Self-similar Fractals: Projections, Sections and Percolationkenneth/SectionsPercolation.pdf · Summary Self-similar sets Hausdor dimension Projections Fractal percolation Sections

Self-similar Fractals: Projections, Sections andPercolation

Kenneth Falconer

University of St Andrews, Scotland, UK

Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation

Summary

• Self-similar sets

• Hausdorff dimension

• Projections

• Fractal percolation

• Sections or slices

• Projections −→ percolation −→ sections

For this talk we will generally work in R2.


Self-similar sets

Example: Middle-third Cantor set (in R)

Made up of 2 copies of itself scaled by a factor 13 .

In particular, if f1, f2 : R→ R are the similarities

f1(x) = 13x ; f1(x) = 1

3x + 23 ,

thenE = f1(E ) ∪ f2(E ). (1)


Self-similar sets

A mapping f : R2 → R2 with

|f (x)− f (y)| = r |x − y | (x , y ∈ R2)

is called a similarity with ratio r . If 0 < r < 1 then f iscontracting.

A family of contracting similarities f1, . . . , fm : R2 → R2 is a(special case of) an iterated function system or IFS.

Theorem (Self-similar sets) Given an IFS of contracting similaritiesthere exists a unique non-empty compact set E ⊂ R2 such that

E =m⋃i=1

fi (E )

called a self-similar set.


Self-similar sets

A mapping f : R2 → R2 with

|f (x)− f (y)| = r |x − y | (x , y ∈ R2)

is called a similarity with ratio r . If 0 < r < 1 then f iscontracting.

A family of contracting similarities f1, . . . , fm : R2 → R2 is a(special case of) an iterated function system or IFS.

Theorem (Self-similar sets) Given an IFS of contracting similaritiesthere exists a unique non-empty compact set E ⊂ R2 such that

E =m⋃i=1

fi (E )

called a self-similar set.


Self-similar sets

Example: (Right-angled) Sierpinski triangle

f1(x , y) = (12x ,12y); f2(x , y) = (12x + 1

2 ,12y); f3(x , y) = (12x ,

12y + 1

2).

E = f1(E ) ∪ f2(E ) ∪ f3(E )


Iterative construction of a self-similar set

Start with a region D and take its images under iterations of thefi . Then E =

⋂∞k=0

⋃1≤i1,...,ik≤m fi1 ◦ fi2 ◦ · · · ◦ fik (D).




⋂∞k=0

⋃1≤i1,...,ik≤m fi1 ◦ fi2 ◦ · · · ◦ fik (D).




⋂∞k=0

⋃1≤i1,...,ik≤m fi1 ◦ fi2 ◦ · · · ◦ fik (D).




⋂∞k=0

⋃1≤i1,...,ik≤m fi1 ◦ fi2 ◦ · · · ◦ fik (D).




⋂∞k=0

⋃1≤i1,...,ik≤m fi1 ◦ fi2 ◦ · · · ◦ fik (D).




⋂∞k=0

⋃1≤i1,...,ik≤m fi1 ◦ fi2 ◦ · · · ◦ fik (D).


Self-similar sets

Self-similar sets: Sierpinski triangle and carpet, snowflake andspiral


More self-similar sets


Yet more self-similar sets


Hausdorff dimension

The Hausdorff dimension of a set E ⊂ R2 is

dimH E = inf{s : for all ε > 0 there is a countable cover

{Ui} of E such that∑i

(diamUi )s < ε

}.


Dimension of self-similar sets

Let f1, . . . , fm : R2 → R2 be contracting similarities and let E bethe associated self-similar set satisfying

E =m⋃i=1

fi (E ). (∗)

Then provided, the union in (∗) is ‘nearly disjoint’ (i.e.strong-separation condition or open set condition holds),

dimH E = s wherem∑i=1

r si = 1,

where ri is the similarity ratio of fi .

E.g. Hausdorff dimension of the Sierpinski triangle is given by3(1/2)s = 1 or s = log 3/ log 2.


Von Koch curves of various dimensions


Marstrand’s projection theorems

Theorem (Marstrand 1954) Let E ⊂ R2 be a Borel set. Then, foralmost all θ ∈ [0, π),

(i) dimH projθE = min{dimH E , 1},(ii) L(projθE ) > 0 if dimH E > 1.

[projθ denotes orthogonal projection onto the line Lθ, dimH isHausdorff dimension, L is Lebsegue measure or ‘length’.]


Bristol 2014: Marstrand, Mattila, Falconer, Davies


Exceptional directions

Marstrand’s theorem tells nothing about which particular directionsmay have projections with dimension or measure smaller thannormal, i.e. when dimH projθE < min{dimH E , 1}, or dimH E > 1and L(projθE ) = 0.

The set shown has dimension log 4/ log(5/2) = 1.51, but withsome projections of dimension < 1.


Projections of specific self-similar sets

How dimH projθE varies with θ has been investigated in certainspecific cases. For example:

Let E be the 1-dimensionalSierpınski triangle, so dimH E = 1.

For projections in direction θ:(a) if θ = p/q is rational,

and p + q 6≡ 0 (mod 3)dimH projθE < 1;

and p + q ≡ 0 (mod 3)projθE contains an interval,

(b) if θ is irrational,dimH projθE = 1 but L(projθE ) = 0.

(Kenyon 1997, Hochman 2014)

Similar investigations have been done for certain other ‘regular’sets.


Self-similar sets with rotations

Write the similarities on R2 asfi (x) = riOi (x) + ti

where 0 < ri < 1 is the scalefactor, Oi is a rotation and tiis a translation. We say thatthe family {f1, . . . , fm} has denserotations if at least one of theOi is a rotation by an irrationalmultiple of π, equivalently ifgroup{O1, . . . ,Om} is dense inSO(2,R).



Theorem (Peres & Shmerkin 2009, Hochman & Shmerkin 2012)Let E ⊂ R2 be a self-similar set defined by a family {f1, . . . , fm} ofsimilarities with dense rotations. Then

dimH projθE = min{dimH E , 1} for all θ.

Proof uses CP chains. Depends on the ergodic behaviour of the‘scenery flow’ of pairs (x , µ) where x ∈ E and µ is a normalisedmeasure, under the transformation (x , µ) 7→ (f −1i (x), (f −1i µ)∗).



Theorem (Peres & Shmerkin 2009, Hochman & Shmerkin 2012)Let E ⊂ R2 be a self-similar set defined by a family {f1, . . . , fm} ofsimilarities with dense rotations. Then

dimH projθE = min{dimH E , 1} for all θ.

Proof uses CP chains. Depends on the ergodic behaviour of the‘scenery flow’ of pairs (x , µ) where x ∈ E and µ is a normalisedmeasure, under the transformation (x , µ) 7→ (f −1i (x), (f −1i µ)∗).


Measure of projections

These methods are unable to show that for a self-similar set Ewith dimH E > 1 all projections have positive measure. However,this is very nearly so in the plane.

Theorem (Shmerkin & Solomyak 2014) Let E ⊂ R2 be theself-similar attractor of an IFS with dense rotations with1 < dimH E < 2. Then L(projθE ) > 0 for all θ except for a set ofθ of Hausdorff dimension 0.

The proof involves a careful analysis of how the Fourier transformof the projections of a natural measure on E varies with θ.


Measure of projections

These methods are unable to show that for a self-similar set Ewith dimH E > 1 all projections have positive measure. However,this is very nearly so in the plane.

Theorem (Shmerkin & Solomyak 2014) Let E ⊂ R2 be theself-similar attractor of an IFS with dense rotations with1 < dimH E < 2. Then L(projθE ) > 0 for all θ except for a set ofθ of Hausdorff dimension 0.

The proof involves a careful analysis of how the Fourier transformof the projections of a natural measure on E varies with θ.


Mandelbrot percolation on a square

• Squares are repeatedly divided into 3× 3 subsquares• Each square is retained independently with probability p (' 0.6).















If p > 1/M2 then Ep 6= ∅ with positive probability, conditional onwhich dimH Ep = 2 + log p/ logM almost surely.


Projections of Mandelbrot percolation

Theorem (Rams & Simon, 2012) Let Ep be the Mandelbrotpercolation set obtained by dividing sqaures into M ×Msubsquares, each square being retained with probability p > 1/M2.Conditional on Ep 6= ∅:

(i) dimH projθEp = min{dimH Ep, 1},

(ii) if p > 1/M then dimH Ep > 1, and, for all θ ∈ [0, π),projθEp contains an interval and in particular L(projθEp) > 0.

Proof depends on a geometrical analysis of how lines intersect thegrid squares.


Projections of Mandelbrot percolation

Theorem (Rams & Simon, 2012) Let Ep be the Mandelbrotpercolation set obtained by dividing sqaures into M ×Msubsquares, each square being retained with probability p > 1/M2.Conditional on Ep 6= ∅:

(i) dimH projθEp = min{dimH Ep, 1},(ii) if p > 1/M then dimH Ep > 1, and, for all θ ∈ [0, π),

projθEp contains an interval and in particular L(projθEp) > 0.

Proof depends on a geometrical analysis of how lines intersect thegrid squares.


Percolation on a self-similar set

Start with a self similar set. At each stage of the iteratedconstruction, retain each component with probability p.


















If p > 1/m then Ep 6= ∅ with positive probability, conditional onwhich dimH Ep = s, where p

∑mi=1 r

si = 1


Projection of percolation on a self-similar set with rotations

Let E ⊂ R2 be a self-similar set defined by a family {f1, . . . , fm} ofsimilarities with dense rotations. Perform the percolation processon E with respect to its hierarchical construction, with each similarcomponent retained independently with probability p, to obtain apercolation set Ep ⊂ E .

Theorem (Jin & F, 2014) Let E have dense rotations (+OSC) andlet p > 1/m. Then Ep 6= ∅ with positive probability, conditional onwhich dimH projθEp = min{dimH Ep, 1} for all θ.

The is a special case of a result on projections (and other smoothimages) of random cascade measures on self-similar sets. Proofuses CP trees on a space of random measures.


Projection of percolation on a self-similar set with rotations

Let E ⊂ R2 be a self-similar set defined by a family {f1, . . . , fm} ofsimilarities with dense rotations. Perform the percolation processon E with respect to its hierarchical construction, with each similarcomponent retained independently with probability p, to obtain apercolation set Ep ⊂ E .

Theorem (Jin & F, 2014) Let E have dense rotations (+OSC) andlet p > 1/m. Then Ep 6= ∅ with positive probability, conditional onwhich dimH projθEp = min{dimH Ep, 1} for all θ.

The is a special case of a result on projections (and other smoothimages) of random cascade measures on self-similar sets. Proofuses CP trees on a space of random measures.


Marstrand’s section theorem

Theorem (Marstrand 1954) Let E ⊂ R2 be a Borel set withdimH E > 1. Then,

(i) for all θ ∈ [0, π)

dimH(E ∩ proj−1θ a) ≤ dimH E − 1 for L-almost all a,

(ii) for L-almost all θ ∈ [0, π)

L{a ∈ Lθ : dimH(E ∩ proj−1θ a) ≥ dimH E − 1

}> 0.



Question: When is


}> 0

for ‘all θ’ rather than ‘almost all θ’ ?

The graph of a function can have dimension as large as 2.However, E ∩ proj−10 a is a single point for all a ∈ L0, sodimH(E ∩ proj−10 a) = 0.



Question: When is


}> 0

for ‘all θ’ rather than ‘almost all θ’ ?

The graph of a function can have dimension as large as 2.However, E ∩ proj−10 a is a single point for all a ∈ L0, sodimH(E ∩ proj−10 a) = 0.


Sections of self-similar sets

Theorem (F & Jin 2014) Let E ⊂ R2 be a self-similar set withdense rotations with 1 < dimH E ≤ 2. Then, for all ε > 0:

(i) L{a ∈ Lθ : dimH(E ∩ proj−1θ a) > dimH E − 1− ε

}> 0

for all θ except for a set of θ of Hausdorff dimension 0.

(ii) If, in addition, E is connected or projθE is an intervalfor all θ, then

dimH

{a ∈ Lθ : dimB(E ∩ proj−1θ a) > dimH E − 1− ε

}= 1

for all θ.

To obtain such results on sections of a self-similar set E we use theresults on projections of percolation subsets of E .





}> 0



dimH


}= 1

for all θ.






}> 0



dimH


}= 1

for all θ.



Percolation to analyse dimensions of deterministic sets

To find the Hausdorff dimension of subsets of the self-similar set E (OSCassumed) it is enough to take covers by ‘basic sets’ of the iterativeconstruction of E , that is sets of the form fi1 ◦ · · · ◦ fik (D).Thus, for F ⊂ E :

dimH F = inf{s : for all ε > 0 there are basic sets {Ui}

with F ⊂⋃i

Ui and∑i

(diamUi )s < ε

}.



We can use (random) percolation sets to test the dimension ofdeterministic (i.e. non-random) subsets of self-similar sets.

Lemma Let E be a self-similar set (with OSC, say) constructediteratively with basic sets {Ui}. Let Ep be obtained by fractalpercolation on the basic sets {Ui}, and suppose for some α > 0

P{Ui survives the percolation process} ≤ c(diamUi)α for all i.

If F ⊂ E and dimH F < α then Ep ∩ F = ∅ almost surely.

– In particular, if F ⊂ E and Ep ∩ F 6= ∅ with positive probability,then dimH F ≥ α.



We can use (random) percolation sets to test the dimension ofdeterministic (i.e. non-random) subsets of self-similar sets.

Lemma Let E be a self-similar set (with OSC, say) constructediteratively with basic sets {Ui}. Let Ep be obtained by fractalpercolation on the basic sets {Ui}, and suppose for some α > 0



– In particular, if F ⊂ E and Ep ∩ F 6= ∅ with positive probability,then dimH F ≥ α.



Lemma Let Ep be obtained from the self-similar set E by fractalpercolation on the basic sets {Ui}, and suppose



Proof Given ε > 0 let I be a family of indices such that

F ⊂⋃i∈I

Ui and∑i∈I

(diamUi)α < ε.

Then

E(#{i ∈ I : Ep∩Ui 6= ∅}

)≤∑i∈I

P{Ui survives} ≤ c∑i∈I

(diamUi)α < cε,

soP(Ep ∩ F 6= ∅

)≤ P

(Ep ∩

⋃i∈I

Ui 6= ∅)< cε.

This is true for all ε > 0, so P(Ep ∩ F 6= ∅

)= 0.



Lemma Let Ep be obtained from the self-similar set E by fractalpercolation on the basic sets {Ui}, and suppose



Proof Given ε > 0 let I be a family of indices such that

F ⊂⋃i∈I

Ui and∑i∈I

(diamUi)α < ε.

Then

E(#{i ∈ I : Ep∩Ui 6= ∅}

)≤∑i∈I

P{Ui survives} ≤ c∑i∈I

(diamUi)α < cε,

soP(Ep ∩ F 6= ∅

)≤ P

(Ep ∩

⋃i∈I

Ui 6= ∅)< cε.

This is true for all ε > 0, so P(Ep ∩ F 6= ∅

)= 0.


Using percolation to analyse sections of self-similar sets

Proposition Let E be a self-similar set (with OSC, say) constructediteratively using a hierarchy of basic sets {Ui}. Let Ep be therandom set obtained by some fractal percolation process on the{Ui} and suppose that for some α > 0


For each θ, ifP{L(projθEp) > 0

}> 0,

thenL{a ∈ Lθ : dimH(E ∩ proj−1θ a) ≥ α

}> 0.



Proof LetS =

{a ∈ Lθ : dimH(E ∩ proj−1θ a) < α

}.

For each a ∈ S , taking F = E ∩ proj−1θ a in the lemma,

Ep ∩ proj−1θ a = Ep ∩ E ∩ proj−1θ a = ∅

almost surely. In other words, for each a ∈ S , a 6∈ projθEp withprobability 1.

By Fubini’s theorem, with probability 1, a 6∈ projθEp for L-almostall a ∈ S .Hence, with positive probability,

0 < L(projθEp) = L((projθEp) \ S

)≤ L

((projθE ) \ S

).



Proof LetS =

{a ∈ Lθ : dimH(E ∩ proj−1θ a) < α

}.

For each a ∈ S , taking F = E ∩ proj−1θ a in the lemma,

Ep ∩ proj−1θ a = Ep ∩ E ∩ proj−1θ a = ∅

almost surely. In other words, for each a ∈ S , a 6∈ projθEp withprobability 1.By Fubini’s theorem, with probability 1, a 6∈ projθEp for L-almostall a ∈ S .Hence, with positive probability,

0 < L(projθEp) = L((projθEp) \ S

)≤ L

((projθE ) \ S

).





}> 0



dimH


}= 1

for all θ.


Sections of Mandelbrot percolation

Theorem (F & Jin 2014) Let Ep be the Mandelbrot percolation setobtained by dividing sqaures into M ×M subsquares, each squarebeing retained with probability p > 1/M2. Then, for all ε > 0,conditional on Ep 6= ∅,

L{a ∈ Lθ : dimH(Ep ∩ proj−1θ a) > dimH Ep − 1− ε

}> 0

for all θ.

Proof. Similar idea, using that the intersection of two independentpercolation sets Ep ∩ Eq has the same distribution as the singlepercolation set Epq.


Sections of Mandelbrot percolation

Theorem (F & Jin 2014) Let Ep be the Mandelbrot percolation setobtained by dividing sqaures into M ×M subsquares, each squarebeing retained with probability p > 1/M2. Then, for all ε > 0,conditional on Ep 6= ∅,

L{a ∈ Lθ : dimH(Ep ∩ proj−1θ a) > dimH Ep − 1− ε

}> 0

for all θ.

Proof. Similar idea, using that the intersection of two independentpercolation sets Ep ∩ Eq has the same distribution as the singlepercolation set Epq.


Thank you!


Documents

Self-similar Fractals: Projections, Sections and Percolationkenneth/SectionsPercolation.pdf · Summary Self-similar sets Hausdor dimension Projections Fractal percolation Sections