FRACTAL GEOMETRY

Katrin GelfertCourse notes

IM-UFRJ 2016 2nd semester

Contents

1. Dimensions 11.1. Hausdorff dimension 11.2. Box-counting dimension 12. Methods to calculate dimension 32.1. Primer on measures 32.2. Local dimension of a measure 42.3. Product sets 52.4. Energy methods, potentials 53. Iterated function systems 63.1. Existence of attractor of IFS 63.2. Coding etc. 63.3. Similarity dimension 73.4. Self-similar sets 83.5. Nonlinear conformal IFS 83.6. Gibbs measures for weakly Holder potential 83.7. The pressure function 103.8. Bowen’s formula 114. Repellers and Markov partition 124.1. Examples 124.2. One-dimensional Markov maps 124.3. Examples – Rational maps 125. Multifractal formalism 135.1. Spectrum of local dimension 13References 15

(X, d) will always be a separable metric space (mostly Rn),for box-counting dimension (X, d) will be totally bounded.

1. Dimensions

1.1. Hausdorff dimension. Given A ⊂ X and r > 0,consider the family U(A, r) of all finite or countable covercovers of A by open balls of radius smaller than r. Givens ≥ 0 define

Hsr (A)

def= inf{

∑i

|Bi|s : Bi ∈ U(A, r)}.

Since r 7→ Hsr (A) is monotone, the following limit exists

(possibly infinite)

Hs(A)def= lim

r→0Hsr (A)

and is called the s-dimensional Hausdorff measure of A.For every s ≥ 0, Hs is a Borel regular measure.

Lemma 1.1. We have

• Hs(∅) = 0• Hs(A) ≤ Hs(B) if A ⊂ B• Hs(

⋃iAi) ≤

∑iH

s(Ai)

Proof. �

Lemma 1.2. For every A ⊂ X there exists a uniques ∈ [0,∞] such that Ht(A) = 0 for every t > s andHt(A) =∞ for every t < s.

Proof. �

The number s defined by the above lemma is the Hausdorffdimension of A and denoted by dimHA

dimHA = inf{s : Hs(A) = 0} = sup{s : Hs(A) =∞}.

Lemma 1.3. We have

i) dimH ∅ = 0ii) dimHA ≤ dimHB if A ⊂ Biii) dimH

⋃iAi = supi dimHAi

iv) dimH Rn = n

Property iii) is also called (countable) stability.

Proof. �

Given two metric spaces (X, dX) and (Y, dY ), we call amap f : X → Y (C,α)-Holder if for every x, y ∈ X wehave dY (f(x), f(y)) ≤ CdX(x, y)α. For any such map wehave the following.

Lemma 1.4. If f is (C,α)-Holder then for every A ⊂ Xand every s ≥ 0 we have Hs/α(f(A)) ≤ Cs/αHs(A).

Proof. �

Corollary 1.5. If f is (C,α)-Holder then for every A ⊂X we have dimH f(A) ≤ 1

α dimHA.In particular, if f is bi-Lipschitz then

dimH f(A) = dimHA.

Example (Middle-α Cantor set). Given α ∈ (0, 1), letCα,0 = I = [0, 1]. For n = 0, 1, 2 . . ., assuming that Cα,nwas already defined and is the disjoint union of 2n closedintervals Ii1...in of length(

1− α2

)ndivide each of them into two subintervals

Ii1...in0 ∪ Ii1...in1 ⊂ Ii1...inobtained by removing from Ii1...in the open interval oflength α times shorter and concentric. Let

Cα,n+1def=

⋃i1...inin+1

Ii1...inin+1

observe that Cα,0 ⊃ Cα,1 ⊃ . . . are nested compact setsand

Cαdef=

∞⋂i=1

Cα,n

is a nonempty compact, so-called middle-α Cantor, set.All Cα are mutually homeomorphic (each is Cantor) sets.

1

2 FRACTAL GEOMETRY

Lebesgue measure of each Cα is 0, since

Leb(Cα,n) = (1− α)n → 0.

1.2. Box-counting dimension. Given a totally boundedset1 A ⊂ X and r > 0 denote by N(A, r) the minimalnumber of sets of diameter r needed to cover A. Givens ≥ 0 and r > 0 define

Msr (A)

def= N(A, r)rs

and let (note that r 7→Msr (A) may not be monotone hence

the limit may not exist)

Ms(A)def= lim inf

r→0Msr (A), M

s(A)

def= lim sup

r→0Msr (A).

Lemma 1.6. We have

• Ms(∅) = Ms(∅) = 0

• Ms(A) ≤Ms(B) and Ms(A) ≤Ms

(B) if A ⊂ B

Lemma 1.7. For every A ⊂ X there exists a uniques ∈ [0,∞] such that M t(A) = 0 for every t > s andM t(A) =∞ for every t < s.

Analogously, Mt(A).

The unique number s defined by the above lemma is thelower box-counting dimension of A and denoted by dimBA

dimBA = inf{s : Ms(A) = 0} = sup{s : Ms(A) > 0}.

Analogously, the upper box-counting dimension dimBA ofA is defined.The following facts are an exercise:

dimBA = lim infr→0

logN(A, r)

− log r,dimBA = lim sup

r→0

logN(A, r)

− log r.

If dimBAdef= dimBA = dimBA, we call this number

the box-counting dimension of A. Other name is (are)Minkowski dimension(s).

Lemma 1.8. We have

i) dimB∅ = dimB∅ = 0ii) dimBA ≤ dimBB and dimBA ≤ dimBB if A ⊂ B

iii) dimHA ≤ dimBA ≤ dimBAiv) dimB(A ∪B) = max{dimBA,dimBB}v) if dimBA,dimBB exist, then dimBA∪B does and

dimB(A ∪B) = max{dimBA,dimBB}vi) dimB ,dimB ,dimB are invariant under bi-Lipschitz

mapsvii) dimBA = 0 if A countable

To show v): If dimBA,dimBB exist, then by iv)

dimB(A ∪B) = max{dimBA,dimBB},

on the other hand, by monotonicity of dimB

dimBA ∪B ≥ max{dimBA,dimBB}.

Inequalities in item iii) can be strict:

Example. For every 0 < a < b < 1 there is A ⊂ R with

dimHA = 0 < a = dimBA < b = dimBA.

Indeed [NN], fix squences (ak)k, ak = e−k, and

Sn =

n∑k=1

bk, Tn =

n∑k=1

akbk, T = limnTn

Let

Adef= {0, a1, . . . , b1,

b1a1 + a2, . . . , b1a1 + b2a2, . . .n∑k=1

bkak + an+1, . . . ,

n∑k=1

bkak + bn+1an+1, . . .}

choose (bk)k ⊂ N so that

a) bk →∞ monotonicallyb)∑∞k=1 akbk < 1

c)

a = lim infn

1

nlogSn, b = lim sup

n

1

nlogSn,

d) exists C > 0 st. T−Tn

anSn≤ C for all n

by c), given r > 0 and n = n(r) ≥ 1 with e−(n+1) < r ≤e−n, all Sn points {a1, . . . ,

∑nk=1 bkak} are e−n-separated.

hence N(A, r) ≥ Sn and

lim infn

logN(A, r)

− log r≥ lim inf

n

logSnn

= a

by d),

N([Tn, T ], r) ≤ T − Tne−(n+1)

=1

e

T − Tnan

≤ C

eSn

and N([0, Tn], r) ≤ Sn. hence

logN(A, r)

− log r≤ (1 + C/e)Sn

n+ 1

and the lim sup of the latter is b.

In general, box-counting dimension is not stable under in-finite sums. The lower box counting dimension is not evenstable under finite sums, that is, item iv) does not have acounterpart for dimBA.

Exercise ([Fal1, Chapter 3]). Find A,B ⊂ R such that

dimBA ∪B > max{dimBA,dimBB}.

Example. Any rectangle R = [a, b]n ⊂ Rn has dimBR =n since covered by Cr−n square boxes of side length r,that is, N(R, r) ≤ Cr−n, and hence dimBR ≤ n.If A ⊂ Rn has d = dimBA < n then Leb(A) = 0. In-deed, with ε = 1

2 (n − d) and small r, there is cover of A

by r−(d+ε) sets of diameter r. Each of them is covered bya set of Lebesgue volume < Crn and hence gives rise toa set of volume < Crn · r−(d+ε). As r was arbitrary, itfollows Leb(A) = 0.Hence, if A ⊂ Rn and Leb(A) > 0 then dimBA ≥ n. Inparticular, dimBR = n.

Corollary 1.9. Any open set U ⊂ Rn has box-countingdimension n. Any set A ⊂ Rn has upper box countingdimension ≤ n.

1A subset of a metric space is totally bounded if for any r > 0, it can be covered by a finite number of balls of diameter r.

FRACTAL GEOMETRY 3

The following is one property of the box-counting dimen-sion(s) which distinguishes them from Hausdorff.

Lemma 1.10. dimBA = dimBA, dimBA = dimBA

Proof. A ⊂ A and item ii) implies dimA ≤ dimA. Toprove the claim it hence suffices to assume dimA <∞.A ⊂

⋃iAi ⇒ A ⊂

⋃iAi ⇒ N(A, r) ≥ N(A, r)

A ⊂⋃iAi ⇒ A ⊂

⋃iAi ⇒ N(A, r) ≥ N(A, r) �

The following examples indicate problems/disadvantagesthe box-counting dimension has.

Example. The countable set Q∩ [0, 1] has box-dimension

1 since Q ∩ [0, 1] = [0, 1].

Example. For A = {0, 1, 1/2, 1/3, . . .} compact andcountable

dimBA = 1/2.

Indeed, given r ∈ (0, 1/2) take k ≥ 1 such that

1

(k + 1)k≤ r < 1

(k − 1)k.

If |U | ≤ r then because

1

k − 1− 1

k=

1

(k − 1)k> r

such U covers at most one point in the family{1, 1/2, . . . , 1/k}. Hence

N(A, r) ≥ k.

On the other hand, k+1 intervals of length r cover [0, 1/k]and the remaining k− 1 points are covered by some k− 1other intervals and hence

N(A, r) ≤ (k + 1) + (k − 1) = 2k.

Taking limits r → 0 hence

log k

log(k + 1)k≥ lim

r

logN(A, r)

− log r≥ log 2k

log(k − 1)k

implies dimBA = 1/2.

The above serves also as an example to the fact that (nocountable stability for dimB)

dimB

⋃i

Ai 6= supi

dimBAi.

29/08/2016

Equivalent definitions. Let A ⊂ X and r > 0.

Denote by N(A, r) the minimal number of open balls ofdiameter r centred at points in A and covering A.

Denote by N(A, r) the maximal number of pairwise dis-joint balls of diameter r centred at points in A.

Lemma 1.11. We have (analogously dimB)

dimBA = lim infr

log N(A, r)

− log r= lim inf

r

log N(A, r)

− log r.

Proof. If {Br(xi) : xi ∈ A}, are some maximal set of ballsthen {B2r(xi)} cover A (which have diameter 4r). Indeed,for x ∈ A there is i such that x ∈ Br(xi), otherwise couldtake {Br(xi)}∪{Br(x)} which would contradict maximal-

ity. Hence, N(A, 4r) ≤ N(A, r).On the other hand, if {Aj} covers A by sets of diameterr and {Br(xi) : xi ∈ A} family of pairwise disjoint balls,then every Aj must cover (at least one) point xi for somei = i(j) and hence Br(xi) ⊃ Aj . Since {Br(xi)} are pair-wise disjoint, there are at least as many Aj as Br(xi),

hence N(A, r) ≤ N(A, r).Finally, note that

N(A, r) ≤ N(A, r) ≤ N(A,r

2)

Take limits. �

There are many more – equivalent – ways to define thesedimensions: If X = Rn then we can choose any r-grid andreplace N(A, r) by the number of grid cubes intersectingA (this gave rise to the name).

Determining the box-counting dimension of the coastline of Great Britain,

Figure by: Prokofiev, https://en.wikipedia.org/wiki/File:Great_Britain_Box.svg#

filelinks

Equivalently, instead of sets of diameter r one can useclosed (open) balls of radius r.

Example. For Cα we have

dimB Cα =log 2

log 21−α

.

Ineed, given rn = ((1−α)/2)n, cover Cα,n by 2n intervalsof length rn, hence N(Cα, rn) ≤ 2n. For any r ∈ [rn+1, rn)we have

N(Cα, r) ≤ N(Cα, rn+1) ≤ 2n+1

Any set of diameter ≤ r intersects at most 2 of those in-tervals which define Cα,n+1, hence

N(Cα, r) ≥1

2· 2n.

Hence,

(n− 1) log 2

(n+ 1) log(2/(1− α))≤ logN(Cα, r)

log 1/r≤ (n+ 1) log 2

n log(2/(1− α)

and taking r → 0 proving the formula.

The above illustrates that box-counting dimension is nota topological concept. All Cα have different dimensionsfor different α, but are all mutually homeomorph.

4 FRACTAL GEOMETRY

Corollary 1.12. Given 0 < α < β < 1, then Cα and Cβare not bi-Lipschitz equivalent. In particular, they are notC1-diffeomorphic.

2. Methods to calculate dimension

2.1. Primer on measures. A function µ : 2X → [0,∞]is an outer measure if

• µ(∅) = 0• µ(A) ≤ µ(B) if A ⊂ B• µ(

⋃∞i=1) ≤

∑∞i=1 µ(Ai)

A set E ⊂ X is µ-measurable if for every A ⊂ X

µ(A) = µ(A ∩ E) + µ(A ∩ Ec).

µ restricted to the family M of all µ-measurable sets iscountable additive complete measure:

• M is a σ-algebra• if µ(A) = 0 then A ∈M (complete)• if A1, A2, . . . ∈ M are pairwise disjoint thenµ(⋃∞i=1Ai) =

∑∞i=1 µ(Ai) (countably additive)

Moreover

• ifA1 ⊂ A2 ⊂ . . . ∈M then µ(⋃∞i=1Ai) = limi µ(Ai)

• if A1 ⊃ A2 ⊃ . . . ∈ M and µ(A1) < ∞ thenµ(⋂∞i=1Ai) = limi µ(Ai)

Recall that

• µ is locally finite if ∀x ∈ X ∃r > 0: µ(Br(x)) <∞• µ is Borel regular if Borel sets are µ-measurable• µ is Radon if is Borel and

– µ(K) <∞ for every K ⊂ X compact– µ(V ) = sup{µ(K) : K ⊂ V,K compact} for

every V ⊂ X open– µ(A) = inf{µ(V ) : V ⊃ A, V open}.

Recall that if X is complete separable metric space and µis Borel regular locally finite then µ is Radon.

2.2. Local dimension of a measure. Given µ a mea-sure, define

dµ(x)def= lim inf

r→0

logµ(Br(x))

log r, dµ(x)

def= lim sup

r→0

logµ(Br(x))

log r

lower and upper local dimension of µ at x.if dµ(x) = dµ(x) = dµ(x) we call it local dimension of µ

at x. if almost everywhere dµ = dµ we say that µ is exactdimensional.we define the Hausdorff dimension of µ by

dimH µdef= inf{dimH Y : µ(Y ) = 1}

Lemma 2.1. For µ finite Borel

dimH µ = ess sup dµ.

We postpone the proof of this fact until we have the so-called mass distribution principle.

Lemma 2.2 ([You]). For µ exact dimensional µ-almosteverywhere dimH µ = dµ(x).

We postpone the proof of the above lemma.Borel measure µ is mass distribution on X if 0 < µ(X) <∞.µ is α-regular if there is C > 0 such that µ(Br(x)) ≤ Crαfor every x and r.

Proposition 2.3 (Mass distribution principle). For α-regular mass distribution µ on A ⊂ X for every r > 0 wehave C2αHα

r (A) ≥ µ(A) and hence dimHA ≥ α.

Proof. For any nonempty E we have µ(E) ≤ C(2|E|)α.hence A ⊂

⋃iAi implies

0 < µ(A) ≤∑i

µ(Ai) ≤ C2α∑i

|Ai|α

which implies the claim. �

Example (Middle α-Cantor). Note

dimH Cα ≤ dimB Cα = β β =log 2

log(2/(1− α)).

Let µ = µα on Cα the measure giving equal mass to anyof the 2n intervals in the construction at level n which areof length rn = ((1− α)/2)n.x ∈ Cα ⇒ Brn(x) contains some Ii1...in and at most inter-sects one more such interval, hence

µ(Brn(x)) ≤ 2 · 2−n = C · rβn,since Brn+1

⊂ Br(x) ⊂ Brn(x) if rn+1 ≤ r < rn, hence

µ(Br(x)) ≤ µ(Brn(x)) ≤ Crβn ≤ C(

2

1− α

)βrβn+1 ≤ C ′rβ .

Hence µ is β-regular and we obtain

dimH Cα ≥ β.

In the case X = Rn we have the following ‘converse’ state-ment. It requires indeed a particular covering result truefor euclidean spaces (Besicovich covering lemma).

Proposition 2.4 (Frostman’s Lemma). Let A ⊂ Rn bea Borel subset such that Hs(A) > 0. Then there is µ as-regular mass distribution (Borel probability measure sup-ported) on A.

Corollary 2.5. If dimHA = s > 0 then for every t ∈ [0, s)there is a t-regular mass distribution on A.

The above is not true for t = s. For example, takeX =

⋃iXi with dimHXi = s − 1/n. Then dimHX =

supi dimHXi = s. For any s-regular µ we have µ(Xn) = 0(by contradiction, from the mass distribution principle).Hence µ(X) ≤

∑i µ(Xi) = 0. Hence, such s-regular mass

distribution cannot exist.

The following very useful result is sometimes also referredto as ‘Mass Distribution Principle’ or ‘Frostman’s Lemma’or ‘Billingsley’s Lemma’ (see discussion [BisPer, Chapter1.4]).

Proposition 2.6. Let A ⊂ Rn be Borel set. Let µ be afinite measure. Then

• If µ(A) > 0 and dµ(x) ≥ s for µ-almost every x ∈ Athen dimHA ≥ s

• If dµ(x) ≤ s for all x ∈ A then dimHA ≤ s.

FRACTAL GEOMETRY 5

The following result implies Lemma 2.2.

Proposition 2.7 ([You]). Let µ be a mass distribution onA ⊂ Rn. If for s ≤ t for all x ∈ A we have

(2.1) s ≤ dµ(x) ≤ dµ(x) ≤ t

then s ≤ dimHA ≤ t.

Proof. Let Ur = {Br(x) : x ∈ A} and define

Mµ(A, d, r)def= inf{

∑i

µ(Ui)d : Ii ∈ Ur,

⋃i

Ui ⊃ A}

Mµ(A, d)def= lim

r→0Mµ(A, d, r)

Mµ(A)def= inf{d : Mµ(A, d) = 0}.

From hypothesis µ(A) > 0 we obtain the following.

Claim. Mµ(A) = 1.

Claim. If ∃ε0 and V such that µ(V ) > 0 and ∀x ∈ V and∀r ∈ (0, ε0)

s ≤ logµ(Br(x))

log r≤ t

or equivalently

rt ≤ µ(Br(x)) ≤ rs

then s ≤ dimH V ≤ t.

Proof. By Mµ(A) = 1, for d > 1 we have Mµ(A, d) = 0.Hence there exist ε1 st. ∀r ∈ (0, ε1) Mµ(A, d, r) < δhence ∀r ∃{Br(xi) : xi ∈ V } cover st.

∑i µ(Br(xi))

d < δby hypothesis, rt ≤ µ(Br(xi)) ≤ rs, hence∑

i

rdt ≤∑i

µ(Br(xi))d < δ

hence Hdt(V ) <∞hence dimH V ≤ dtas d > 1 was arbitrary, dimH V ≤ t.similar, for any r-cover of V∑

i

|Br(xi)|s ≥∑i

µ(Br(xi)) ≥ µ(V ) > 0

implies dimH V ≥ s. �

Given n ≥ 1 let

ANdef= {x ∈ A : s− 1

N≤

logµ(B1/N (x))

log 1/N≤ t+ 1

N∀n ≥ N}

check that AN are measurableAN ⊂ AN ′ if N ≤ N ′A =

⋃N AN since µ(A) = limN µ(AN )

By Claim for every N ≥ 1 we have

s− 1

N≤ dimHAN ≤ t+

1

N

Countable stability of dimH implies for every N ≥ 1 wehave

s− 1

N≤ dimHA ≤ t+

1

N

hence the assertion. �

Proof of Lemma 2.1. for adef= ess sup dµ let

Z = {x : dµ(x) ≤ a}.

then µ(Z) = 1. By Proposition 2.6 dimH µ ≤ dimH Z ≤ a.for ε > 0 let

Zε = {x : dµ(x) ≥ a− ε}.

definition of ess sup gives µ(Zε) > 0. By Proposition 2.6(or 2.7) considering the mass distribution µ|Zε

on Zε andby Z ⊃ Zε

dimH Z ≥ dimH Zε ≥ a− εas ε was arbitrary this proves the lemma. �

Exercise. Show that the conclusion of Proposition 2.7holds true provided there exists a sequence (εn)n, εn → 0and log εn+1/εn → 1 such that (2.1) holds true for everyx ∈ A and every εn in place of r.

there is also the following type of result.

Proposition 2.8 ([Fal1, Proposition 4.9]). Let µ be amass distribution on A ⊂ Rn. For s > 0 and C > 0we have

a) ∀x ∈ A, lim suprµ(Br(x))

rs < C⇒ 0 < µ(A) ≤ C ·Hs(A) ⇒ dimHA ≥ s

b) ∀x ∈ A, lim suprµ(Br(x))

rs > C⇒ C ·Hs(A) ≤ 2sµ(Rn) <∞ ⇒ dimHA ≤ s

2.3. Product sets. Assume X = Rn.

Proposition 2.9. For bounded sets A,B ⊂ Rn we have

dimBA+ dimBB ≤ dimBA×BdimBA×B ≤ dimBA+ dimBB

Proof. Taking grids, it follows from

N(A×B, r) = N(A, r) ·N(B, r)

Now take lim inf and lim sup, respectively. �

Proposition 2.10. For bounded sets A,B ⊂ Rn we have

dimHA+ dimHB ≤ dimHA×B ≤ dimHA+ dimBB.

Corollary 2.11. If dimHA = dimBA then

dimHA×B = dimBA×B = dimHA+ dimHB.

Proof of Proposition 2.10. Let a = dimHA, b = dimH B.

Let ε > 0. Note Ha−ε(A) > 0, Hb−ε(B) > 0.By Proposition 2.4 (Frostman’s Lemma), there are µε1 andµε2 a (a− ε)-regular and (b− ε)-regular mass distributionon A and B, respectively. Then λε = µε1 × µε2 is massdistribution on A×B.Note that λε is (a + b − 2ε)-regular. Indeed, WithBr(x, y) = Br(x)×Br(y)

λε(Br(x, y)) = µε1(Br(x)) · µε2(Br(y))

≤ C1ra−ε · C2r

b−ε = C1C2ra+b−2ε

By Proposition 2.3, proving dimHA×B ≥ a+ b− 2ε andthus

dimHA×B ≥ a+ b.

6 FRACTAL GEOMETRY

On the other hand, let b′ = dimBB. Let ε > 0. AsHa+ε(A) = 0, there is a cover A ⊂

⋃iAi with∑

i

|Ai|a+ε < ε.

We cover each Ai by N(B, |Ai|) sets Aij of diameter |Ai|.If ε was small, we have

N(B, |Ai|) · |Ai|b′+ε < 1.

Hence {Ai ×Aij} cover A×B and∑i

∑j

|Ai ×Aij |a+b′+2ε

=∑i

|Ai|a+ε ·N(B, |Ai|)|Ai|b′+ε

<∑i

|Ai|a+ε < ε

Thus Ha+b′+2ε(A×B) = 0. Hence

dimHA×B ≤ a+ b′.

�

Example. There is X ⊂ [0, 1] with dimHX × X >2 dimHX. (see [Hoc])

2.4. Energy methods, potentials. Let µ be mass dis-tribution on Rn, s ≥ 0.Define s-potential2 of µ at x ∈ X by

φs(x)def=

∫1

d(y, x)sdµ(y).

Define the s-energy of µ by

Is(µ)def=

∫φs(x) dµ(x) =

∫∫1

d(x, y)sdµ(x)dµ(y)

and the s-dimensional capacity of a set A ⊂ Rn the num-ber

caps(A)def=

(infµIs(µ)

)−1,

where the infimum is taken over all mass distributionson A (Borel probability measures supported on A). If

Is(µ) =∞ for all µ, then we let caps(A)def= 0.

Proposition 2.12 ([Fro]). For A ⊂ Rn compact

1) If Hs(A) > 0 then capt(A) > 0 for all t < s2) If caps(A) > 0 then Hs(A) =∞.

In particular,

dimHA = inf{s : caps(A) = 0}.

Proof. For 1), if Hs(A) > 0 by Proposition 2.4 (Frost-man’s Lemma) there is s-regular mass distribution µ on

A, i.e. µ(B) ≤ C|B|s. For t < s and x ∈ A, if |A| ≤ 2k

then ∫A

1

d(y, x)tdµ(y)

=

∞∑n=−k

∫2−(n+1)≤d(y,x)<2−n

1

d(y, x)tdµ(y)

≤∞∑

n=−k

µ(B2−n(x))2t(n+1)

≤ C∞∑

n=−k

2−ns2nt ≤ C ′ <∞

Hence, It(µ) ≤ C ′µ(A) <∞, implies capt(A) > 0.For 2): If caps(A) > 0, by definition there is µ on A withIs(µ) <∞ and hence given C large

A′def= {x ∈ A :

∫A

1

d(y, x)sdµ(y) <∞}

has µ(A′) > 0. Writing again sums∫A

1

d(y, x)sdµ(y)

≥∞∑

n=−k

µ({2−(n+1) ≤ d(y, x) < 2−n})2ns

=

∞∑n=−k

(µ(B2−n(x))− µ(B2−(n+1)(x))) 2ns

= C1 +

∞∑n=−k

((2ns − 2(n−1)s)µ(B2−n(x))

)= C1 + C2

∞∑n=−k

2nsµ(B2−n(x))

For every x ∈ A′, the integral, and hence the latter sum,is finite. This implies

limn→∞

µ(B2−n(x))

2−ns= 0

Hence, by Proposition 2.8 item a), for any C > 0 we haveµ(A)/C ≤ Hs(A) and hence Hs(A) =∞. �

3. Iterated function systems

Let (X, d) be a metric space. Given A ⊂ X and r > 0, let

Ardef= {x ∈ X : d(x, a) < r for some a ∈ A}.

On the space C of compact nonempty subsets of X, con-sider the Hausdorff distance

dH(A,B)def= inf{r : A ⊂ Br, B ⊂ Ar}.

This defines a metric on C, (C, dH) is compact (complete)if (X, d) is.

A map F : X → X is a contraction if there is L ∈ [0, 1) sothat for all x, y ∈ X

d(F (x), F (y)) ≤ Ld(x, y).

2For s = 1 and X = R3, this is essentially the Newton gravitational potential.

FRACTAL GEOMETRY 7

3.1. Existence of attractor of IFS. A a finite family{Fi}Ni=1 (of contractions) is a (contracting) iterated func-tion system (IFS) on (X, d).

Example. α-Cantor in X = [0, 1]

Let Φ: C→ C defined by

Φ(A)def=

N⋃i=1

Fi(A).

By the following theorem, there is a unique fixed point ofthis map, the attractor of the IFS.

Theorem 3.1 ([Hut]). There exists unique K ∈ C suchthat Φ(K) = K. Moreover, for every A ∈ C we haveΦn(A)→ K exponentially fast (in dH).

Proof. Banach fixed point theorem on (C, dH) �

One calls K in Theorem 3.1 the (global) attractor or limitset of the contracting IFS.There is also a “probabilistic” version of the above theo-rem.

Theorem 3.2 ([Hut]). let (p1, . . . , pN ) be a probabilityvector. Then there is a unique Borel probability measureµ on K such that

µ =

N∑i=1

pi(Fi)∗µ.

Moreover, if pi > 0 for all i then suppµ = K.

3.2. Coding etc. Given i = (i1i2 . . . in) ∈ {1, . . . , N}n,write

Fi1...indef= Fi1 ◦ . . . ◦ Fin

since by invariance Fi(K) ⊂ K, we have

Fi1...in(K) ⊂ Fi1...in−1,

that is, these sets are nested and by contraction,diamFi1...in(K) ≤ Ln diamK. hence, for any infinite se-quence i = (i1i2 . . .), Fi1...in(K) shrinks to a point: forany x ∈ K, let

h(i)def=⋂n≥1

Fi1...in(x).

This defines a map h : Σ+N → K, where Σ+

N = {1, . . . , N}N.

Consider on Σ+N the metric

d(i, k)def= 2−n,

where n is largest integer with i1 . . . in = k1 . . . kn. Notethat (Σ+

N , d) is compact.Given n ≥ 0, we consider finite sequences (i1 . . . in) ∈{1, . . . , N}n, where {1, . . . , N}0 stands for the empty se-quence. Let

Σ∗def=⋃n≥0

{1, . . . , N}n,

Given (i1 . . . in) ∈ Σ∗ we define

[i1 . . . in]def= {k ∈ Σ+

N : i1 = k1, . . . , in = kn}and call it the cylinder associated to (i1 . . . in). the as-sociated cylinder to the empty sequence {1, . . . , N}0 is[∅] = Σ+

N .

A subset S ⊂ Σ∗ is called a section of Σ∗ if for everyi ∈ Σ+

N there exists a unique s = (s1 . . . sn) ∈ S, wheren = n(i) such that i ∈ [s1 . . . sn], that is, i has a “uniqueprefix in S”.Given (i1 . . . in) ∈ Σ∗, we call the set

Fi1...in(K)

an nth generation cylinder set.

Remark 3.3. An important observation is that {[s] : s ∈Σ∗ is a cover of Σ∗N whose elements are either disjointor contained in each other. For any section S ⊂ Σ∗ thefamily of cylinders {[s] : s ∈ S} is a cover of Σ+

N by pair-

wise disjoint sets. On the other hand, any cover of Σ+N by

pairwise disjoint sets defines a section.Note that in general the cylinder sets (even of equal level)Fi1...in(K) are not disjoint.

Lemma 3.4. For every i, k ∈ Σ+N and every n ≥ 1

d(h(i), h(k)) < Ln diamK.

In particular, h is Holder continuous.

Proof. Fix x ∈ K. If i 6= k let n be largest withi1 . . . in = k1 . . . kn, i.e. d(i, k) = 2−n. For ` > n,

d(Fi1...i`(x), Fk1...k`(x))

= d(Fi1...in(Fin+1...i`(x)), Fi1...in(Fkn+1...k`(x)))

< Lnd(Fin+1...i`(x), Fkn+1...k`(x))

≤ Ln diamK = d(i, k)α diamK, α =|logL|log 2

As ` ≥ 1 was arbitrary (large)

d(h(i), h(k)) < Ln diamK ≤ diamK · d(i, k)α.

�

Consider the shift map σ : Σ+N → Σ+

N defined by

σ(i1i2 . . .)def= (i2i3 . . .)

and its ‘inverse’ σk : Σ+N → Σ+

N , defined by

σk(i1i2 . . .)def= (k i1i2 . . .)

Lemma 3.5. h ◦ σk = Fk ◦ h

if {Fi}Ni=1 is a contracting IFS on (X, dX) with attractorKX and {Gi}Ni=1 is a contracting IFS on (Y, dY ) with at-tractor KY , then by the above there is a unique morphismφ : KX → KY , that is a continuous onto map such thatφ ◦ Fi = Gi ◦ φ for every i = 1, . . . , N .

8 FRACTAL GEOMETRY

3.3. Similarity dimension. Assume now that X = Rnand consider an IFS {Fi}Ni=1 of contractions of Rn. ForF : Rn → Rn let

r(F )def= sup

x,y∈Rn

‖F (x)− F (y)‖‖x− y‖

Note that F is contraction iff r(F ) < 1, which is thencalled the contraction ratio of F .Given a contracting IFS {Fi}Ni=1 of linear maps with con-traction ratios ri = r(Fi), its similarity dimension sdimKis the unique solution d of the equation

N∑i=1

rdi = 1.

This equation is called Moran’s formula.

Theorem 3.6. Let K ⊂ Rn be the attractor of a contract-ing IFS {Fi}Ni=1. Then

dimBK ≤ sdimK.

Proof. Given ε > 0, let Sε ⊂ Σ∗ denote the set of finitesequences (i1 . . . in) such that

ri1 · . . . · rin <ε

|K|≤ ri1 · . . . · rin−1 .

Let

rdef= min

iri = min

ir(Fi)

and observe that

(3.1) r · ε

|K|≤ ri1 · . . . · rin .

Note that all associate cylinder sets Fi1...in(K) haveroughly the same maximal diameter (this type of coveris sometimes called Moran cover).

Claim. Sε is a section of Σ∗.

By the claim, {[s] : s ∈ Sε} is a cover of Σ+N by dis-

joint(!) sets. Hence {Fs(K) : s ∈ Sε} covers K. For everys = (i1 . . . in) ∈ Sε we have

|Fs(K)| ≤ ri1...in · |K| = ri1 · . . . · rin · |K| <ε

|K||K| = ε.

To estimate N(K, ε) we estimate #Sε.Let µ be the product measure on Σ+

n with marginal(rd1 , ldots, r

dN ), where d = sdimK is the similarity dimen-

sion of K. Note that for every s = (i1 . . . in) ∈ Sε

µ([s]) = rdi1 · . . . · rdin = (ri1 · . . . · rin)d <

(ε

|K|

)dHence

rd(

ε

|K|

)d≤ µ([s]) <

(ε

|K|

)d.

Recalling again that {[s] : s ∈ Sε} partitions Σ+n , the mea-

sure µ satisfies ∑s∈Sε

µ([s]) = 1

and thus we have #Sε ≤ (mins∈Sεµ([s]))−1. Thus

N(K, ε) ≤ |Sε| ≤ (mins∈Sε

µ([s]))−1 ≤ |K|d 1

rdε−d

thus

dimBK = limε→0

logN(K, ε)

− log ε≤ d

proving the theorem. �

3.4. Self-similar sets. A linear map F : Rn → Rn is asimilarity if there is r > 0 such that for every x, y ∈ Rn

d(F (x), F (y)) = r · d(x, y).

Definition. The attractor of a contracting IFS of simi-larities {Fi}i is a self-similar set.

Definition. An IFS {Fi}i satisfies the open set condi-tion (OSC) if there is a nonempty open set U such thatFi(U) ⊂ U and Fi(U) ∩ Fj(U) = ∅ for all i 6= j. If,

moreover also Fi(U)∩Fj(U) = ∅ for all i 6= j then we saythat the IFS {Fi}i satisfies the strong separation condition(SSC).

Example. The IFS {x2 ,x2 + 1

2} has attractor K = [0, 1]and with U = (0, 1) satisfies the OSC, but not the SSC.The IFS {x3 ,

x3 + 1

3} has also attractor K = [0, 1], but doesnot satisfy OSC (hence neither SSC).

Remark 3.7. Note that for an IFS satisfying the SSCthe coding map h is a bijection. If we consider on Σ+

N themetric

ρ(i, k)def= |Fi1...in(K)|,

where n = n(i, k) ≥ 1 is the largest integer such thatin = kn. For (Σ+

N , ρ) and h : Σ+N → K we then have the

following:If the IFS satisfies the OSC then h is Lipschitz.If the IFS satisfies the SSC then h is bi-Lipschitz.

Theorem 3.8. If K is the self-similar set of a contractingIFS of similarities {Fi}i satisfying the OSC then

dimHK = dimBK = sdimK.

Proof. Given ε > 0, let Sε be defined as in the proof ofTheorem ??. Consider also the there constructed measureµ. Recall that µ([s]) ≤ (ε/|K|)d and |Fs(K)| < ε for everys ∈ Sε, where we used the notation for s = (i1 . . . in) ∈ Sε

Fs(K) = (Fi1 ◦ . . . ◦ Fin)(K).

We will use the following

Claim. There exists M ≥ 1 such that for every ε > 0 andfor every x ∈ Rn, the ball Bε(x) intersects at most M setsof the family {Fs(K) : s ∈ Sε}.

Now let ν = h∗µ be the projection of µ to K. For everyx ∈ Rn together with the claim

ν(Bε(x)) =∑

s∈Sε : Fs(K)∩Bε(x) 6=∅

µ([s])

≤M · 1

|K|dεd.

Hence, ν is an d-regular mass distribution on K and byProposition 2.3 it follows dimHK ≥ d.

FRACTAL GEOMETRY 9

Proof of the Claim. By the OSC, for different s ∈ Sε, thesets Fs(U) are disjoint. Note that by (3.1)

|Fs(U)| < ε

|K||U |

and that (recalling that Fi are similarities) by (3.1) (recallK ⊂ Rn)

Vol(Fs(U)) ≥(r · ε

|K|

)nVol(U).

Note that if Bε(x) ∩ Fs(U) 6= ∅ then

Bε(1+|U |)(x) ⊃ Fs(U),

but as the latter sets are pairwise disjoint, there are atmost

Mdef=

⌊Vol(Bε(1+|U |)(x))

(r · ε|K| )

n Vol(U)

⌋+ 1

such sets with nonempty intersection (note that M doesnot depend on ε). �

This proves the theorem. �

Proposition 3.9 ([?]). Under the above hypothesis, thefollowing facts are equivalent: (1) the IFS satisfies theOSC, (2) Hd(K) > 0.

3.5. Nonlinear conformal IFS. A map F : Rn → Rn isconformal if there exists a function r : Rn → Rn such thatfor every x ∈ Rn we have F (x) = r(x) · I(x), where I(x)is an isometry. For an differentiable isometry we simplywrite F ′ to denote its derivative.In the following we will assume that {Fi}Ni=1 is a con-tracting IFS of C1+α conformal diffeomorphisms (for someα > 0). Note that then the contracting ratios ri associ-ated to Fi, respectively, are Holder continuous functions(with Holder constant α).

Example. Schottky groups, Julia sets

3.6. Gibbs measures for weakly Holder potential.Denote by M = M(Σ) the space of σ-invariant Borel prob-ability measures on Σ. Call µ ∈ M a Gibbs measure (inthe sense of Bowen [Bo]) for φ if there are P ∈ R andC > 1 such that for every n ≥ 1 and every i ∈ Σ andevery k ∈ [i1 . . . in]

C−1 ≤ µ([i1 . . . in])

e−Pn+Snφ(k)≤ C.

This should be read as

µ([i1 . . . in]) � 1

ZnexpSnφ(k), Zn

def= enP .

Definition. Consider Σ = Σ+N and φ : Σ → R. Given

n ≥ 1 denote

Vnφdef= max{|φ(i)− φ(k)| : k ∈ [i1 . . . in], i ∈ Σ}.

We say that φ is weakly Holder if there exists are b > 0and a ∈ (0, 1) such that for every n

Vnφ < bαn.

We say that φ has summable variation if∑n Vnφ <∞.

Theorem 3.10. For every φ weakly Holder there exists aunique (invariant probability) Gibbs measure µφ, and µφis mixing, hence ergodic.

Proof. We study the transfer operator L which acts onC(Σ): given h ∈ C(Σ) define Lh ∈ C(Σ) by

(Lh)(i)def=

∑k∈σ−1(i)

eφ(k)h(k).

Note that

(Lnh)(i) =∑

k∈σ−n(i)

eSnφ(k)h(k),

where

Snφ = φ+ φ ◦ σ + . . .+ φ ◦ σn−1

(induction on n)

Theorem 3.11 (Perron-Frobenius Theorem [?]). Thereexists λ > 0, h ∈ C(Σ), h > 0, and ν ∈M such that

Lh = λh, L∗ν = λν, ν(h) = 1.

Moreover, for every g ∈ C(Σ) we have

limn→∞

‖λ−nLng − ν(g)h‖ = 0.

Proof. The operator L acts on the (Banach) space of con-tinuous functions C(Σ). It is a positive positive operator,that is, if h ≥ 0 then Lh ≥ 0.Consider the conjugate operator L∗ : C∗(Σ) → C∗(Σ),where C∗(Σ) denotes the space of all bounded linear func-tionals on C(Σ). This space is equipped with the norm‖F‖ = sup{F (h) : h ∈ C(Σ), ‖h‖ ≤ 1}, which makes it aBanach space.As conjugate to a positive operator, L∗ is also positive, ittransforms measures into measures.The operator G defined by

G(µ)def=

L∗µ

(L∗µ)(1)

acts as G : M→M.By the Krylov-Bogolyubov Theorem, M is compact in theweak∗ topology. By the Schauder -Tychonoff Theorem3,G has a fixed point ν ∈M:

ν = G(ν) =L∗ν

(L∗ν)(1).

With

λdef= (L∗ν)(1)

we have λ > 0 and L∗ν = λν.It remains to show that there is h ∈ C(Σ), h > 0, withν(h) = 1. Study how the operator λ−1L acts. Let

Bndef= exp

∞∑k=n+1

2bαk

and consider the space H ⊂ C(Σ) defined by

Hdef={h ∈ C(Σ):

h ≥ 0, ν(h) = 1, h(k) ≤ Bnh(i) if k ∈ [i1 . . . in]}.

Claim. There is h ∈ H such that λ−1Lh = h.

3Schauder–Tychonoff Theorem: If K is a nonempty K is a compact convex subset of a locally convex space and T is a continuousmapping of K into itself such that T (K) is contained in a compact subset of K, then T has a fixed point.

10 FRACTAL GEOMETRY

Indeed, one checks λ−1LH ⊂ H. Further, one checksthat H is equicontinuous. By the Arzela-Ascoli TheoremH is compact. The claim then follows by the Schauder-Tychonoff Theorem.For details see [Bo]. �

Denote the measure in Theorem 3.11 by µφdef= hν.

Claim. µφ is σ-invariant.

Indeed,

((Lh) · f)(i) =∑

k∈σ−1(i)

eφ(k)h(k) · f(i)

=∑

k∈σ−1(i)

eφ(k)h(k) · f(σ(k))

= L(h · (f ◦ σ))(i)

We have

〈µφ, f〉 = 〈hν, f〉 = 〈ν, h · f〉 = 〈ν, λ−1Lh · f〉= λ−1〈ν,L(h · (f ◦ σ))〉= λ−1〈L∗ν, h · (f ◦ σ)〉 = 〈ν, h · (f ◦ σ)〉= 〈µφ, f ◦ σ〉

proving invariance.This also shows that µφ being invariant is equivalent with

(3.2) λ−1Lh = h.

Since φ is weakly Holder, for every i ∈ Σ, n ≥ 1, andk ∈ [i1 . . . in] we have

|Snφ(k)− Snφ(i)| <∞∑n=0

varn φ <∞.

Claim. µφ is Gibbs

Indeed, let i ∈ Σ, n ≥ 1, k ∈ [i1 . . . in].

(Ln(hχ[i1...in]))(k) =∑

j∈σ−n(k)

eSnφ(j)h(j)χ[i1...in](j)

for j = i1 . . . ink1k2 . . . = eSnφ(j)h(j)

≤ eSnφ(i)C‖h‖with j = i1 . . . ink1k2 . . ., with the analogous lower bound... h > 0 .Since

µφ([i1 . . . in]) = 〈ν, hχ[i1...in]〉 = λ−n〈ν,Ln(hχ[i1...in])〉,the Gibbs property follows, where P = log λ.

Claim. P is uniquely defined by φ.

Proof. Indeed, let µφ be a Gibbs measure for φ, that is,for every i, n ≥ 1, k ∈ [i1 . . . in]

C−1e−Pn+Snφ(k) ≤ µφ([i1 . . . in]) ≤ Ce−Pn+Snφ(k).

Since µφ is probability,

1 =∑

[i1...in]

µφ([i1 . . . in])

Hence

C−1e−Pn∑

[i1...in]

eSnφ(i) ≤∑

[i1...in]

µφ([i1 . . . in]) = 1

hence for any k

C−1e−Pn∑

i∈σ−n(k)

eSnφ(i) ≤ 1

and hence,

P = limn→∞

1

nlog

∑i∈σ−n(k)

eSnφ(i)

and this expression is indeed independent of µφ and k (butonly depends on φ). �

P = P (φ) is called pressure of φ (with respect to σ).Recall that µ ∈M is ergodic if for every Borel A ⊂ Σ withA = σ−1(A) either µ(A) = 1 or µ(A) = 0. Recall thatµ ∈M is mixing if for every Borel A,B ⊂ Σ

limn→∞

µ(A ∩ σ−n(B)) = µ(A)µ(B).

Note that mixing implies ergodic. Note that these prop-erties only need to be checked in a basis which generatesthe topology (e.g. cylinder sets).

Claim. µφ is mixing hence ergodic.

Proof. Recall that for f, g ∈ C(Σ)

(Lnf)(i) =∑

k∈σ−n(i)

eSnφ(k)f(k),

and((Lnf) · g)(i) = Ln(f · (g ◦ σn))

It suffices to show mixing for cylinder sets of the type

[im . . . in] = {j : jm = im, . . . , jn = in}and since µφ is σ-invariant and hence

µφ([im . . . in]) = µφ(σ−m([im . . . in]))

=⋃

(i1...im−1)

µφ([i1 . . . im−1im . . . in])

it suffices to show mixing for cylinder sets of the type

[i1 . . . im].

Let A = [i1 . . . ik], B = [j1 . . . j`]. Let n ≥ k, `µφ(A ∩ σ−n(B)) = µφ(χA · χσ−n(B))

= µφ(χA · (χB ◦ σn))

= ν(hχA · (χB ◦ σn))

= λ−n(L∗)nν(hχA · (χB ◦ σn))

= ν(λ−nLn(hχA · (χB ◦ σn)))

= ν(λ−nLn(hχA) · (χB))

With this

|µφ(A ∩ σ−n(B))− µφ(A)µφ(B)|= |ν(λ−nLn(hχA) · (χB))− ν(hχA) · ν(hχB)|

=∣∣∣ν(λ−nLn(hχA) · (χB)− ν(hχA) · h · χB

)∣∣∣≤ ‖λ−nLn(hχA)− ν(hχA) · h‖ · ν(B)

Since for n ≥ k we have χA ∈ Bk by Theorem 3.11 wehave

‖λ−nLn(hχA)− ν(hχA) · h‖ → 0.

which finishes the proof of the claim. �

FRACTAL GEOMETRY 11

Claim. µφ is unique.

Proof. Indeed, by contradiction, assume that there are µφand µ′φ Gibbs (for the same φ and hence with the same

P ). Then

µφ([i1 . . . in]) � 1

ZnexpSnφ(k) � µ′φ([i1 . . . in])

(� “up to some constant factor”). Hence for every A ⊂ ΣBorel µφ(A) � µ′φ(A). Hence the measures are equiv-alent. Hence by the Radon-Nikodym theorem, there ish′ ∈ L1(µφ) such that µ′φ = h′µφ. Moreover, h′ is uniquelydefined up to a µφ-null set. As both Gibbs measures µφand µ′φ are (assumed to be) invariant,

µ′φ = σ∗µ′φ = σ∗(h

′µφ) = (h′ ◦ σ−1)σ∗µφ = (h′ ◦ σ−1)µφ

However, by uniqueness of h′ we conclude h′ = h′ ◦ σ−1(up to a µφ-null set), that is, h′ is µφ-almost everywhereinvariant. since µφ is ergodic, h′ is µφ-almost everywhereconstant c. Since 1 = µ′φ(Σ) = µφ(h′) = c we conclude

µ′φ = µφ. �

This proves the theorem. �

3.7. The pressure function. Note that any Gibbs mea-sure µφ is an equilibrium state for φ, that is, an invariantprobability measure such that

h(µφ) +

∫φdµφ = sup

µ∈M

(h(µ) +

∫φdµ

),

where h(µ) is the entropy of µ.

Note that the fact that there is some universal C > 0 suchthat for every i ∈ Σ, n ≥ 1, k ∈ [i1 . . . in], we have

|Snφ(k)− Snφ(i)| < C,

we can take k = (i1 . . . in)Z periodic. Hence we have

P (φ) = limn→∞

1

nlog

∑i=σn(i)

eSnφ(i)

For φ ≡ 0, we have

P (0) = supµ∈M

h(µ) = htop(σ)

= limn→∞

1

nlog card{i = σn(i)}

For φ : Σ→ R continuous, one can also define (one of) itspressure (with respect to σ), though taking the lim sup totake into account that the limit may not exist

P (φ)def= lim sup

n→∞

1

nlog

∑(i1...in)

emaxk∈[i1...in] Snφ(k).

Exercise. Show that in this definition for continuous φ itis irrelevant if we take Snφ at periodic points or if we takethe max or if we take any other point in the cylinder.Hint: Use the modulus of continuity of φ, e.g. considerthe numbers

εndef= max

(i1...in)max

k,j∈[i1...in]|φ(k)− φ(k)|.

Lemma 3.12. For φ : Σ → R continuous and satisfyingφ < 0 the function t 7→ P (t · φ) has a unique zero.

Proof. Indeed, with

P (tφ) = lim supn

1

nlog

∑k∈σ−n(i)

eSnφ(k)

and by negativity of φ, there are c1, c2 > 0 such that−c1 < φ < −c2 and hence we have

−c1δ < P ((t+ τ)φ)− P (tφ) < −c2δ.Hence t 7→ P (tφ) is monotonically decreasing and theBowen’s equation

P (tφ) = 0

has a unique zero. �

3.8. Bowen’s formula. Given a conformal contractingIFS {Fi}Ni=1 of C1+α maps.Consider the potential

φ(i)def= log F ′i1(π(σ(i)))

Corollary 3.13. The potential φ is weakly Holder contin-uous.

Proof. We have

Vnφ = maxk∈[i1...in]

|φ(k)− φ(i)| = maxk∈[i1...in]

∣∣∣∣logF ′k1(π(σ(k)))

F ′i1(π(σ(i)))

∣∣∣∣= maxx,y∈Fi1...in (K)

∣∣log F ′i1(y)− log F ′i1(x)∣∣

≤ C1 maxx,y∈Fi1...in (K)

‖y − x‖α

≤ C1rn diamKα

Now let b = C1 diamK. �

Lemma 3.14 (Bounded distortion). There exists C > 1such that for every n ≥ 1, every (i1 . . . in), and for everyx, y ∈ Rn we have

(Fi1 ◦ . . . ◦ Fin)′(x)

(Fi1 ◦ . . . ◦ Fin)′(y)≤ C.

Proof. By the chain rule

(Fi1 ◦ . . . ◦ Fin)′(x)

= F ′i1((Fi2 ◦ . . . ◦ Fin)(x)) · . . . · F ′i2(Fin(x)) · F ′in(x)

By the uniform continuity, there exists r ∈ (0, 1) such thatfor every x ∈ K

|F ′(x)| ≤ r.Hence, for every k = 2, . . . , n

‖(Fik ◦ . . . ◦ Fin)(x)− (Fik ◦ . . . ◦ Fin)(y)‖ < rk‖x− y‖Hence, by Holder continuity of F and F ′ > 0 we have

|log(Fi1 ◦ . . . ◦ Fin)′(x)− log(Fi1 ◦ . . . ◦ Fin)′(y)|≤ C1|(Fi1 ◦ . . . ◦ Fin)′(x)− (Fi1 ◦ . . . ◦ Fin)′(y)|

≤ C1

n∑k=1

|F ′ik(Fik+1...in(x))− F ′ik(Fik+1...in(y))|

≤ C1

n∑k=1

rk‖x− y‖α ≤ C1

∞∑k=1

rk‖x− y‖α

=C1

1− r‖x− y‖α

The claimed property is proved. �

12 FRACTAL GEOMETRY

Corollary 3.15. There exists C > 1 such that for everyi ∈ Σ, n ≥ 1, k ∈ [i1 . . . in] we have

|Snφ(k)− Snφ(i)| < C.

Theorem 3.16. Let K be the attractor of a contractingconformal IFS {Fi} of C1+α maps satisfying the OSC. Theunique solution t of Bowen’s equation is the Hausdorff di-mension of K.Moreover, t = dimBK.Moreover, the Gibbs measure is equivalent to the t-dimensional Hausdorff measure and hence 0 < Ht(K) <∞.

Proof. Let d be the unique solution of Bowen’s equation.For s > d we have P (sφ) < 0 by Lemma 3.12. Hence,there exists N = N(sφ, ε) ≥ 1 such that for every n ≥ N∑

i=σn(i)

es·Snφ(i) < e−nε < 1.

Since the IFS is contracting, taking N large enough, wecan also assume that for every n ≥ N for every i

diamFi1...in(K) < ε.

By the bounded distortion, for any i ∈ Σ we have

diamFi1...in(K) � diamK · (F ′i1...in(π(i)))

= diamK · eSnφ(i)

Hence {Fi1...in(K) : (i1 . . . in)} is a ε-cover of K such that∑i=σn(i)

(diamFi1...in(K))s �∑

i=σn(i)

es·Snφ(i) <∞.

Hence Hs(K) < ∞ which implies dimHK ≤ s. As s > dwas arbitrary, dimHK ≤ d follows.Let ν = µdφ be the Gibbs measure for dφ. Recall that

π[i1 . . . in] = Fi1...in(K).

Also (ν is Gibbs! and P (dφ) = 0)

ν([i1 . . . in]) � edSnφ(k)

hence

ν([i1 . . . in]) � diamπ([i1 . . . in])d

We now proceed as before (when constructing the Morancover). �

4. Repellers and Markov partition

A contracting IFS is in some sense the “converse” to a re-peller of an expanding map. This correspondence is trans-lated using the concept of a Markov partition.Let f : Rn → Rn be a continuous map. Let K ⊂ M beforward invariant, that is, f(K) ⊂ K.

Definition. We call a forward invariant set K a repellerfor f is there exists an open neighborhood U of K suchthat

K =⋂n≥0

f−n(U)

(sometimes K satisfying this property is said to be isolatedor locally maximal). We say that f is uniformly expandingon K if there exists λ > 1 such that for every x ∈ K forevery n ≥ 1 for every v ∈ TxM we have ‖dfnx (v)‖ ≥ λn|v|.

A set which is compact, forward invariant, isolated anduniformly expanding is an expanding repeller.

Definition. A finite collection {M1, . . . ,MN} of closedsubsets of K is a Markov partition if:

• K =⋃Ni=1Mi

• Mi = intMi (relative to K)• intMi ∩ intMj = ∅ if i 6= j• f(Mi) ⊃Mj if f(intMi) ∩ intMj 6= ∅.

For any Markov partition, we associate its transition ma-trix A = (aij)

Ni,j=1

aijdef=

{1 if f(intMi) ∩ intMj 6= ∅0 otherwise

Given an expanding repeller with a Markov partitionand associated transition matrix, there exists a semi-conjugation π : Σ+

A → K between the subshift of finite

type σ : Σ+A → Σ+

A and f : K → K.

Theorem 4.1. For K ⊂M a expanding repeller of C1+α

map f there exist Markov partition(s) of arbitrarily smalldiameter.

[PU] considers a much more general case of distance ex-panding maps.

If K is the attractor of a IFS {Fi}Ni=1 of bijective con-tractions satisfying the SSC, that is, there is a neighbor-hood U of K such that F1(U), . . . , FN (U) are pairwisedisjoint, then K is the repeller for f : U → U given byf(x) = F−1i (x) if x ∈ Fi(U).On the other hand, given a C1 repeller K ⊂ M forf : M →M with Markov partition {M1, . . . ,MN}, then inparticular f is a local diffeomorphism. The local inversesdefine a contracting IFS.

4.1. Examples.

1. Cookie cutter2. logistic map f : [0, 1] → [0, 1] : x 7→ 4x(1 − x)

(though not expanding)

4.2. One-dimensional Markov maps.

4.3. Examples – Rational maps. Consider a rationalmap f : C → C of degree at least 2 on the Riemannsphere C

f(z) =p(z)

q(z).

A n-periodic point x (this means that fn(z) = z) is saidto be repelling if |(fn)′(z)| > 1.The Julia set J = J(f) ⊂ C of f is the closure of the set ofrepelling periodic points of f (see [M] for lots of details).In particular, J is compact, nonempty, and f -invariantJ = f(J) = f−1(J).Consider the particular case of maps fc : C→ C

fc(z) = z2 + c,

where c ∈ C is some parameter. Note that

f−1c (z) = ±(z − c)1/2, z 6= c

and for U ⊂ C \ {c}, f−1c (U) = V1 ∪ V2 and fc maps V1/2diffeomorphically into U .0 is a critical point for fc, that is, f ′c(0) = 0.

FRACTAL GEOMETRY 13

Lemma 4.2. If |c| > 14 (5 + 2

√6) then J(fc) is completely

disconnected and the attractor of an contracting IFS {f−1c }satisfying the SSC. For |c| � 1 we have

dimH J(fc) = dimB J(fc) ∼2 log 2

log 4|c|.

Proof. Consider the (circle) curve Cdef= {|z| = |c|} and D

the interior of the circumscribed disk D = {|z| < c}.Note that

f−1c (C) = {(ceiθ − c)1/2 : 0 ≤ θ ≤ 4π}Indeed,

((ceiθ − c)1/2)2 + c = (ceiθ − c) + c = ceiθ

This is a figure-8 with self-intersection at 0. Taking z = −cit is easy to see that

f−1c (C) ⊂ V def= {|z| = |2c|1/2}.

|c| > 2 implies f−1c (C) ⊂ D. Indeed, if |z| > |c| then

|fc(z)| = |z2 + c| ≥ |z2| − |c| ≥ |c2| − |c| > |c|The interior of each of the loops of f−1c (C) is mapped byfc bijectively onto D.Define F1, F2 : D → D being the two branches of f−1c (z),z ∈ D.By the choice of V we have

F1(D) ∪ F2(D) ⊂ V ⊂ D.Hence

F1(V ) ∪ F2(V ) ⊂ Vand F1(V )∩F2(V ) = ∅. That is, the IFS {F1, F2} satisfiesthe SSC.The IFS is contracting in V . Indeed,

|Fi(z)− Fi(w)| = |(z − c)1/2 − (w − c)1/2|

=|z − w|

|(z − c)1/2 + (w − c)1/2|and hence

1

2

1

(|c|+ |2c|1/2)1/2≤ |Fi(z)− Fi(w)|

|z − w|≤ 1

2

1

(|c| − |2c|1/2)1/2

the latter being < 1 is |c| > 14 (5 + 2

√6).

By Theorem ??, there exists a unique nonempty compactset K ⊂ V satisfying

K = F1(K) ∪ F2(K).

Because F1(V ) ∩ F2(V ) = ∅, K is totally disconnected.What remains to prove is that K = J(fc). Indeed, notethat K contains the fixed points

z1/2 =1

2±√

1

4− c.

Note thatf−kc (V ) ⊂ . . . ⊂ f−1c (V ) ⊂ V

implies

J(fc) =⋃k≥1

f−kc (z1/2) ⊂ V.

Since J = J(fc) is a nonempty and compact subset of Vsatisfying

J = f−1c (J)

or, equivalently

J = F1(J) ∪ F2(J),

we have J = K.Taking

r1 = r2 =1

2

1

(|c| − |2c|1/2)1/2

and determining, by Moran’s formula s such that

2rs = 1

or equivalently

s =2 log 2

log 4(|c|+ |2c|1/2)

the claim follows. �

Lemma 4.3. If |c| < 14 then J(fc) is a simple closed

curve.

The Julia set in the latter lemma is a quasi-circle.

Definition. A set C ⊂ C is a quasi-circle if

• C is homeomorphic to a circle.• with s = dimH C = s we have 0 < Hs(C) <∞.• There are constants a, b, r > 0 such that for any setU ⊂ C, |U | ≤ r there is a mapping ϕ : U → C suchthat

a|x− y| ≤ |U | · |ϕ(x)− ϕ(y)| ≤ b|x− y|, x, y ∈ C

(small parts U of C are roughly similar, i.e. quasi-self-similar, to C).

Theorem 4.4 ([FM]). Two quasi-circles C1, C2 are bi-Lipschitz equivalent iff dimH C1 = dimH C2.

Corollary 4.5. For |c1|, |c2| < 14 , the two Julia sets

J(fc1) and J(fc2) are bi-Lipschitz equivalent iff they havethe same Hausdorff dimension.

5. Multifractal formalism

How “big” are subsets of points (of an attractor, of an ex-panding repeller) with some special geometry or dynami-cal property?History: Eggleston, Billingsley – points with certain fre-quency of digits in their binary decimal expansionHentschel, Procaccia – mathematical physics, studying the“support” of physically relevant measures (e.g. SRB)

14 FRACTAL GEOMETRY

5.1. Spectrum of local dimension. the following canbe done for expanding Markov maps of the interval andHolder continuous potentials. we will look at a much sim-pler case

Example (affine cookie cutter). C0, C1 ⊂ [0, 1] disjoint

intervals, fi : Ci → [0, 1] bijective C1 with |fi′|def= r−1i > 1

ψ : C0 ∪ C1 → R locally constant with

eψ0 + eψ1 = 1

the (p1, p2) = (eψ0 , eψ1)-Bernoulli measure is the Gibbsmeasure of a piecewise constant normalized potential

it is very convenient to work with the normalized poten-tial φ−P (φ) instead of φ which has the very same Gibbsmeasure but has the advantage that its pressure is 0. thecondition on the potential guarantees precisely that

P (ψ) = 0 = log∑`=1,2

eψ` =1

nlog

∑(`1...`n)

eψ`1+...+ψ`n

consider the associated IFS Fi = (f |Ci)−1 and cylinders

Fi1...in(K) = (Fi1 ◦ . . . ◦ Fin)([0, 1])

consider higher order Markov rectangles

C`1...`n = C`1 ∩ f−1(C`2) ∩ . . . ∩ f−(n−1)(C`n)

= {x ∈ K : fk(x) ∈ Cik , k = 1, . . . , n}= F`n...`1(K)

given x ∈ K and n ≥ 1 denote by

Cn(x)

the nth level cylinder which contains x. note that – upto a countable (hence 0-dimensional) set of points - this iswell defined.note that we cover K by – essentially disjoint – intervals

K =⋃

(`1...`n)

C`1...`n

We recover the entropy of ν (w.r.t. to the Markov parti-tion)

h(ν) = −∑`=1,2

µ(C`) logµ(C`) = −∑`=1,2

ψ`eψ`

we also have∫ψ dν =

∑`=1,2

ψ`µ(C`) =∑`=1,2

ψ`eψ`

Note that, in particular, we have

h(ν) +

∫ψ dν = 0 = P (ψ),

that is, ν is an equilibrium state for ψ.

|C`1...`n | = r`1 · . . . · r`nby the Gibbs property, and using that the potential is nor-malized, or simply using the Bernoulli property, we have

ν(C`1...`n) = p`1 · . . . · p`n = eψ`1+...+ψ`n

We would like to know the local dimension spectrum of ν,that is, the set of values of

dν(x) = limr→0

log ν(Br(x))

log r

and the level sets and their “size”

L(β) = {x : dν(x) = β}.

Note that for ν-almost every x

dν(x) = limr

log ν(Br(x))

log r= lim

n

log ν(Cn(x))

log|Cn(r)|

=limn

1n log ν(Cn(x))

limn1n log|(fn)′(x)|−1

=hν(x)

χ(x)

=h(ν)

χ(ν)(= dimH ν),

where χ(x) denotes the Lyapunov exponent and hν(x) thelocal entropy of ν.but this doesn’t mean that there are no other values forthe local dimension. in fact, the other level sets are “large”in terms of dimension.

Lemma 5.1. For q ∈ R let β = β(q) such that∑`=1,2

(eψ`)q rβ` =∑`=1,2

pq` rβ` = 1

This function β : R → R is real analytic, decreasing, andconvex (and strict convex iff log p1/ log r1 6= log p2/ log r2).We have

limq→−∞

β(q) =∞, limq→∞

β(q) = −∞.

Proof. With F (q, β) = pq1rβ1 + pq2r

β2 − 1 = 0

β′(q) = − ∂F/∂q∂F/∂β

= −∑` pq`rβ` log p`∑

` pq`rβ` log r`

and∑`

pq`rβ`

(β′′(q) log r` + (log p` + β′(q) log r`)

2)

= 0

hence β′′ ≥ 0 and β is convex,and β′′ = 0 iff log p1/ log r1 6= log p2/ log r2. �

FRACTAL GEOMETRY 15

The function

f(α)def= inf

q(β(q) + αq)

is the Legendre transform of β.if β′ ∈ [−αmin,−αmax] then f : [−αmin,−αmax] → R, andfor α ∈ (−αmin,−αmax) there is a unique q = q(β) with

f(α) = β(q) + αq = β(q)− β′(q)q.

Case q = 0. with rβ1 +rβ2 = 0 we recover Moran’s formula,that is,

β = dimHK = β(0) = f(α(0))

df

dα=dβ

dq

dq

dα+ 1 · q + α

dq

dα= −α dq

dα+ q + α

dq

dα= q = 0

hence dimHK = maxα f(α).

Case q = 1. with p1 + p2 = 0 we have β(1) = 0 and

f(α) = β(1) + α · 1 = α

df

dα=dβ

dq

dq

dα+ 1 · q + α

dq

dα= −α dq

dα+ 1 + α

dq

dα= 1

hence f(α) ≤ α.

Given q, determine β = β(q) and consider the Borel mea-sure

νq(I)def= ν(I)q · |I|β

observedνq (x) = q dν(x) + β

Then for α = −β′(q) we have

νq(D(α)) = 1

(see Lemma 5.3 below) and for every x ∈ D(α) we have

dνq (x) = q dν(x) + β = q α+ β = −qβ′(q) + β = f(α)

and hence by the Mass Distribution Principle

dimH D(α) = f(α)

Theorem 5.2. The dimension spectrum,

α 7→ dimH D(α), D(α) = {x : dν(x) = α}of ν the Bernoulli measure (p1, p2) is the Legendre trans-form of the implicitly defined function β∑

`=1,2

pq` rβ` = 1.

this result extends to C1+ε Markov expanding repellersand a Holder continuous potential ψ where β is implicitlydefined by

P (qψ − β log |f ′|) = 0

What remains to prove is the following lemma.

Lemma 5.3. For α = −β′(q) we have νq(D(α)) = 1.

To prove this lemma we need the following classical resultfrom probability theory.

Lemma 5.4 (Borel-Cantelli). If µ is a probability measureand E1E2, . . . a sequence of measurable sets satisfying∑

n≥1

µ(En) <∞,

then the “set of all outcomes that are repeated infinitelymany times must occur with probability zero”, that is, wehave

µ( ⋂n≥1

⋃k≥n

Ek)

= 0.

Proof. (See [Fal2].) Considering the expression implicitlydefining β = β(q), consider the function Φ: R2 → R de-fined by

Φ(q, β)def=∑`=1,2

pq`rβ` .

Then Φ(q, β(q)) = 1.

Fix q and let α = −β′(q). For every ε > 0 and δ = δ(ε)sufficiently small we have

(5.1) Φ(q + δ, β(q) + (−α+ ε)δ) < 1.

Let

Endef= {x : ν(Cn(x)) ≥ |Cn(X)|α−ε}.

Note that then for n ≥ 1

νq(En) = νq({x : 1 ≤ ν(Cn(x))δ|Cn(X)|(ε−α)δ})

=

∫En

1 dνq

≤∫En

ν(Cn(x))δ|Cn(X)|(ε−α)δ dνq

≤∫ν(Cn(x))δ|Cn(X)|(ε−α)δ dνq(x)

=∑

(`1...`n)

ν([`1 . . . `n])δ|[`1 . . . `n]|(ε−α)δνq([`1 . . . `n])

=∑

(`1...`n)

(p`1 · · · p`n)δ+q(r`1 · · · r`n)(ε−α)δ+β

=( ∑`=1,2

pq+δ` r(ε−α)δ+β`

)n< γn

by (5.1) for some γ = γ(q, ε, δ), but not depending on n.Note that we have x ∈ En for only finitely many times iff

(5.2) lim infn→∞

log ν(Cn(x))

log|Cn(x)|> α− ε.

16 FRACTAL GEOMETRY

With the above estimates

νq({x : ν(Cn(x)) ≥ |Cn(X)|α−ε for some n ≥ 1})

=⋃n≥1

νq(En)

<∑n≥1

γn =γ

1− γ<∞.

Hence, by the Borel-Cantelli lemma 5.4, relation (5.2) istrue for νq-almost every x.Analogously, we show that for νq-almost every x

lim supn→∞

log ν(Cn(x))

log|Cn(x)|< α+ ε.

Since ε was arbitrary, for νq-almost every x we have

dν(q) = limε→0

log ν(Bε(x))

log ε= limn→∞

log ν(Cn(x))

|Cn(x)|= α.

the lemma is proved. �

References

[Ba] V. Baladi, Positive Transfer Operators and Decay of Cor-relations, World Scientific, 2000.

[BisPer] Ch. Bishop and Y. Peres, Fractals in Probability and Anal-ysis, 2016.

[Bo] R. Bowen, Equilibrium States and the Ergodic Theory ofAnosov Diffeomorphisms, Lecture Notes in Mathematics

470, Springer-Verlag Berlin Heidelberg, 2008.

[Mat] P. Mattila, Geometry of Sets and Measures in EuclideanSpaces.

[Fal1] K. Falconer, The Geometry of Fractal Sets

[Fal2] K. Falconer, Fractal Geometry[FM] K. Falconer and D. T. Marsh, Classification of quasi-

circles by Hausdorff dimension, Nonlinearity 2 (1989),

489–493.[Fro] O. Frostman, Potential dequilibre et capacite des ensem-

bles avec quelques applications a la theorie des fonctions,

Meddel. Lunds Univ. Math. Sen. 3 (1935), 1–118.[Hoc] M. Hochman, .

[Hut] Hutchinson, (1988).

[M] J. Milnor, Dynamics in One Complex Variable, Introduc-tory Lectures, Friedr. Vieweg and Sohn, Braunschweig,

1999.[PW] Pesin, Weiss

[PU] Przytycki and M. Urbanski, Conformal Fractals: Ergodic

Theory Methods, London Mathematical Society LectureNotes Series 371, Cambridge, 2010.

[You] L.-S. Young, Dimension, entropy and Lyapunov expo-

nents, 1982.[NN] This is not my original idea, but I do not find the original

reference.