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A Selective Survey of Self-Similar Sets and Suchlike Structures

Kenneth Falconer

University of St Andrews, Scotland

Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures

Kenneth Falconer, Dan Mauldin & Toby O’Neil – 1997

Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures

Bagpipes!

Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures

Carn Liath

Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures

Iterated function systems

A family S1, . . . , Sm of contractions on D ⊆ RN, i.e. |Si(x) − Si(y)| ≤ ci|x − y| x, y ∈ RN, ci < 1 is called an iterated function system (IFS). Given an IFS there exists a unique, non-empty compact set E satisfying E =

m

• i=1

Si(E), called the attractor of the IFS. [Hutchinson (1981)] If the Si are similarities E is called a self-similar set.

Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures

Some self-similar sets

Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures

Dimension

Let E ⊆ RN. For δ > 0 we let Nδ(E) be the least number of sets of diameter at most δ that can cover E. We define the box counting dimension or box dimension of E by dimB E = lim

δ→0

log Nδ(E) − log δ (assuming this exists, otherwise lower and upper limits define the lower and upper box dimensions). Roughly speaking Nδ(E) ∼ δ− dimB E as δ → 0. We will also mention Hausdorff dimension dimH E, defined via Hausdorff measures Hs. Finding the dimensions of fractal attractors has attracted much interest. Let’s recall some classes of attractor where formulae are known.

Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures

Self-similar sets

If the Si are similarities of ratio ri, i.e. |Si(x) − Si(y)| = ri|x − y|, the self-similar set E, satisfying E = m

i=1 Si(E), has

dimH E = dimB E = s where

m

• i=1

r s

i = 1,

provided the open set condition holds, that is there exists an non-empty bounded open set O such that ∪m

i=1Si(O) ⊂ O.

[Moran (1946), Hutchinson (1981)]

Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures

Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures

Statistically self-similar sets

We can randomise these constructions in a natural way: Let (S1, . . . , Sm) be a random m-tuple of contracting similarities. For each i = (i1, i2, . . . , ik) ∈ {1, 2, . . . , m}k; k = 0, 1, 2, . . . let (Si,1, . . . , Si,m) be an independent realisation of (S1, . . . , Sm). Starting with some compact set A = A∅, define a hierarchy of sets Ai = Ai1,...,ik with Ai,1 ≡Si,1(Ai), · · ·,Ai,m ≡ Si,m(Ai) (essentially) disjoint subsets of

• Ai. We get a

statistically self-similar set E =

∞

• k=0
• i1,...,ik

Ai1,...,ik.

Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures

A random von Koch curve Suppose that the (S1, . . . , Sm) have (random) scaling ratios (R1, . . . , Rm). Then with probability 1, dimH E = dimB E = s where E

• m
• i=1

Rs

i

• = 1.

[Mauldin, Graf, Williams (∼1986)] + exact dimension

Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures

Graph-directed sets

Let G be a directed graph on m vertices (multiple edges allowed). For each directed edge (i, j) ∈ G let Si,j : RN → RN be a contracting similarity of ratio ri,j. This defines m graph-directed self-similar sets E1, . . . , Em such that Ei =

• (i,j)∈G

Si,j(Ej) (i = 1, 2, . . . , m).

Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures

Under reasonable conditions (G strongly connnected, open set condition), for i = 1, . . . , m, dimH Ei = dimB Ei = s where ρ[as

i,j] = 1,

where ρ[bi,j] is the spectral radius of the matrix with entries bi,j. [Mauldin & Williams (1988)]

Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures

Self-conformal sets

Now let S1, . . . , Sm be conformal contractions on D ⊆ RN. The attractor E is called a self-conformal set. Examples include sets of continued fractions, e.g. the attractor of S1(x) = 1 + 1/x; S2(x) = 2 + 1/x is

• x = a0 +

1 a1+ 1 a2+ 1 a3 + · · · : ai ∈ {1, 2}

• ,

and certain Julia sets, e.g. repellers of the complex mapping f (z) = z2 + c for suitable c ∈ C.

Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures

Dimension calculations go back to Bowen’s formula which uses the thermodynamic formalism to express the dimension as a zero of a pressure functional P. For a function f on a compact domain D and a H¨

• lder function ψ : D → R, the pressure is given by

P(ψ) = 1 k log lim

k→∞

• x∈Fixf k

exp

• ψ(x) + ψ(fx) + · · · + ψ(f k−1x)
• .

Let f be defined on each Si(D) by the inverses of the Si (suitable separation conditions) and let ψ(x) = −s log |f ′(x)|. Then dimH E = dimB E = s where P(−s log |f ′|) = 0, with 0 < Hs(E) < ∞. [Bowen (1979)] This can be interpreted as a limiting case of the Moran-Hutchinson formula: For some fixed x ∈ D let Φs

k ≡

• i1...ik

|(Si1 ◦ · · · ◦ Sik)′(x)|s, Note that |(Si1 ◦ · · · ◦ Sik)′(x)| is essentially the diameter of Si1 ◦ · · · ◦ Sik(D). Using bounded distortion and submultiplicativity, lim

k→∞(Φs k)1/k ≡ Φs

exists and dimH E = dimB E = s where Φs = 1 .

Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures

Infinite self-conformal systems

Now let S1, S2, . . . be a sequence of conformal contractions on some compact subset D of RN. The limit set E =

∞

• k=0
• i1,...,ik

Si1 ◦ · · · ◦ Sik(A), need not be compact, and there may be no sets or several sets satisfying F = ∞

i=1 Si(F).

Apollonian gasket - limit set of an infinite conf. IFS, dim ∼ 1.3057

One can extend the definition of ‘pressure’ P(s) to infinite systems. Then dimH E = inf{s : P(s) < 0}. In general, dimH E = dimP E. However, if P(s) = 0 for some s then dimH E = dimB E = dimP E = s with 0 < Hs(E), Ps(E) < ∞. [Mauldin, Urbanski ∼1998, ...]

Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures

Self-affine sets

Now let the Si be affine contractions, i.e. Si(x, y) = Ti(x, y) + (ci, di) where Ti is a linear mapping and (ci, di) is a translation. Some self-affine sets - J Miao

Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures

Some more self-affine sets

Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures

A problem ...

If λ > 0 then dimH E ≥ 1

Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures

A problem ...

If λ = 0 then dimH E = log 2

log 3 < 1

Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures

Dimensions of self-affine carpets

A carpet on R2 is the attractor of an IFS of the form Si(x, y) = (aix + ci, biy + di) i = 1, . . . , m (i.e. contractions that map a square to rectangles).

Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures

A generic formula for the dimension of carpets E defined by Si(x, y) = (aix + ci, biy + di) i = 1, . . . , m (with |ai|, |bi| < 1

2) is given by

dimH E = dimB E = max

• s1, s2 :

m

• i=1

as1

i

bi ai max{s1−1,0} = 1,

m

• i=1

bs2

i

ai bi max{s2−1,0} = 1

• ,

valid for Lebesgue a.a. (c1, d1, . . . , cm, dm) ∈ R2m. [F, Solomyak] Note that the formula depends on whether the contractions in the x-direction or in the y-direction dominate. [In fact this generic formula also holds where Si(x, y) = (aix + eiy + ci, biy + di) i = 1, . . . , m ] [F, Miao]

Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures

Exceptional set of translations

When does the generic formula give the right answer? What can we say about the set of translations (c1, d1, . . . , cm, dm) for which the dimension of the attractor is smaller than the affinity dimension? For t ≤ s = the almost sure value of the dimension, write E(t) = {(c1, d1, . . . , cm, dm) ∈ R2m : dimH Eω < t}. Then: (a) dimH E(t) ≤ 2m − c(s − t) where c > 0 is a constant (b) dimF E(t) ≤ 2t, where dimF denotes Fourier dimension, i.e. dimF A = sup{t : ∃ µ on A s.t. ˆ µ(x) = O(|x|−t/2) as |x| → ∞} [F & Miao 2008]

Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures

Carpets where dim E is not the generic value

Bedford McMullen self-affine carpets

Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures

dimH E = 1 log p log

• p
• j=1

Nlog p/ log q

j

• dimB E = log N

log q +

• 1 − log p

log q

• dimB
• projE
• In general, dimH E = dimB E and the dimensions depend on the

positions of the rectangles selected to give the IFS mappings.

Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures

Gatzouras-Lalley carpet Bara´ nski carpet

Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures

The Bedford-McMullen formula (and others) suggest that the dimension of the projections onto the axes has a significant role in the box dimension of the attractor. This was formalised for a very general class of carpets [Feng, Wang 2005]. dimB E = max

• s1, s2 :

m

• i=1

as1

i

bi ai max{s1−dimB(proj1E),0} = 1,

m

• i=1

bs2

i

ai bi max{s2−dimB(proj2E),0} = 1

• Note that with this index modification using the dimension of

projections, provides a sure rather than an almost sure result.

Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures

Recently box-like sets were considered by Jonathan Fraser - here the affine mappings taking the square onto the rectangles may include rotations through 90◦, 180◦, 270◦ and reflections. Provided that at least one Si involves a 90◦ or 270◦ rotation, let p = dimB projE where proj is projection onto a principal axis. Then there is a formula for dimB E which depends on p along the lines of that for Feng-Wang carpets, but more complicated since the linear parts of the Si no longer commute.

Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures

Symmetries and enumeration of self-similar sets

Let D be the unit square, with 3 subsquares D1, D2, D3 of side 1

2.

Let Si : D → Di be similarity mappings of ratio 1

2.

Let E be the attractor of the IFS {S1, S2, S3} so that E = 3

i=1 Si(E).

There are 8 possible choices for each Si (rotations through 0◦, 90◦, 180◦

• r 270◦, and reflections about 4 lines), giving 83 = 512 different IFSs.

Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures

Here are two possible attractors:

• How many different attractors can we get? - different IFSs need not

give different attractors

• What are their symmetries?

Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures

Lemma

The IFSs {S1, S2, S3} and {S′

1, S′ 2, S′ 3} both give rise to the same

attractor E if and only if S′

i ∈ Si symE

for all i = 1, 2, 3, where symE is the group of symmetry transformations of E. In particular, exactly |symE|3 of these IFSs have a given attractor E.

Proof.

E =

3

• i=1

Si(E) =

3

• i=1

S′

i (E)

⇔ Si(E) = S′

i (E)

for all i ⇔ E = S−1

i

S′

i (E)

for all i ⇔ S−1

i

S′

i ∈ symE

for all i ⇔ S′

i ∈ Si symE

for all i.

Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures

Symmetries and groups

To use the above lemma we need to know what symE is. What we really want is a characterisation of the IFSs {Si, S2, S3} whose attractor has a given symmetry group H. Write G0 = D4 for the group of symmetries of the square and G = {i, reflection in ր} for the symmetries of the ‘L’ shaped figure D1, D2, D3. A symmetry g ∈ G induces a permutation g ∗ of (D1, D2, D3), that is of (1, 2, 3). E.g reflection in ր gives

• 1

2 3 1 3 2

• Kenneth Falconer

A Selective Survey of Self-Similar Sets and Suchlike Structures

Lemma

For an IFS {S1, S2, S3} we have that g ∈ symE if and only if S−1

g ∗(i) g Si ∈ symE for all i = 1, 2, 3.

• Proof. g ∈ symE

⇔ g Si(E) = Sg ∗(i)(E) for all i ⇔ S−1

g ∗(i) g Si(E) = E

for all i ⇔ S−1

g ∗(i) g Si ∈ symE

for all i.

Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures

This leads us to define, for H a subgroup of G and an IFS {S1, S2, S3} ≡ {Si}: H′({Si}) = {g ∈ G : S−1

g ∗(i) g Si ∈ H for all i}.

Then H′ is a subgroup of G.

Theorem

Let {S1, S2, S3} ≡ {Si} be an IFS. Then symE = H where H is the maximal subgroup of G such that H ≤ H′({Si}). Thus, given a possible symmetry group H, we can look for IFSs {S1, S2, S3} ≡ {Si} such that H is the maximal subgroup with H ≤ H′({Si}). This reduces the problem of enumerating the different attractors with given symmetries to a problem in group theory that is computationally tractable. Of course this type of analysis generalises to many other families of IFSs.

Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures

Examples

There are just 8 attractors with mirror symmetry about the diagonal line, each of these is the attractor of 8 IFSs; these account for 8 × 8 = 64 IFSs (figs from Jurgens, Pietgen & Saupe)

Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures

The remaining 448 attractors have no symmetry, and each of these arise from just 1 IFS.

Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures

Constructions with 6 squares G0 = D4 (order 8), G = D2 (order 4). There are 86 = 262144 IFSs. 4 attractors have order 4 symmetry; 32 have horizontal symmetry; 32 have vertical symmetry; 32 have 180◦ rotational symmetry; 239616 have no symmetry; giving 239716 different attractors.

Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures

Hexagonal constructions G0 = D6 (order 12), G = D3 (order 6). 123 = 1728 IFSs. 2 attractors have order 6 symmetry; 0 have only rotational symmetry; 6 have reflectional symmetry about each of the 3 axes; 32 have 180◦ rotational symmetry; 1152 have no symmetry; giving 1172 different attractors.

Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures

Cube-based constructions. G0 is the group of symmetries of the cube (order 48), G = S3 (order 6). There are 484 = 5308416 IFSs. 8 attractors have order 6 symmetry; 32 have rotational symmetry about the leading diagonal; 360 have reflectional symmetry in each of the 3 diagonal planes; 5278176 have no symmetry; giving 5279296 different attractors.

Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures

Thank you!

Kenneth Falconer A Selective Survey of Self-Similar Sets and Suchlike Structures