(

A

)

=

N

j=¹

e∈Γi j

p_e

◦ μ

_j

_◦

f_e⁻¹

(

A

). 2

Two paths are said to becomparableif one of them is a prefix of the other one; otherwise, they are incomparable. The following measure separation property, proved by Fan and Lau [10, Theorem 2.2], illustrates the importance of SOSC.

Proposition 2.3.Suppose a graph IFS

{

^fe

;

^e

∈ Γ }

satisfies the strong open set condition with open sets

{

^Ui

}

1^iN. Let

μ

ibe Markov measures on E_i,1^{iN. Then}

(i)

μ

i

(

U_i

) =

^1,

(ii)

μ

i

(

EI

∩

^EJ

) =

⁰for any incomparable I

,

J

∈ Σ

_i^∗.

Question.Does the inverse of Proposition2.3hold? That is, does the measure separation property imply SOSC?

2.3. Stationary Markov measures. Recall thatM

= (

mi j

)

1ⁱ,j^N is the associated matrix of graph

(

V

, Γ )

. ThenMis a non- negative primitive matrix. Let

(

c1

, . . . ,

cN

)

be the left eigenvector of M corresponding to the maximal eigenvalue

λ

. By Perron–Frobenius Theorem,

λ >

0 and

(

c1

, . . . ,

cN

)

is a positive vector. Let us assume thatc1

+ · · · +

^cN

=

^1.

Fore

∈ Γ

i j, set pe

=

^cj

c_i

λ

⁻¹

.

It is easy to see that

b(e)=ipe

=

^{1 for each}i. Therefore

{

^pe

;

^e

∈ Γ }

is a probability weight. LetP1

, . . . ,

PN be the measures on

Σ

₁^N

, . . . , Σ

_N^Ndefined by formula (2.2), and let

μ

1

, . . . , μ

N be the Markov measures on E₁

, . . . ,

E_N defined by the weight

{

^pe

;

^e

∈ Γ }

. We shall call

μ

1

, . . . , μ

N thestationary Markov measuresof the graph IFS

{

^fe

;

^e

∈ Γ }

^.

Lemma 2.4.Suppose the SOSC holds for a graph IFS

{

^fe

;

^e

∈ Γ }

. Then for any I

∈ Γ

_{i j}^k, we have

μ

i

fI

(

Ej

) =

^cj ci

λ

^−k

.

Proof. It is obvious that

μ

i

(

EI

)

Pi

( [

^I

] )

.

The strong open set condition implies that

μ

i

(

E_I

∩

^EJ

) =

0 for any incomparable I

,

J

∈ Σ

_i^∗ (Proposition 2.3). Hence

μ

i

(

EI

) = μ

i

EI

\ ∪

EJ

; |

J

| = |

I

|,

J

=

I

,

J

∈ Σ

_i^∗

P

i

[

I

] .

Therefore

μ

i

fI

(

Ej

) = μ

i

(

EI

) = P

i

[

I

] =

pe₁

· · ·

pe_k

=

^cj ci

λ

^−k

. 2

3. Pseudo-norm

Let A be a d

×

^dreal expanding matrix with

|

^detA

| =

q. With respect to A, [20] defines a pseudo norm

ω

onR^{d as} follows (see also [12]).

Denote B

(

x

,

r

)

the open ball with centerxand radiusr. Then V

=

^A

(

B

(

0

,

1

)) \

^B

(

0

,

1

)

is an annular region. Choose any 0

< δ <

¹₂ and anyC^∞ function

φ

_δ

(

x

)

with support inB

(

0

, δ)

such that

φ

_δ

(

x

) = φ

_δ

( −

^x

)

and

φ

_δ

(

x

)

dx

=

1, define a pseudo norm

ω (

x

)

inR^{d by}

ω (

x

) =

n∈Z q^−n/dh

Aⁿx

,

(3.1)

whereh

(

x

) = χ

V

∗ φ

_δ

(

x

)

is the convolution of the characteristic function

χ

V and

φ

_δ

(

x

)

. We list some basic properties of

ω (

x

)

.

Proposition 3.1.(See [12].) The

ω (

x

)

is a C^∞function onR^dand satisfies (i)

ω (

x

)

^0;

ω (

x

) =

⁰if and only if x

=

^0.

(ii)

ω (

x

) = ω (−

^x

)

.

(iii)

ω (

Ax

) =

^q1/d

ω (

x

) ω (

x

)

for all x

∈

R^d.

(iv) There exists

β

such that

ω (

x

+

^y

) β

max

{ ω (

x

), ω (

y

)}

^{for any x}

,

y

∈

R^d.

Item (iii) says that matrix A is a similitude in this weak norm, a fact plays a central role in [12] as well as in our paper.

The next proposition shows that the pseudo-norm

ω (

x

)

is comparable with the Euclidean norm

|

^x

|

^through

λ

max and

λ

min, the maximal and minimal modulus of the eigenvalues of A.

Proposition 3.2.(See [12].) For any0

< ε < λ

min

−

1, there exists C

>

0 (depends on

ε

)such that C⁻¹

|

^x

|

^{lnq/dln(λmax+ε)}

ω (

x

)

^C

|

^x

|

^{lnq/dln(λmin−ε)}

,

if

|

^x

| >

1

,

C⁻¹

|

^x

|

^{lnq/dln(λmin−ε)}

ω (

x

)

^C

|

^x

|

^{lnq/dln(λmax+ε)}

,

if

|

^x

|

¹

.

Let E be a subset of R^d. Define diamωE

=

^sup

{ ω (

x

−

^y

)

: x

,

y

∈

^E

}

^{be the}

ω

-diameter of E. Now we can define a Hausdorff measure of E with respect to the pseudo norm

ω (

x

)

.

H^s_ω,δ

(

E

) =

^inf

_∞

i=¹

(

diam_ωE_i

)

^s: E

⊂

i

E_i

,

diam_ωE_i

δ

.

SinceH^s_ω,δ

(

E

)

is increasing when

δ

tends to 0, we can define H^s_ω

(

E

) =

^lim

δ→⁰H_ω^s_,δ

(

E

).

It is shown thatH^s_ω

(

E

)

is an outer measure and is a regular measure on the family of Borel subsets onR^d. It is translation invariant and has the scaling property; precisely,

H^s_ω

(

E

+

x

) =

H^s_ω

(

E

)

and H^s_ω

A⁻¹E

=

q^−s/dH^s_ω

(

E

).

(3.2)

A Hausdorff dimension with respect to the pseudo norm thus can be defined to be dim_ωE

=

^inf

s

;

H^s_ω

(

E

) =

⁰

=

^sup

s

;

H^s_ω

(

E

) = ∞ .

The relation of dimωE and the classical Hausdorff dimension dimH

(

E

)

has been studied in [12].

4. Uniform discreteness and OSC

We prove Theorem 1.1 in this section. Our proof is an analogue of [12], where the basic idea belongs to [31].

Proof of Theorem 1.1. (i)

⇒

(ii). Suppose the open set condition holds andU1

, . . . ,

UN are open sets such that

N

j=1

e∈Γi j

fe

(

Uj

) ⊂

^Ui

,

1

ⁱ

^N

,

(4.1)

and the left-hand side of (4.1) is a disjoint union.

First we note that f_I

(

U_j

) ⊂

^Uiholds forI

∈ Γ

_{i j}ⁿ.

Let I

=

^e1

· · ·

^en and I

=

^e1

· · ·

^en be two elements of

Γ

_{i j}ⁿ. Using (4.1) repeatedly, one can show that fI

(

Uj

)

and fI

(

Uj

)

belong toU_iand they are disjoint. Hence A⁻ⁿ

(

U_j

+

^dI

) ∩

^A−n

(

U_j

+

^dI

) = ∅

and so that

(

U_j

+

^dI

) ∩ (

U_j

+

^dI

) = ∅.

^{It follows} thatdI

−

^dI

=

0 and thus #Dⁿ_{i j}

=

^#

Γ

_{i j}ⁿ.

Set

η

j

=

^infx∈R^d

{|

^x

|

^:

(

U_j

+

^x

) ∩

^Uj

= ∅}

^{, then}

η

j

>

0 since U_j is an open set. Put r

=

^min

{ η

j: 1 ^jN

}

^{. Since}

(

U_j

+

^dI

) ∩ (

U_j

+

^dI

) = ∅

, we conclude that

|

^dI

−

^dI

|

^rforI

,

I

∈

Dⁿ_{i j}^.

(ii)

⇒

(iii). This is the difficult part of this theorem.

We define the

δ

-parallel body of a set E with respect to the weak norm

ω

as follows:

[

^E

]

^δ

= {

^x

∈

R^d:

ω (

x

−

^y

) < δ

for somey

∈

^E

} .

(Usually

[

^E

]

δ stands for the

δ

-parallel body of E. Here we use an unusual notation to avoid confusion.)

Pick any

δ >

0 and fix it. For any J

∈ Γ

_{i j}ⁿ, define GJ

=

fJ

[

Ej

]

^δ

,

thenGJ

= [

^EJ

]

^q−n/dδ by Proposition 3.1(iii). We define V

(

J

)

, neighbors of J, to be a collection of paths giving by V

(

J

) =

I

∈ Σ

_i^∗:

|

I

| =

nandEI

∩

GJ

= ∅ .

Note thatEI

∩

^GJ

= ∅

if and only if

(

Et(I)

+

^dI

) ∩ ( [

^Ej

]

^δ

+

^dJ

) = ∅

^. We shall show that #V

(

J

)

have a uniform bound, namely,

γ ₌

sup

#V

(

J

)

: J

∈ Σ

^∗

< +∞ .

Let R be a real number such that

1^jNE_j

⊂

^B

(

0

,

R

),

and let

η

be a positive number such that _N

j=¹

[

^Ej

]

^δ

⊂

^B

(

0

, η )

. Then for J

∈ Γ

_{i j}ⁿ,

#V

(

J

) =

^#

I

∈ Σ

_i^∗:

|

^I

| =

ⁿ

, (

Et(I)

+

^dI

) ∩

[

^Ej

]

^δ

+

^dJ

= ∅

^#

I

∈ Σ

_i^∗:

|

^I

| =

ⁿ

, (

E_t(I)

+

^dI

) ∩

B

(

0

, η ) +

^dJ

= ∅

^sup

x∈R^d

#

I

∈ Σ

_i^∗:

|

^I

| =

ⁿ

, (

E_t(I)

+

^dI

) ∩

^B

(

x

, η ) = ∅

N

j=1

sup

x∈R^d

#

dI:dI

∈

^B

(

x

, η ₊

R

) ∩

^Dni j

< ∞,

by the uniform discreteness of Dⁿ_{i j}. Therefore

γ < +∞

^.

Let J

∈ Σ

^∗ be a path such that #V

(

J

) = γ

attains the maximum. We claim that if I J

∈ Σ

^∗, then V

(

I J

) =

^{I V}

(

J

) :=

I L: L

∈

^V

(

J

) .

For any L

∈

^V

(

J

)

, we have EL

∩

^GJ

= ∅

^{, hence E}I L

∩

^GI J

=

^fI

(

EL

) ∩

^fI

(

GJ

) = ∅

. It follows that I L

∈

^V

(

I J

)

and so that I V

(

J

) ⊂

^V

(

I J

)

. By the maximality of #V

(

J

)

, we obtainI V

(

J

) =

^V

(

I J

)

. The claim is proved.

Let us assume that J

∈ Γ

_hjⁿ. Let

δ

=

_β^δ, where the constant

β

is as in Proposition 3.1(iv). Define

U_i

=

^∞

n=¹

I∈Γ_ihⁿ f_{I J}

[

^Ej

]

^δ

,

1

ⁱ

^N

.

(4.2)

ClearlyU_i

∩

^Ei

= ∅

. We shall show these open sets satisfy SOSC.

Fore

∈ Γ

ik, we have fe

(

U_k

) =

^∞

n=1

I∈Γ_khⁿ feI J

[

Ej

]

^δ

⊂

^∞

n=2

I∈Γ_ihⁿ f_IJ

[

Ej

]

^δ

⊂

Ui

.

So

N

k=¹

e∈Γik

fe

(

U_k

) ⊂

^Ui

.

(4.3)

It remains to show that the left-hand side of (4.3) is a disjoint union.

Suppose this is false. Then fe

(

U_k

) ∩

^fe

(

U_k

) = ∅

^{for somee}

∈ Γ

ik, e

∈ Γ

ik

,

e

=

^e (it may happen that k

=

^{k). By the} construction of the open sets, there exist two paths I1 fromktohandI2fromk tohsuch that

feI₁J

[

Ej

]

^δ

∩

feI₂J

[

Ej

]

^δ

= ∅.

Suppose yis a point in this intersection, then y

=

^feI1J

(

y₁

) =

^feI₂J

(

y₂

)

for some y1

,

y2

∈ [

^Ej

]

^{δ. Choosez}1

,

z2

∈

^Ej such that

ω (

y₁

−

^z1

) < δ

, ω (

y₂

−

^z2

) < δ

.

Without loss of generality, we assume that

|

^I1

| |

^I2

|

^. On one hand, as fI

(

x

) =

^A−|I|

(

x

+

^dI

),

we have

ω

feI₁J

(

z1

) −

f_eI2J

(

z2

) = ω

feI₁J

(

z1

) −

y

+

y

−

f_eI2J

(

z2

)

= ω

A^−|eI1J|

(

z1

−

^y1

) +

^A−|eI2J|

(

y2

−

^z2

)

< β

max

δ

q^−|eId1J|

, δ

q^−|e

_I 2J|

d by Proposition 3.1(iv)

= βδ

q^−|eId1J|

=

q^−|eId1J|

δ.

On the other hand, for any L

∈ Σ

_k^∗ with length

|

^L

| = |

^I1J

|

^{, clearlyeL}

∈ /

^V

(

eI1J

) =

^eI1V

(

J

)

; so we have f_eL

(

Et(L)

) ∩

^feI₁J

[

^Ej

]

^δ

= ∅ .

(4.4)

As

|

^eL

| = |

^eI1J

| |

^eI2J

|

^{, we have} f_eI2J

(

Ej

) ⊂

L

f_eL

(

Et(L)

),

whereLruns over the paths in

Σ

_k^∗ with length

|

^L

| = |

^I1J

|

. Therefore, the point feI2J

(

z2

) ∈

^feI2J

(

Ej

)

must belong to some cylinder of the form f_eL

(

E_t(L)

)

; as a result, it does not belong to f_eI1J

([

^Ej

]

^δ

)

by (4.4). It follows that

ω

feI₁J

(

z1

) −

feI₂J

(

z2

)

q^−|eId1J|

δ,

which is a contradiction.

(iii)

⇒

(i) is trivial. 2