Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2012, Article ID 807405, 10 pages
doi:10.1155/2012/807405
Research Article
The Projection Pressure for Asymptotically Weak
Separation Condition and Bowen’s Equation
Chenwei Wang1 and Ercai Chen1, 2
1
2
School of Mathematics, Nanjing Normal University, Nanjing 210046, China
Center of Nonlinear Science, Nanjing Normal University, Nanjing 210093, China
Correspondence should be addressed to Ercai Chen, ecchen@njnu.edu.cn
Received 1 March 2012; Accepted 15 April 2012
Academic Editor: Francisco Solis
Copyright q 2012 C. Wang and E. Chen. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
Let {Si }li1 be a weakly conformal iterated function system on Rd with attractor K. Let π be the
canonical projection. In this paper we define a new concept called “projection pressure” Pπ ϕ
for ϕ ∈ CΣ and show the variational principle about the projection pressure under AWSC.
Furthermore, we check that the zero of “projection pressure” still satisfies Bowen’s equation. Using
the root of Bowen’s equation, we can get the Hausdorff dimension of the attractor K.
1. Introduction
Let {Si : X → X}li1 be a family of contractive maps on a nonempty closed set X ⊂ Rd .
Following Barnsley 1, we say that {Si }li1 is an iterated function system IFS on X. Hutchinson
2 showed that there is a unique nonempty compact set K ⊂ X, called the attractor of {Si }li1 ,
such that K ∪li1 Si K.
There are many references to compute the Hausdorff dimension of K or the Hausdorff
dimension of multifractal spectrum, such as, 3–5. Thermodynamic formalism played a
significant role when we try to compute the Hausdorff dimension of K, especially the
Bowen’s equation. Usually, we call PJ tψ 0 the Bowen’s equation, where PJ is the
topological pressure of the map f : J → J, and ψ is the geometric potential ψz log |f z|.
The root of Bowen’s equation always approaches the Hausdorff dimension of some sets.
In 6, Bowen first discovered this equation while studying the Hausdorff dimension of
quasicircles. Later Ruelle 7, Gatzouras and Peres 8 showed that Bowen’s equation gives
the Hausdorff dimension of J whenever f is a C1 conformal map on a Riemannian manifold
and J is a repeller. According to the method for calculating Hausdorff dimension of cookiecutters presented by Bedford 9, Keller discussed the relation between classical pressure
2
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and dimension for IFS 10. He concluded that if {Si }li1 is a conformal IFS satisfying
the disjointness condition with local energy function ψ, then the pressure function has a
unique zero root t0 dimH K. In 2000, using the definition of Carathe’ odory dimension
characteristics, Barreira and Schmeling 11 introduced the notion of the u-dimension dimu Z
for positive functions u, showing that dimu Z is the unique number t such that PZ −tu 0.
On the progress of calculating dimH K, 3–5 depend on the open set condition and
separable condition. In fact, there are a lot of examples that do not satisfy this disjointness
condition. Rao and Wen once discussed a kind of self-similar fractal with overlap structure
called λ-Cantor set 12.
In order to study the Hausdorff dimension of an invariant measure μ for conformal
and affine IFS with overlaps, Feng and Hu introduce a notion projection entropy see 13,
which plays the similar role as the classical entropy of IFS satisfying the open set condition,
and it becomes the classical entropy if the projection is finite to one.
Bedford pointed out that the Bowen’s equation works if three elements are present:
i conformal contractions, ii open set conditions, and iii subshift of finite-type Markov
structure. Chen and Xiong 14 proved that subshift of finite-type Markov structure can
be replaced by any subshift structure. In 15, 16, the authors defined projection pressure for
two different types of IFS. In this paper, we consider projection pressure under asymptotically
weak separation condition AWSC and check that Bowen’s equation still holds.
2. The Projection Pressure for AWSC: Definition and
Variational Principle
Let {Si }li1 be an IFS on a closed set X ⊂ Rd . Denote by K its attractor. Let Σ {1, . . . , l}N
associated with the left shift σ. Let Mσ Σ denote the space of σ-invariant measure on Σ,
endowed with the weak-star topology, CX the space of real-valued continuous functions of
X, and π : Σ → K be the canonical projection defined by
{πx} ∞
Sx1 ◦ Sx2 ◦ · · · ◦ Sxn K,
where x {xi }∞
i1 .
2.1
n1
A measure μ on K is called invariant resp., ergodic for the IFS if there is and invariant
resp., ergodic measure ν on Σ such that μ ν ◦ π −1 .
Let Ω, F, ν be a probability space. For a sub-σ-algebra A of F and f ∈ L1 Ω, F, ν,
we denote by Eν f|A the conditional expectation of f given A. For countable F-measurable
partition ξ of Ω. We denote by Iν ξ|A the conditional information of ξ given A, which is
given by the formular:
2.2
Iν ξ | A − XA log Eν XA | A,
A∈ξ
where XA denote the characteristic function on A.
The conditional entropy of ξ given A, written Hν ξ|A is defined by the formula
Hν ξ|A Iν ξ|Adν.
The above information and entropy are unconditional when A N, the trivial σalgebra consisting of sets of measure zero and one, and in this case we write
Iν ξ | A Iν ξ,
Hν ξ | N Hν ξ.
2.3
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3
Now we consider the space Σ, BΣ, m, where BΣ is the Borel σ-algebra on Σ and
m ∈ Mσ Σ. Let P denote the Borel partition:
P
j :1≤j≤l
2.4
of Σ, where j {xi ∞
i1 ∈ Σ, x1 j}. Let I denotes the σ-algebra:
I B ∈ BΣ : σ −1 B B .
2.5
For convenience, we use γ to denote the Borel σ-algebra BRd of Rd . For f ∈ CX,
n
denote f
supx∈X fx and Sn fx n−1
i0 fσ x, for all x ∈ X. Let Σn {b : b x1 , x2 , . . . , xn , xi ∈ Σ, i 1, . . . , n}.
Definition 2.1. For any m ∈ Mσ Σ, we call
hπ σ, m Hm P | σ −1 π −1 γ − Hm P | π −1 γ
2.6
the projection entropy of m under π w.r.t {Si }, and we call
hπ σ, m, x Em f | I x
2.7
the local projection entropy of m at x under π w.r.t {Si }li1 , where f denote the function
Im P|σ −1 π −1 γ − Im P|σ −1 γ.
It is clear that hπ σ, m hπ σ, m, xdmx.
The following Lemma 2.2 gives the relation between the projection entropy and the
classical entropy and the basic properties of the new entropy which are similar to the classical
entropy’s. For more details we can see Theorem 2.2 in 13.
Lemma 2.2. Let {Si }li1 be an IFS. Then
i For any m ∈ Mσ Σ, one has 0 ≤ hπ σ, m ≤ hσ, m, where hσ, m denotes the classical
measure-theoretic entropy of m associated with σ.
ii The map m → hπ σ, m is affine on Mσ Σ. Furthermore if m νdPν is the ergodic
decomposition of m, one has
hπ σ, m hπ σ, νdPν.
2.8
iii For any m ∈ Mσ Σ, one has
1 n−1
Im P0 | π −1 γ x hσ, m, x − hπ σ, m, x,
n→∞n
lim
2.9
for m-a.e. x ∈ Σ, where hσ, m, x denotes the local entropy of m at x, that is, hσ, m, x Im P | σ −1 BΣx.
4
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Definition 2.3. Let k ∈ N and ν ∈ Mσ k Σ. Define
hπ σ k , ν Hν P0k−1 | σ −k π −1 γ − Hν P0k−1 | π −1 γ .
2.10
The term hπ σ k , ν can be viewed as the projection measure-theoretic entropy of ν
w.r.t. the IFS {Si1 ◦ · · · ◦ Sik : 1 ≤ ij ≤ l for 1 ≤ j ≤ k}. The following lemma exploits the
−i
connection between hπ σ k , ν and hπ σ, m, where m 1/k k−1
i0 ν ◦ σ .
Lemma 2.4. Let k ∈ N and ν ∈ Mσ k Σ. Set m 1/k
hπ σ, ν 1/khπ σ k , ν 1/khπ σ k , m.
k−1
i0
ν ◦ σ −i . Then m is σ-invariant, and
Proof. See Proposition 4.3 in 13.
Definition 2.5. An IFS {Si }li1 on a compact set X ⊂ Rd is said to satisfy the asymptotically weak
separation condition AWSC, if
1
log tn 0,
n→∞n
2.11
tn sup # Su : u ∈ {1, . . . , l}n , x ∈ Su K ,
2.12
lim
where tn is given by
x∈Rd
here K is the attractor of {Si }li1 .
Lemma 2.6. Let {Si }li1 be an IFS with attractor K. Suppose that Ω is a subset of {1, . . . , l} such that
there is a map g: {1, . . . , l} → Ω so that
Si Sgi
i 1, . . . , l.
2.13
∞
Let ΩN , σ denote the one-side full shift over Ω. Define G: Σ → ΩN by xj ∞
j1 → gxj j1 . Then
i K is also the attractor of {Si }i∈Ω . Moreover, if one lets π : ΩN → K denote the canonical
projection w.r.t. {Si }i∈Ω , then one has π π ◦ G.
σ , ν.
ii Let m ∈ Mσ Σ. Then ν m ◦ G−1 ∈ Mσ ΩN . Furthermore, hπ σ, m hπ Proof. See Lemma 4.23 in 13.
Lemma 2.7. Let {Si }li1 be an IFS with attractorK ⊂ Rd . Assume that
#{1 ≤ i ≤ l : x ∈ Si K} ≤ k
for some k ∈ N and each x ∈ Rd . Then hπ σ, m ≥ hσ, m − log k for any m ∈ Mσ Σ.
Proof. See Lemma 4.21 in 13.
2.14
Discrete Dynamics in Nature and Society
5
Lemma 2.8. Let a1 , a2 , . . . , ak be given real numbers. If pi ≥ 0 and
log pi ≤ log ki0 eai and equality holds iff pi eai / kj1 eaj .
k
i0
pi 1, then
k
i0
pi ai −
Proof. See Lemma 9.9 in 17.
For convenience, for n ∈ N, write Σn {1, . . . , l}n . According to Lemma 2.6 there is a
set Ωn ⊂ Σn and a map g : Σn → Ωn such that Su Sgu for u ∈ Σn . Let ΩN
n , T denote
and
ξ
denote
the
natural
generator.
Let
the one-sided full shift space over the alphabet ΩN
n
n
N
G : Σ → Ωn be defined by
∞
xi ∞
i1 −→ g xjn1 · · · xj1n j0 .
2.15
Theorem 2.9. Suppose an IFS {Si }li1 satisfies the AWSC with attractor K and f : Σ → R is
continuous. Then
⎛
1⎝
log
lim
n→∞n
u∈ξ
⎞
sup e
n
Sn fx ⎠
x∈G−1 u
sup hπ σ, m fdm : m ∈ Mσ Σ .
2.16
Proof. We divided the proof into two steps.
Step 1.
⎛
1⎝
lim inf
log
n→∞ n
u∈ξ
⎞
sup e
n
Sn fx ⎠
x∈G−1 u
≥ sup hπ σ, m fdm : m ∈ Mσ Σ .
2.17
For arbitrary n ∈ N, m ∈ Mσ Σ, then m ∈ Mσ n Σ. By Lemma 2.8, we have
log
sup e
u∈ξn x∈G−1 u
Sn fx
≥
−1
m ◦ G u
u∈ξn
Hm◦G−1 ξn −1
sup Sn fx − log m ◦ G u
x∈G−1 u
m ◦ G−1 u ◦ sup Sn fx
u∈ξn
x∈G−1 u
≥ Hm◦G−1 ξn Sn fxdm
−1
n fdm.
h T, m ◦ G
By Lemma 2.2i and Lemma 2.6ii, divided by n yields
1
log
n
u∈ξ
sup e
n
x∈G−1 u
Sn fx
h T, m ◦ G−1
fdm
≥
n
hπ T, m ◦ G−1
≥
fdm
n
2.18
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Discrete Dynamics in Nature and Society
hπ σ n , m
fdm
n
hπ σ, m fdm.
2.19
By the arbitrariness of m and n, we have Step 1.
Step 2.
⎛
1⎝
sup hπ σ, m fdm : m ∈ Mσ Σ ≥ lim sup
log
n→∞ n
u∈ξ
⎞
sup eSn fα ⎠.
n
x∈G−1 u
2.20
By the continuity of f, for arbitrary > 0, there exists N ∈ N such that for arbitrary aN ∈ ΣN
and any x, y ∈ aN , we have
fx − f y < .
2.21
Now, for any n ∈ N and anN ∈ ΣnN
SnN fx − SnN f y ≤ n 2N f ,
∀x, y ∈ anN .
2.22
Define a Bernoulli measure ν on ΩN
nN by
νu supx∈G−1 u eSnN fx
v∈ξn
νw1 , . . . , wk supy∈G−1 v eSnN fy
k
νwi ,
,
2.23
wi ∈ ξn k ∈ N.
i1
Then ν can be viewed as a σ nN -invariant measure on Σ by viewing ΩN
as a subset of Σ. By
n
ν ◦ σ −i ∈ Mσ Σ.
Lemma 2.6, we have hπ σ nN , ν hπ T, ν. Define μ 1/n N nN−1
i0
We have
hπ σ, μ fdμ
σ nN , ν
SnN fdν
nN
nN
1
hπ T, ν SnN fdν
nN
1
hT, ν − log tnN SnN fdν
≥
nN
hπ
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1
Hν ξnN − log tnN SnN fdν
nN
⎞
⎛
log tnN
1 ⎝ ≥
−νu log νu νu inf SnN fx ⎠ −
−1
n N u∈ξ
nN
x∈G u
7
nN
⎛
⎞
1 ⎝ ⎠
sup SnN fx − n − 2N f −νu log νu νu
≥
n N u∈ξ
x∈G−1 u
nN
−
log tnN
nN
1
log
nN
u∈ξ
nN
sup e
x∈G−1 u
SnN fx
n 2N f log tnN
.
−
nN
2.24
Let k n N and let n → ∞, then k → ∞. We have
1
sup hπ σ, μ fdμ, m ∈ Mσ Σ ≥ lim sup log
k→∞ k
u∈ξ
nN
sup eSk fx − .
x∈G−1 u
2.25
Since is arbitrary, we finish the proof of Step 2.
Definition 2.10. If an IFS {Si }li1 satisfies AWSC with attractor K and f ∈ CΣ. We call
⎞
⎛
1
2.26
sup eSn fπx ⎠
Pπ f lim ⎝log
n→∞n
u∈ξ x∈G−1 u
n
the projection pressure of f under π w.r.t. {Si }li1 .
It is clearly that, if f 0, we have the same result of Lemma 9.1 in 13.
Corollary 2.11. limn → ∞ log #{Su : u ∈ Σn }/n sup{hπ σ, m : m ∈ Mσ Σ}.
3. Application for Projection Pressure
Definition 3.1. {Si : X → X}li1 is called a C1 IFS on a compact set X ⊂ Rd if each Si extends
to a contracting C1 -diffeomorphism Si : U → Si U ⊂ U on an open set U ⊃ X.
For any d × d real matrix M, we use M
to denote the usual norm of M and M the
smallest singular value of M, that is,
M
max |Mv| : v ∈ Rd , |v| 1 ,
M min |Mv| : v ∈ Rd , |v| 1 .
3.1
8
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Definition 3.2. The IFS {Si }li1 is conformal if for every i ∈ {1, 2, . . . , l}, 1 Si : U → Si U is
0 for all x ∈ U, and 3 |Si xy| Si x
|y| for all x ∈ U, y ∈ Rd .
C1 , 2 Si x
/
Definition 3.3. Let {Si }li1 be a C1 IFS. For x xj ∞
j1 ∈ Σ, the upper and lower Lyapunov
exponents of {Si }li1 at x are defined, respectively, by
1
log Sx1 ,...,xn πσ n x,
n
1
λx −lim sup logSx1 ,...,xn πσ n x,
n→∞ n
λx −lim inf
n→∞
3.2
where Sx1 ,...,xn πσ n x denote the differential of Sx1 ,...,xn : Sx1 ◦ Sx2 ◦ · · · ◦ Sxn at πσ n x. When
λx λx, the common value, denoted as λx, is called the Lyapunov exponents of {Si }li1
at x.
It is easy to check that both λ and λ are positive-valued σ-invariant functions on Σ i.e.,
λ λ ◦ σ and λ λ ◦ σ.
Definition 3.4. A C1 IFS {Si }li1 is said to be weakly conformal if
1
log Sx1 ,...,xn πσ n x − logSx1 ,...,xn πσ n x
n
3.3
converges to 0 uniformly on Σ as n tends to ∞.
l
If IFS {S
weakly conformal,
by Birkhoff’s ergodic theorem, we can conclude
i }i1 is
λxdm − log Sx1 πσx
dm − log Sx1 πσx dm.
Lemma 3.5. Let K be the attractor of a weakly conformal IFS {Si }li1 . Then we have
dimH K dimB K
sup dimH μ : μ m ◦ π −1 , m ∈ Mσ Σ, m is ergodic
max dimH μ : μ m ◦ π −1 , m ∈ Mσ Σ
sup
hπ σ, m
: m ∈ Mσ Σ .
λ dm
3.4
3.5
3.6
3.7
Proof. See Theorem 2.13 in 13.
Theorem 3.6. Let {Si x}li1 be a weakly conformal IFS satsifying AWSC. Let ψx log Sx1 πσx
: Σ → R and π : Σ → K be the canonical projection. Then dimH K is the unique
root of Pπ tψ 0.
Discrete Dynamics in Nature and Society
9
Proof. According to Theorem 2.9, we have
Pπ tψ sup hπ σ, m tψ dm, m ∈ Mσ Σ .
3.8
Let t0 sup{hπ σ, m/ λ dm : m ∈ Mσ Σ}, according to 3.7 t0 dimH K.
First we show Pπ tψ is decreased with respect
to t. If 0 ≤ t1 ≤ t2 , then for any m ∈
Mσ Σ, we have hπ σ, m t1 ψ dm ≥ hπ σ, m t2 ψ dm. Hence according to variational
principle, we have Pπ t1 ψ
≥ Pπ t2 ψ.
As t0 ≥ hπ σ, m/ λ dm for all m ∈ Mσ Σ, hπ σ, m t0 ψ dm ≤ 0 for all m ∈ Mσ Σ,
whence Pπ t0 ψ ≤ 0.
However, Pπ 0 > 0, the existence of a positive
zero t1 for t → Pπ tψ follows from the
intermediate value theorem, that is, sup{hπ σ, m t1 ψ dm, m ∈ Mσ Σ} 0.
For all > 0 there
is a m ∈ Mσ Σ such that
hπ σ, m t1 ψ dm ≥ −. Thus, t1 ≤
hπ σ, m/ −ψ dm / −ψ dm ≤ sup{hπ σ, m/ −ψ dm, m ∈ Mσ Σ} / −ψ dm, let
→ 0 we have t1 ≤ t0 . And t0 ≤ t1 as Pπ t1 ψ 0 implies hπ σ, m t1 ψ dμ ≤ 0 for all
μ ∈ Mσ Σ. So any root of Pπ tψ 0 is equal to dimH K.
Acknowledgments
The author Acknowledges the support by the National Natural Science Foundation of China
10971100 and National Basic Research Program of China 973 Program 2007CB814800.
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