Dimension and measures on sub-self-affine sets

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by  Antti Käenmäki

DIMENSION AND MEASURES ON SUB-SELF-AFFINE SETS
ANTTI KA¨ENMA¨KI AND MARKKU VILPPOLAINEN
Abstract. We showthatinatypicalsub-self-affineset, the Hausdorffandthe
7 Minkowskidimensionscoincideandequalthezeroofanappropriatetopological
1 pressure. This gives a partial positive answer to the question of Falconer. We
0
also study the properties of the topological pressure and the existence and the
2
uniqueness of natural measures supported on a sub-self-affine set.

n
a
J

0
1. Introduction
3
An iterated function system (IFS) on Rd is a finite collection of strictly con-
]
S tractive mappings f ,...,f : Rd Rd. For such a system there exists a unique
1 κ
D nonempty compact set E Rd sa→tisfying
. ⊂
h κ
t
a E = f (E), (1.1)
i
m
i=1
[
[
see Hutchinson [17]. When the mappings are similitudes, the set E satisfying (1.1)

1 is called self-similar, and if they are affine, then E is called self-affine. There are
v
many works focusing on calculating the dimension and measures of these sets, see,
1
1 for example [17, 21, 7, 8, 16, 12, 2, 20].
6 Falconer [10] introduced a generalization of self-similar sets by relaxing the
8
equality in (1.1) to inclusion. He termed a compact set satisfying such an in-
0
. clusion sub-self-similar. The same generalization can, of course, be done with
1
0 self-affine sets. If the mappings f ,...,f are affine, then any nonempty compact
1 κ
7 set E Rd satisfying
1 ⊂ κ
:
v E f (E)
i
i ⊂
X i=1
[
r is called sub-self-affine. These sets include many interesting examples, such as
a
sub-self-similar sets, graph directed self-affine sets, unions of self-affine sets, and
topological boundaries of self-affine sets. The reader is referred to [10, 2] for a
§
more comprehensive list of examples.
For sub-self-similar sets satisfying a sufficient separation condition, the open set
condition, Falconer [10] proved that the Hausdorff and the Minkowski dimensions
Date: January 31, 2017.
2000 Mathematics Subject Classification. Primary 28A80; Secondary 37C45.
Key words and phrases. Sub-self-affine set, equilibrium measure, Hausdorff dimension.
AK acknowledges the support of the Academy of Finland (project #114821).
1
2 ANTTI KA¨ENMA¨KIANDMARKKUVILPPOLAINEN
coincide andequal thezeroofanappropriatetopologicalpressure. InTheorem 5.2,
we show that the same result is true for a typical sub-self-affine set and moreover,
such a set carries an invariant measure of full Hausdorff dimension. This gives
a partial positive answer to the question of Falconer [10]. The proof of Theorem
5.2 is based on the existence of an equilibrium measure. The existence of such
measures on a self-affine set was proved by Ka¨enma¨ki [19]. To our knowledge, it is
the first proof for the existence of an ergodic equilibrium measure that is not based
on the existence of the Gibbs-type measure. Recall also the question of Falconer
[9, 6]. Cao, Feng, and Huang [5] have later studied the variational principle in
§
a more general setting. The uniqueness of the equilibrium measure was implicitly
asked in [19]. In Example 6.2, we answer this question in the negative. Sufficient
conditions for the uniqueness can be found in Lemma 3.5 and Theorem 3.6.
We also study the behavior of the topological pressure. There are at most
countably many points where the pressure is not differentiable. We exhibit a
nondifferentiable pressure in Example 6.5. A sufficient condition for the existence
of the derivative is given in Theorem 4.4.
2. Setting and preliminaries
Throughout the article, we use the following notation: Let 0 < α < 1 and I
be a finite set with cardinality κ := #I 2. Put I∗ = ∞ In and I∞ = IN.
For each i I∗, there is n N such th≥at i = (i ,...,i )n=1 In. We call this
1 n
∈ ∈ S ∈
n the length of i and we set i = n. The length of elements in I∞ is infinite.
| |
Moreover, if i I∗ and j I∗ I∞, then by the notation ij we mean the element
∈ ∈ ∪
obtained by juxtaposing the terms of i and j. For i I∗ and K I∞, we define
∈ ⊂
[i;K] = ij : j K and we call the set [i] := [i;I∞] a cylinder set of level i .
{ ∈ } | |
If j I∗ I∞ and 1 n < j , we define j to be the unique element i In for
n
∈ ∪ ≤ | | | ∈
which j [i]. If j I∗ and n j then j = j. We also set i− = i . We
n |i|−1
∈ ∈ ≥ | | | |
say that the elements i,j I∗ are incomparable if [i] [j] = .
∈ ∩ ∅
Defining
αmin{k−1:i|k6=j|k}, i = j
i j = 6
| − | 0, i = j
(
whenever i,j I∞, the couple (I∞, ) is a compact metric space. We call
∈ | · |
(I∞, ) a symbol space and an element i = (i ,i ,...) I∞ a symbol. If there is
1 2
|·| ∈
no danger of misunderstanding, let us also call an element i I∗ a symbol. Define
∈
the left shift σ by setting
σ(i ,i ,...) = (i ,i ,...).
1 2 2 3
It is easy to see that σ is a continous transformation on the symbol space. By the
notation σ(i ,...,i ), we mean the symbol (i ,...,i ) In−1. Observe that to be
1 n 2 n
∈
precise in our definitions, we need an “empty symbol”, that is, a symbol with zero
length.
DIMENSION AND MEASURES ON SUB-SELF-AFFINE SETS 3
The singular values A = α (A) α (A) > 0 of an invertible matrix
1 d
A Rd×d are the squakre kroots of the≥ei·g·e·n≥values of the positive definite matrix
∈
A∗A, where A∗ isthetranspose ofA. Given such amatrixA, wedefine thesingular
value function to be
ϕt(A) = α (A) α (A)α (A)t−l,
1 l l+1
···
where 0 t < d and l = t is the integer part of t. When t d, we set
≤ ⌊ ⌋ ≥
ϕt(A) = det(A) t/d for completeness.
For eac|h i I|, fix an invertible matrix A Rd×d such that A α. Clearly
i i
the products∈A = A A are also inverti∈ble for all i Inkandkn≤ N. If we
i i1··· in ∈ ∈
let α = min α (A ) > 0, it follows that
i∈I d i
ϕt(A )αδ|i| ϕt+δ(A ) ϕt(A )αδ|i| (2.1)
i i i
≤ ≤
for all t,δ 0 and i I∗. According to [28, Corollary V.1.1] and [7, Lemma 2.1],
≥ ∈
we have
ϕt(A ) ϕt(A )ϕt(A ) (2.2)
ij i j
≤
for all t 0 and i,j I∗.
≥ ∈
For K I∞ we set K = i In : i K . If σ(K) K, then it follows that
n n
⊂ { | ∈ ∈ } ⊂
ij K = i K and j K (2.3)
n+m n m
∈ ⇒ ∈ ∈
for all i In and j Im. The converse does not necessarily hold: choose
∈ ∈
I = 0,1 anddefineK = (0,0,...),(1,1,...) I∞ foratrivialcounterexample.
We a{lso s}et K = ∞ K {. Now (2.2) togethe}r⊂with (2.3) implies that
∗ n=1 n
ϕSt(Ai) ≤ ϕt(Ai|n)ϕt(Aσn(i))
i∈XKn+m i∈XKn+m
ϕt(A )ϕt(A ) = ϕt(A ) ϕt(A )
i j i j
≤
i∈Kn i∈Kn j∈Km
jX∈Km X X
for all t 0 and n,m N. This observation allows us to give the following
≥ ∈
definition. Given a set K I∞ with σ(K) K and t 0, we define the
⊂ ⊂ ≥
topological pressure to be
P (t) = lim 1 log ϕt(A ).
K n→∞ n i
i∈Kn
X
The limit above exists by the standard theory of subadditive sequences. Further-
more, it holds that PK(t) = infn∈N n1 log i∈Knϕt(Ai). To simplify the notation,
we let P(t) = P (t).
I∞
P
Lemma 2.1. Suppose that for each i I there is an invertible matrix A Rd×d.
i
∈ ∈
If K I∞ is a nonempty set with σ(K) K, then the function P : [0, )
K
R is c⊂ontinuous, and convex on the connec⊂ted components of [0, ) 1,.∞..,d→.
∞ \ { }
Furthermore, if A < 1 for all i I, then P is strictly decreasing and there
i K
k k ∈
exists a unique t 0 for which P (t) = 0.
K
≥
4 ANTTI KA¨ENMA¨KIANDMARKKUVILPPOLAINEN
Proof. If we let α = min α (A ) > 0 and α = max α (A ), then (2.1) implies
i∈I d i i∈I 1 i
δlogα P (t) P (t+δ) δlogα
K K
− ≤ − ≤ −
for all t,δ 0. It follows that P is continuous. Furthermore, if the matrices are
K
≥
contractive, then δlogα > 0 and P is strictly decreasing with lim P (t) =
K t→∞ K
−
. Since P (0) = lim 1 log#K 0, the last claim is immediate.
−∞ K n→∞ n n ≥
To prove the claimed convexity, take γ (0,1) and 0 t < t < d. Letting
1 2
∈ ≤
l = t , l = t , and l = γt +(1 γ)t , we have for each i K
1 1 2 2 1 2 ∗
⌊ ⌋ ⌊ ⌋ ⌊ − ⌋ ∈
α (A )l1−t1α (A ) α (A )
α (A ) α (A )α (A )t1−l = ϕt1(A ) l1+1 i l1+1 i ··· l i
1 i ··· l i l+1 i i α (A )l−t1
l+1 i
and
α (A )t2−l
α (A ) α (A )α (A )t2−l = ϕt2(A ) l+1 i .
1 i ··· l i l+1 i i α (A ) α (A )α (A )t2−l2
l+1 i ··· l2 i l2+1 i
Hence
ϕγt1+(1−γ)t2(A ) = α (A ) α (A )α (A )t1−l γ
i 1 i l i l+1 i
···
(cid:0)α (A ) α (A )α (A )t2−l(cid:1)1−γ
1 i ··· l i l+1 i (2.4)
(cid:0) ϕt1(A )αl1+1(Ai)l−t1 γ ϕt2(cid:1)(A ) αl+1(Ai)t2−l 1−γ.
i i
≤ α (A )l−t1 α (A )t2−l
(cid:18) l+1 i (cid:19) (cid:18) l2+1 i (cid:19)
The convexity on the connected components of [0,d] 1,...,d is now an imme-
\{ }
diate consequence of H¨older’s inequality. Indeed, choosing 0 t < t < d such
1 2
that t = t , we obtain from (2.4) that for each n N ≤
1 2
⌊ ⌋ ⌊ ⌋ ∈
γ 1−γ
ϕγt1+(1−γ)t2(A ) ϕt1(A ) ϕt2(A ) .
i i i
≤
iX∈Kn (cid:18)iX∈Kn (cid:19) (cid:18)iX∈Kn (cid:19)
The claim follows since the infimum of a family of convex functions is convex. The
convexity on [d, ) follows by a similar reasoning. (cid:3)
∞
Remark 2.2. Observe that (2.4) implies
P γt +(1 γ)t γP (t )+(1 γ)P (t )+dlog(α/α)
K 1 2 K 1 K 2
− ≤ −
for all 0 t1(cid:0)< t2 < d and γ(cid:1) (0,1). It follows from [6, 2] that there exists a
≤ ∈ §
convex function P˜ : [0, ) R such that P (t) P˜ (t) d log(α/α) for all
K ∞ → | K − K | ≤ 2
0 t < d.
≤
Suppose that for each i I there is an invertible matrix A Rd×d with A
i i
α < 1. For a = (a ,...,∈a ) Rdκ, where a Rd is a tra∈nslation vectkor akn≤d
1 κ i
κ = #I, we define a projection∈mapping π : I∞∈ Rd by setting
a
→
∞
π (i) = A a
a i|n−1 in
n=1
X
DIMENSION AND MEASURES ON SUB-SELF-AFFINE SETS 5
as i = (i ,i ,...). The mapping π is clearly continuous. The collection of con-
1 2 a
tractive affine mappings A + a is called an affine iterated function system
i i i∈I
{ }
(affine IFS).
If K I∞ is a nonempty compact set with σ(K) K, then we define E =
K
⊂ ⊂
π (K) and call this set sub-self-affine. We also set E = E and call this set
a I∞
self-affine. The compact set E satisfies
K
E (A +a )(E ). (2.5)
K i i K
⊂
i∈I
[
This is an immediate consequence of the fact that
∞
π (ii) = (A +a ) A a = (A +a )π (i)
a i i i|n−1 in i i a
n=1
X
whenever i K and i K . The converse is also true. Namely, if E′ is a compact
1
set satisfying∈(2.5), then∈forthecompact set K = ∞ i I∞ : σn(i) π−1(E′)
n=0{ ∈ ∈ a }
we clearly have σ(K) K and π (K) E′. To see that E′ π (K), pick x E′
a a 0
⊂ ⊂ T ⊂ ∈
and use (2.5) repeatedly to discover a symbol i = (i ,i ,...) I∞ such that for
1 2
each n N there exists x E′ so that ∈
n
∈ ∈
x = (A +a ) (A +a )(x ) = A x +A a + +a .
0 i1 i1 ··· in in n i|n n i|n−1 in ··· i1
Since A x 0 as n , we have x = π (i). That i K follows from the
| i|n n| → → ∞ 0 a ∈
fact that π σn(i) = x for all n N. See also [10, Proposition 2.1] and [1, 3].
a n
∈ §
Sincetheself-affinesetE satisfies(2.5)withequality, itisalsocalledaninvariant
(cid:0) (cid:1)
set or attractor of the affine IFS A +a . It follows from [17, 3.1] that E is
i i i∈I
{ } §
the only nonempty compact set satisfying such an equality. If there is no danger
of misunderstanding, the image of a cylinder set E = π ([i]) will also be called a
i a
cylinder set.
Recalling Lemma 2.1, we define the singularity dimension to bethe unique t 0
≥
for which P (t) = 0. See also [7, Proposition 4.1], [10, Proposition 3.2], and [20,
K
2]. Inspecting the proof of [7, Theorem 5.4], we see that the singularity dimension
§
serves as an upper bound for the upper Minkowski dimension of E .
K
3. Equilibrium measures
We denote the collection of all Borel probability measures on I∞ by (I∞).
M
We set (I∞) = µ (I∞) : µ is σ-invariant , where the σ-invariance of µ
σ
M { ∈ M }
means that µ([i]) = µ σ−1([i]) = µ([ii]) for all i I∗. Observe that if
i∈I ∈
µ (I∞), then µ(A) = µ σ−1(A) for all Borel sets A I∞ by [3, Theorem
∈ Mσ (cid:0) (cid:1) P ⊂
5.4]. Furthermore, we set (I∞) = µ (I∞) : µ is ergodic , where the
σ σ
E(cid:0) (cid:1){ ∈ M }
ergodicity of µ means that µ(A) = 0 or µ(A) = 1 for every Borel set A I∞ with
⊂
A = σ−1(A). Recall from [29, Theorem 6.10] that the set (I∞) is compact and
σ
M
convex with ergodic measures as its extreme points.
6 ANTTI KA¨ENMA¨KIANDMARKKUVILPPOLAINEN
For K I∞ we also set (K) = µ (I∞) : spt(µ) K and (K) =
σ σ σ
⊂ M { ∈ M ⊂ } E
µ (I∞) : spt(µ) K . Here spt(µ) is the support of µ, that is, the smallest
σ
{ ∈ E ⊂ }
closed set F I∞ for which µ(I∞ F) = 0. Observe that if K I∞ is a nonempty
⊂ \ ⊂
compact set, then (K) is nonempty by [29, Corollary 6.9.1]. It is also compact
σ
M
and convex with (K) as the set of its extreme points.
σ
E
Remark 3.1. (1) Let K I∞ be a nonempty compact set with σ(K) K. If µ
⊂ ⊂ ∈
(I∞) satisfies µ(I∞ K) > 0, then the invariance of µ yields µ(K ) = µ(I∞ K),
σ 0
Ewhere K = ∞ σ−n\(I∞ K) I∞ K. Furthermore, since σ−1(K ) = K\, the
0 n=1 \ ⊂ \ 0 0
ergodicity of µ yields µ(I∞ K) = 1.
T \
(2) There exists a measure µ (I∞) and a nonempty compact set K I∞
σ
∈ E ⊂
with σ(K) K so that µ(K) = 0. Namely, let I = 0,1 and define µ (I∞)
⊂ { } ∈ M
by setting µ( (0,1,0,1,...) ) = 1 = µ( (1,0,1,0,...) ). Since µ([i]) = µ([0i])+
{ } 2 { }
µ([1i]) for all i I∗, the measure µ is invariant. Furthermore, let A I∞ be
∈ ⊂
a Borel set with A = σ−1(A) and 0 < µ(A) < 1. It follows that µ(A) = 1. If
2
(0,1,0,1,...) A = σ−1(A), then also (1,0,1,0,...) A, which clearly cannot
∈ ∈
be the case. Similarly the other way around. Hence µ is ergodic. Choosing now
K = (0,0,...),(1,1,...) (or K = i I∞ : σn−1(i ) = (0,1) for all n N ),
n+1
{ } { ∈ | 6 ∈ }
we have found the desired compact set.
We define a concave function H: [0,1] R by setting H(x) = xlogx when
→ −
0 < x 1 and H(0) = 0. Notice that 0 H(x) 1 for all x [0,1]. The
≤ ≤ ≤ e ∈
following lemma is an immediate consequence of [19, Lemmas 2.3 and 2.2].
Lemma 3.2. If µ (I∞), then
∈ M
n−1
1 H µ([i]) 1 H µ σ−j([i]) + 3k log#I
n ≤ kn ◦ n
iX∈In (cid:0) (cid:1) Xj=0 iX∈Ik (cid:0) (cid:1)
for all integers 0 < k < n. In addition, if t 0 and for each i I there is an
invertible matrix A Rd×d, then ≥ ∈
i
∈
n−1
1 µ([i])logϕt(A ) 1 µ σ−j([i])logϕt(A )+ 3k logα−t
n i ≤ kn ◦ i n
iX∈In Xj=0 iX∈Ik
for all integers 0 < k < n. Here α = min α (A ) > 0.
i∈I d i
If µ (I∞), then we define the entropy of µ by setting
σ
∈ M
h(µ) = lim 1 H µ([i]) .
n→∞ n
i∈In
X (cid:0) (cid:1)
In addition, if for each i I there is an invertible matrix A Rd×d, then for every
i
∈ ∈
t 0 we define the t-energy of µ by setting
≥
Λt(µ) = lim 1 µ([i])logϕt(A ).
n→∞ n i
i∈In
X
DIMENSION AND MEASURES ON SUB-SELF-AFFINE SETS 7
The above limits exist since
limsup 1 H µ([i]) inf 1 H µ([i]) ,
n→∞ n iX∈In (cid:0) (cid:1) ≤ k∈N k iX∈Ik (cid:0) (cid:1)
limsup 1 µ([i])logϕt(A ) inf 1 µ([i])logϕt(A )
n→∞ n iX∈In i ≤ k∈N k iX∈Ik i
by the invariance of µ and Lemma 3.2.
Suppose K I∞ is a nonempty compact set and µ (K). Since, trivially,
⊂ ∈ M
µ([i]) = 0forallI∗ K ,wemaywriteLemma3.2andthedefinitionsofentropyand
∗
\
t-energy by using K instead of In. For our purposes the assumption µ (K)
n
∈ M
is natural, see Remark 3.1. Observe that the mapping µ Λt(µ) defined on
7→
(K) is the infimum of continuous and affine mappings. Since also the mapping
σ
M
µ h(µ)definedon (K)isuppersemicontinuous andaffine(see[29, Theorems
σ
7→ M
8.1 and 8.2] or [19, proof of Theorem 4.1]), the mapping µ h(µ)+Λt(µ) defined
7→
on (K) is upper semicontinuous and affine for each t 0. Finally, it is easy to
σ
M ≥
see that 0 h (µ) log#I and tlogα Λt(µ) tlogα for all µ (K) and
K σ
≤ ≤ ≤ ≤ ∈ M
t 0, where α = min α (A ) > 0 and α = max α (A ).
i∈I d i i∈I 1 i
≥If µ (I∞) and t 0, then for each n N and C In Jensen’s inequality
n
∈ M ≥ ∈ ⊂
implies
µ([i]) logµ([i])+logϕt(A ) log ϕt(A )
i j
− −
i∈Cn (cid:18) j∈Cn (cid:19)
X X (3.1)
µ([i]) µ([i])
= β(i)H H β(i) [0, 1],
β(i) ≤ β(i) ∈ e
iX∈Cn (cid:18) (cid:19) (cid:18)iX∈Cn (cid:19)
where β(i) = ϕt(A )/ ϕt(A ) for all i C . In particular, if K I∞ is a
i j∈Cn j ∈ n ⊂
nonempty compact set with σ(K) K, then (3.1) applied with C = K yields
n n
P ⊂
P (t) h(µ)+Λt(µ)
K
≥
for all µ (K) and t 0. A measure µ (K) is called a (K,t)-equilibrium
σ σ
∈ M ≥ ∈ M
measure if
P (t) = h(µ)+Λt(µ).
K
To simplify the notation, we speak about t-equilibrium measures when K = I∞.
From now on, without mentioning it explicitly, we assume that for each i I there
is an invertible matrix A Rd×d. ∈
i
∈
Theorem 3.3. If K I∞ is a nonempty compact set with σ(K) K and t 0,
⊂ ⊂ ≥
then there exists an ergodic (K,t)-equilibrium measure.
Proof. To prove the theorem, we use an approach similar to that in [19, Theorem
2.6]. For each n N we define a Borel probability measure
∈
ϕt(A )δ
ν = i∈Kn i ihi,
n ϕt(A )
P i∈Kn i
P
8 ANTTI KA¨ENMA¨KIANDMARKKUVILPPOLAINEN
where δ is the Dirac measure and for each i K the symbol h I∞ is chosen
ihi ∈ n i ∈
such that ih K. Now with the measure ν (K), we have an equality in
i n
∈ ∈ M
(3.1) (when C is chosen to be K ). Furthermore, we set
n n
n−1
µ = 1 ν σ−j
n n n ◦
j=0
X
for all n N. By going into a subsequence, if necessary, it follows that µ
n n
∈ { }
converges weakly tosome µ (K). According to Lemma 3.2andtheconcavity
σ
∈ M
of H, we have
1 H ν ([i]) 1 H µ ([i]) + 3k log#I (3.2)
n n ≤ k n n
iX∈Kn (cid:0) (cid:1) iX∈Kk (cid:0) (cid:1)
for all integers 0 < k < n. Letting α = min α (A ) > 0, Lemma 3.2 also implies
i∈I d i
1 ν ([i])logϕt(A ) 1 µ ([i])logϕt(A )+ 3k logα−t (3.3)
n n i ≤ k n i n
iX∈Kn iX∈Kk
for all integers 0 < k < n. Now (3.1) (with the equality), (3.2), and (3.3) yield
1 log ϕt(A ) 1 H µ ([i]) + 1 µ ([i])logϕt(A )+ 3k log#Iα−t
n i ≤ k n k n i n
iX∈Kn iX∈Kk (cid:0) (cid:1) iX∈Kk
for all integers 0 < k < n. Letting n along the chosen subsequence, we get
→ ∞
P (t) 1 H µ([i]) + 1 µ([i])logϕt(A ).
K ≤ k k i
iX∈Kk (cid:0) (cid:1) iX∈Kk
Recall also[22, Theorem1.24]andthefactthatcylinders arebothopenandclosed.
Letting k , we have shown that µ is a (K,t)-equilibrium measure.
→ ∞
Since the mapping µ h(µ)+Λt(µ) defined on (K) is upper semicontinu-
σ
7→ M
ous and affine, the set of all (K,t)-equilibrium measures is compact and convex.
Moreover, for a (K,t)-equilibrium measure µ, by Choquet’s Theorem ([23, 3]),
§
there exists a Borel probability measure τ on (K) such that
µ σ
E
h(µ)+Λt(µ) = h(η)+Λt(η) dτ (η).
µ
ZEσ(K)
(cid:0) (cid:1)
This implies the existence of an ergodic (K,t)-equilibrium measure. See also [19,
(cid:3)
Theorem 4.1].
If K I∞ is a nonempty compact set with σ(K) K, then µ (K)
σ
⊂ ⊂ ∈ M
is called a (K,t)-semiconformal measure if it satisfies the following Gibbs-type
property: there exists a constant c 1 such that
≥
c−1e−|i|PK(t)ϕt(A ) µ([i]) ce−|i|PK(t)ϕt(A ) (3.4)
i i
≤ ≤
DIMENSION AND MEASURES ON SUB-SELF-AFFINE SETS 9
for all i K . To simplify the notation, we speak about t-semiconformal measures
∗
∈
when K = I∞. For the motivation of the term “semiconformal”, the reader is
referred to [8, Lemma 1]. Since a (K,t)-semiconformal measure µ satisfies
h(µ)+Λt(µ) lim 1 µ([i]) logµ([i])+logc−1enPK(t)µ([i]) = P (t),
≥ n→∞ n − K
i∈Kn
X (cid:0) (cid:1)
a (K,t)-semiconformal measure is always a (K,t)-equilibrium measure.
We will discover that semiconformal measures may or may not exist: If for given
t 0 there exists a constant D 1 such that
≥ ≥
D−1ϕt(A )ϕt(A ) ϕt(A )
i j ij
≤
for all i,j I∗, then there exists an ergodic t-semiconformal measure. Take
∈
account of Remark 6.1. Moreover, if a t-semiconformal measure is ergodic, then it
is the only t-semiconformal measure. These facts follow from [21, Theorem 2.2] by
a minor modification. More precisely, in [21] it was assumed that the parameter
t is an exponent, but an examination of the proof reveals that this fact is not
required. In Example 6.4, it is shown that semiconformal measures do not always
exist. Also, even if there exists a semiconformal measure, it is not necessarily
ergodic, see Example 6.2.
Remark 3.4. Suppose K I∞ is a nonempty compact set with σ(K) K. If a
⊂ ⊂
(K,t)-semiconformal measure exists when P (t) = 0, then it is easy to see that
K
t(E ) < . Here t denotesthet-dimensional Hausdorffmeasure, see[22, 4.3].
K
H ∞ H §
Indeed, it follows from [7, proof of Proposition 5.1] that there exists a constant c,
depending only on the dimension of the ambient space, such that
t(E ) climinf ϕt(A ).
K i
H ≤ n→∞
i∈Kn
X
From this, the claim follows immediately. Moreover, if P (t) = 0 for 0 t 1,
K
≤ ≤
then the existence of a (K,t)-semiconformal measure implies t(E ) < . Here
K
P ∞
t denotes the t-dimensional packing measure, see [22, 5.10]. This fact follows
P §
from [11, Proposition 2.2(d)], since for every x E and 0 < r < diam(E )
K K
∈
µ π−1 B(x,r) µ([i ]) ϕt(A )
◦ a |n c i|n = c > 0,
rt ≥ diam(E )t ≥ α (A )t
(cid:0) (cid:1) i|n 1 i|n
where x = π (i) for some i K, µ is a t-semiconformal measure, and n is the
a
∈
smallest integer for which E B(x,r). It seems difficult to extend this to the
i|n ⊂
case t > 1.
The following two results give sufficient conditions for the uniqueness of the
equilibrium measure. A self-affine set carrying two different ergodic equilibrium
measures is given in Example 6.2.
10 ANTTI KA¨ENMA¨KIANDMARKKUVILPPOLAINEN
Lemma 3.5. Suppose that there is an invertible matrix A Rd×d such that A = A
i
∈
for all i I. Then for each t 0, the uniform Bernoulli measure is the unique
∈ ≥
t-equilibrium measure.
Proof. In this case, for a given n N, the value of ϕt(A ) is independent of the
i
∈
specific i In chosen. Using this, one easily finds that
∈
P(t) = logκ+ lim 1 logϕt(An),
n→∞ n
and, for any µ (I∞),
σ
∈ M
Λt(µ) = lim 1 logϕt(An).
n
n→∞
Applying Jensen’s inequality, we have
1 H µ([i]) H( 1 ) = n logκ
κn ≤ κn κn
i∈In
X (cid:0) (cid:1)
with equality if and only if µ([i]) = 1 for every i In whenever n N. Hence
κn ∈ ∈
h(µ) logκ for every µ (I∞) and h(µ) = logκ if and only if µ is the uniform
σ
≤ ∈ M (cid:3)
Bernoulli measure. The proof is finished.
We remark that generalizing the result of Lemma 3.5 for an arbitrary nonempty
set K I∞ satisfying σ(K) K seems to be difficult. The following theorem
⊂ ⊂
shows that if a (K,t)-semiconformal measure is ergodic, then it is the only (K,t)-
equilibrium measure.
Theorem 3.6. Suppose that K I∞ is a nonempty compact set with σ(K) K,
⊂ ⊂
t 0, and c 1. If µ (K) satisfies µ([i]) c−1e−|i|PK(t)ϕt(A ) for all
σ i
≥ ≥ ∈ M ≥
i K , then any (K,t)-equilibrium measure is absolutely continuous with re-
∗
∈
spect to µ. Moreover, if µ lies in the convex hull of a countable family of ergodic
(K,t)-equilibrium measures, then the closure of the convex hull is precisely the set
of all (K,t)-equilibrium measures. In particular, if µ is itself an ergodic (K,t)-
equilibrium measure, then it is the only (K,t)-equilibrium measure.
Proof. Toprovethefirstclaim, wefollowsomeoftheideasofBowen[4]. Assume to
thecontrarythatthereexistsaBorelsetA I∞ suchthatµ(A) = 0andν(A) > 0,
⊂
where ν is a (K,t)-equilibrium measure. Choose 0 < ε < exp 2(logc+2)/ν(A) .
− e
Now there exists a sequence of symbols i such that
k
{ } (cid:0) (cid:1)
∞
A [i ] =: C˜
k
⊂
k=1
[
and µ(C˜) = µ(C˜ A) < ε. Since ν(C˜) ν(A) > 0, there exists N N large
\ ≥ ∈
enough such that ν(C˜ ) ν(A)/2, where
N
≥
N
C˜ = [i ].
N k
k=1
[