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.
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
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)
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
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.
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
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.
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
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 )
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
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).
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
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).
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
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).
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
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).
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
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).
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
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).
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
Self-similar sets
Self-similar sets: Sierpinski triangle and carpet, snowflake andspiral
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
More self-similar sets
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
Yet more self-similar sets
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
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
}.
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
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.
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
Von Koch curves of various dimensions
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
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’.]
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
Bristol 2014: Marstrand, Mattila, Falconer, Davies
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
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.
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
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.
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
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).
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
Self-similar sets with rotations
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)∗).
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
Self-similar sets with rotations
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)∗).
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
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.
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
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.
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
Mandelbrot percolation on a square
• Squares are repeatedly divided into 3× 3 subsquares• Each square is retained independently with probability p (' 0.6).
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
Mandelbrot percolation on a square
• Squares are repeatedly divided into 3× 3 subsquares• Each square is retained independently with probability p (' 0.6).
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
Mandelbrot percolation on a square
• Squares are repeatedly divided into 3× 3 subsquares• Each square is retained independently with probability p (' 0.6).
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
Mandelbrot percolation on a square
• Squares are repeatedly divided into 3× 3 subsquares• Each square is retained independently with probability p (' 0.6).
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
Mandelbrot percolation on a square
• Squares are repeatedly divided into 3× 3 subsquares• Each square is retained independently with probability p (' 0.6).
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
Mandelbrot percolation on a square
If p > 1/M2 then Ep 6= ∅ with positive probability, conditional onwhich dimH Ep = 2 + log p/ logM almost surely.
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
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.
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
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.
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
Percolation on a self-similar set
Start with a self similar set. At each stage of the iteratedconstruction, retain each component with probability p.
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
Percolation on a self-similar set
Start with a self similar set. At each stage of the iteratedconstruction, retain each component with probability p.
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
Percolation on a self-similar set
Start with a self similar set. At each stage of the iteratedconstruction, retain each component with probability p.
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
Percolation on a self-similar set
Start with a self similar set. At each stage of the iteratedconstruction, retain each component with probability p.
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
Percolation on a self-similar set
Start with a self similar set. At each stage of the iteratedconstruction, retain each component with probability p.
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
Percolation on a self-similar set
Start with a self similar set. At each stage of the iteratedconstruction, retain each component with probability p.
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
Percolation on a self-similar set
If p > 1/m then Ep 6= ∅ with positive probability, conditional onwhich dimH Ep = s, where p
∑mi=1 r
si = 1
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
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.
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
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.
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
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.
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
Marstrand’s section theorem
Question: When is
L{a ∈ Lθ : dimH(E ∩ proj−1θ a) ≥ dimH E − 1
}> 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.
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
Marstrand’s section theorem
Question: When is
L{a ∈ Lθ : dimH(E ∩ proj−1θ a) ≥ dimH E − 1
}> 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.
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
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 .
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
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 .
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
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 .
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
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
}.
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
Percolation to analyse dimensions of deterministic sets
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.
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
Percolation to analyse dimensions of deterministic sets
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.
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
Percolation to analyse dimensions of deterministic sets
Lemma Let Ep be obtained from the self-similar set E by fractalpercolation on the basic sets {Ui}, and suppose
P{Ui survives the percolation process} ≤ c(diamUi)α for all i.
If F ⊂ E and dimH F < α then Ep ∩ F = ∅ almost surely.
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.
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
Percolation to analyse dimensions of deterministic sets
Lemma Let Ep be obtained from the self-similar set E by fractalpercolation on the basic sets {Ui}, and suppose
P{Ui survives the percolation process} ≤ c(diamUi)α for all i.
If F ⊂ E and dimH F < α then Ep ∩ F = ∅ almost surely.
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.
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
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
P{Ui survives the percolation process} ≤ c(diamUi)α for all i.
For each θ, ifP{L(projθEp) > 0
}> 0,
thenL{a ∈ Lθ : dimH(E ∩ proj−1θ a) ≥ α
}> 0.
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
Using percolation to analyse sections of self-similar sets
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
).
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
Using percolation to analyse sections of self-similar sets
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
).
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
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.
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
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.
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
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.
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation
Thank you!
Kenneth Falconer Self-similar Fractals: Projections, Sections and Percolation