Projections of Fractals - Old and New Kenneth Falconer University - - PowerPoint PPT Presentation

Projections of Fractals - Old and New

Kenneth Falconer

University of St Andrews, Scotland, UK

Kenneth Falconer Projections of Fractals - Old and New

Marstrand’s projection theorems

Theorem (Marstrand 1954) Let E ⊂ R2 be a Borel set. For all θ ∈ [0, π) (i) dimHprojθE ≤ min{dimHE, 1}. For almost all θ ∈ [0, π), (ii) dimHprojθE = min{dimHE, 1}, (iii) L(projθE) > 0 if dimHE > 1. [projθ denotes orthogonal projection onto the line Lθ, dimH is Hausdorff dimension, L is Lebsegue measure or length on Lθ.]

Kenneth Falconer Projections of Fractals - Old and New

Energy characterisation of Hausdorff dimension

That dimHprojθE ≤ min{dimHE, 1} for all θ follows since projection is a Lipschitz map which cannot increase dimension. Marstrand’s lower bound proof was geometric and intricate. Kaufman’s (1968) potential theoretic proof has become the standard approach for such problems. This depends on the following energy characterisation of Hausdorff

• dimension. Let M(E) be the set of probability measures on E

dimH = sup

• s :

1 C (s)(E) ≡ inf

µ∈M(E)

dµ(x)dµ(y) |x − y|s < ∞

• = sup
• s : C (s)(E) > 0
• .

Kenneth Falconer Projections of Fractals - Old and New

Box-counting dimensions of projections

The box-counting dimension of a non-empty and compact E ⊂ R2 is dimBE = lim

r→0

log Nr(E) − log r where Nr(E) is the least number of sets of diameter r covering E. [Taking lower/upper limits gives the lower/upper box dimensions.] Is there a Marstrand-type theorem for box-dimensions of projections? For E ⊂ R2 dimBE 1 + 1

2dimBE ≤ dimBprojθE ≤ min{dimBE, 1}

(almost all θ ∈ [0, π)) and examples show that these bounds are best possible.

Kenneth Falconer Projections of Fractals - Old and New

Box-counting dimensions of projections

The box-counting dimension of a non-empty and compact E ⊂ R2 is dimBE = lim

r→0

log Nr(E) − log r where Nr(E) is the least number of sets of diameter r covering E. [Taking lower/upper limits gives the lower/upper box dimensions.] Is there a Marstrand-type theorem for box-dimensions of projections? For E ⊂ R2 dimBE 1 + 1

2dimBE ≤ dimBprojθE ≤ min{dimBE, 1}

(almost all θ ∈ [0, π)) and examples show that these bounds are best possible. Even so, dimBprojθE and dimBprojθE must be constant for almost all θ; for a messy argument and indirect value see (F & Howroyd, 1996, 2001). Using capacities things become much simpler.

Kenneth Falconer Projections of Fractals - Old and New

Box-counting dimensions of projections

Define potential kernels φs

r(x) by

φs

r(x) = min

• 1,

r |x|

• s

(x ∈ R2 or Rn) The capacity C s

r (E) of a compact E ⊂ Rn w.r.t. φs r is

1 C s

r (E) =

inf

µ∈M(E)

φs

r(x − y)dµ(x)dµ(y),

where M(E) are the probability measures on E. The infimum is attained by some equilibrium measure µ0 ∈ M(E), and moreover

• φs

r(x − y)dµ0(y) ≥

1 C s

r (E)

(x ∈ E), with equality for µ0-almost all x ∈ E.

Kenneth Falconer Projections of Fractals - Old and New

Box-counting dimensions of projections

Then for E ⊂ Rn c1C s

r (E) ≤ Nr(E) ≤

c2 log(1/r) C s

r (E)

if s = n c2 C s

r (E)

if s > n (1), (c1, c2 depend on n, s, diamE). In particular for E ⊂ Rn lim

r→0

log C n

r (E)

− log r = lim

r→0

log Nr(E) − log r = dimBE (we can replace dimB and lim by either dimB and lim, or by dimB and lim). Note: Inequalities (1) fail if 0 < s < n.

Kenneth Falconer Projections of Fractals - Old and New

Box-counting dimensions of projections

Theorem Let E ⊂ R2 be non-empty compact. For all θ ∈ [0, π) (i) dimB projθE ≤ lim

r→0

log C 1

r (E)

− log r For almost all θ ∈ [0, π), (ii) dimB projθE = lim

r→0

log C 1

r (E)

− log r [We can replace dimB and lim by either dimB and lim, or by dimB and lim.]

Kenneth Falconer Projections of Fractals - Old and New

Box-counting dimensions of projections

Theorem Let E ⊂ R2 be non-empty compact. For all θ ∈ [0, π) (i) dimB projθE ≤ lim

r→0

log C 1

r (E)

− log r ≡ dim1

BE.

For almost all θ ∈ [0, π), (ii) dimB projθE = lim

r→0

log C 1

r (E)

− log r ≡ dim1

BE.

[We can replace dimB and lim by either dimB and lim, or by dimB and lim.] We call dims

BE := lim r→0

log C s

r (E)

− log r (E ⊂ R2 or Rn), using capacity with respect to the kernel φs

r(x) = min

• 1,

r

|x|

• s

, the s-box-dimension profile of E, which should be thought of as the ’box-dimension of E when regarded from an s-dimensional viewpoint’.

Kenneth Falconer Projections of Fractals - Old and New

Box-counting dimensions of projections

Lower bound proof: Let F ⊂ R be compact, ν a probability measure on F, and Ir(F) the intervals [ir, (i + 1)r), (i ∈ Z) that intersect F. 1 =

I∈Ir (F)

ν(I) 2 ≤ Nr(F)

• I∈Ir (F)

ν(I)2 ≤ Nr(F)

• I∈Ir(F)

(ν×ν){(w, z) ∈ I×I} ≤ Nr(F)(ν × ν){(w, z) : |w − z| ≤ r}. (1)

Kenneth Falconer Projections of Fractals - Old and New

Box-counting dimensions of projections

Lower bound proof: Let F ⊂ R be compact, ν a probability measure on F, and Ir(F) the intervals [ir, (i + 1)r), (i ∈ Z) that intersect F. 1 =

I∈Ir (F)

ν(I) 2 ≤ Nr(F)

• I∈Ir (F)

ν(I)2 ≤ Nr(F)

• I∈Ir(F)

(ν×ν){(w, z) ∈ I×I} ≤ Nr(F)(ν × ν){(w, z) : |w − z| ≤ r}. (1) Let µ be an equilibrium measure on E ⊂ R2, with projections µθ onto Lθ.

• (µθ × µθ){(w, z):|w − z| ≤ r}dθ =
• (µ×µ){(x, y):|projθx−projθy| ≤ r}dθ

=

• L{θ:|projθ(x−y)| ≤ r}dµ(x)dµ(y) ≤ c
• φ1

r (x−y)dµ(x)dµ(y) =

c C 1

r (E).

Kenneth Falconer Projections of Fractals - Old and New

Box-counting dimensions of projections

Lower bound proof: Let F ⊂ R be compact, ν a probability measure on F, and Ir(F) the intervals [ir, (i + 1)r), (i ∈ Z) that intersect F. 1 =

I∈Ir (F)

ν(I) 2 ≤ Nr(F)

• I∈Ir (F)

ν(I)2 ≤ Nr(F)

• I∈Ir(F)

(ν×ν){(w, z) ∈ I×I} ≤ Nr(F)(ν × ν){(w, z) : |w − z| ≤ r}. (1) Let µ be an equilibrium measure on E ⊂ R2, with projections µθ onto Lθ.

• (µθ × µθ){(w, z):|w − z| ≤ r}dθ =
• (µ×µ){(x, y):|projθx−projθy| ≤ r}dθ

=

• L{θ:|projθ(x−y)| ≤ r}dµ(x)dµ(y) ≤ c
• φ1

r (x−y)dµ(x)dµ(y) =

c C 1

r (E).

If

k 2skC 1 2−k(E)−1 < ∞ then there are Mθ < ∞ for a.a. θ such that

2sk(µθ × µθ){(w, z):|w − z| ≤ 2−k} ≤ Mθ (k ∈ N), so from (1), 1 ≤ N2−k(projθE) 2−skMθ. Hence if dim

1 B(E) > s then dimB(projθE) ≥ s for almost all θ.

Kenneth Falconer Projections of Fractals - Old and New

Exceptional directions

Marstrand’s theorem tells nothing about which particular directions may have projections with dimension or measure smaller than usual, i.e. when dimHprojθE < min{dimHE, 1}, or if dimHE > 1 when L(projθE) = 0. The set shown has dimension log 4/ log(5/2) = 1.51, but with some projections of dimension < 1.

Kenneth Falconer Projections of Fractals - Old and New

Exceptional directions

The set of exceptional directions can’t be ‘too big’: Theorem (Kaufman, 1968) If E ⊂ R2 and dimHE ≤ 1, dimH{θ : dimHprojθE < dimHE} ≤ dimHE. – follows from an energy estimate Theorem (F, 1982) If E ⊆ R2 and dimHE > 1, dimH{θ : L(projθE) = 0} ≤ 2 − dimHE. – proof uses Fourier transforms. Theorem (F, Howroyd, 1997) For compact E ⊂ R2 and let 0 ≤ s ≤ 1. Then dimH{θ ∈ [0, π) such that dimBprojθE < dims

BE} ≤ s.

Kenneth Falconer Projections of Fractals - Old and New

General problem

Theorem (Marstrand 1954) Let E ⊂ R2 be a Borel set. For all θ ∈ [0, π) (i) dimHprojθE ≤ min{dimHE, 1}. For almost all θ ∈ [0, π), (ii) dimHprojθE = min{dimHE, 1}, (iii) L(projθE) > 0 if dimHE > 1. General problem: find sets or classes of sets for which there are no exceptional directions for Marstrand’s theorem, i.e. where (ii) or (iii) hold for all θ, or at least where the exceptional directions can be identified. We will consider self-similar and self-affine sets and their random counterparts.

Kenneth Falconer Projections of Fractals - Old and New

Self-similar sets

Given an iterated function system of contracting similarities f1, . . . , fm : R2 → R2 there exists a unique non-empty compact E ⊂ R2 called a self-similar set such that E =

m

• i=1

fi(E). (∗) We assume that the union (∗) is disjoint or perhaps ‘nearly disjoint’ (i.e. OSC), so dimHE = dimBE = s, where s is the similarity dimension, given by m

i=1 r s i = 1.

Kenneth Falconer Projections of Fractals - Old and New

Self-similar sets

Given an iterated function system of contracting similarities f1, . . . , fm : R2 → R2 there exists a unique non-empty compact E ⊂ R2 called a self-similar set such that E =

m

• i=1

fi(E). (∗) We assume that the union (∗) is disjoint or perhaps ‘nearly disjoint’ (i.e. OSC), so dimHE = dimBE = s, where s is the similarity dimension, given by m

i=1 r s i = 1.

Write the similarities as fi(x) = riOi(x) + ti where 0 < ri < 1 is the scale factor, Oi is a rotation and ti is a translation.

Kenneth Falconer Projections of Fractals - Old and New

Self-similar sets

Given an iterated function system of contracting similarities f1, . . . , fm : R2 → R2 there exists a unique non-empty compact E ⊂ R2 called a self-similar set such that E =

m

• i=1

fi(E). (∗) We assume that the union (∗) is disjoint or perhaps ‘nearly disjoint’ (i.e. OSC), so dimHE = dimBE = s, where s is the similarity dimension, given by m

i=1 r s i = 1.

Write the similarities as fi(x) = riOi(x) + ti where 0 < ri < 1 is the scale factor, Oi is a rotation and ti is a translation. The family {f1, . . . , fm} has dense rotations if at least one of the Oi is a rotation by an irrational multiple of π, equivalently if group{O1, . . . , Om} is dense in SO(2, R). Otherwise {f1, . . . , fm} has finite rotations.

Kenneth Falconer Projections of Fractals - Old and New

finite rotations dense rotations Self-similar sets

Kenneth Falconer Projections of Fractals - Old and New

Self-similar sets with finite rotations

Theorem (Farkas 2014) Let E ⊂ R2 be a self-similar set defined by a family {f1, . . . , fm} of similarities with finite rotations and satisfying OSC. Then there is at least one value of θ such that dimHprojθE < dimHE

Kenneth Falconer Projections of Fractals - Old and New

Self-similar sets with dense rotations

Theorem (Peres & Shmerkin 2009, Hochman & Shmerkin 2012) Let E ⊂ R2 be a self-similar set defined by a family {f1, . . . , fm} of similarities with dense rotations. Then dimHprojθE = min{dimHE, 1} for all θ.

Kenneth Falconer Projections of Fractals - Old and New

Self-similar sets with dense rotations

Theorem (Peres & Shmerkin 2009, Hochman & Shmerkin 2012) Let E ⊂ R2 be a self-similar set defined by a family {f1, . . . , fm} of similarities with dense rotations. Then dimHprojθE = min{dimHE, 1} for all θ. Theorem (Shmerkin & Solomyak 2014) Let E ⊂ R2 be the self-similar attractor of an IFS with dense rotations with 1 < dimHE < 2. Then L(projθE) > 0 for all θ except (perhaps) for a set of θ of Hausdorff dimension 0.

Kenneth Falconer Projections of Fractals - Old and New

Self-similar sets with dense rotations

Theorem (Peres & Shmerkin 2009, Hochman & Shmerkin 2012) Let E ⊂ R2 be a self-similar set defined by a family {f1, . . . , fm} of similarities with dense rotations. Then dimHprojθE = min{dimHE, 1} for all θ. Theorem (Shmerkin & Solomyak 2014) Let E ⊂ R2 be the self-similar attractor of an IFS with dense rotations with 1 < dimHE < 2. Then L(projθE) > 0 for all θ except (perhaps) for a set of θ of Hausdorff dimension 0. Theorem (Rapaport 2017) There exists a self-similar E ⊂ R2 defined by an IFS {fi(x) = rO(x) + ti}4

i=1 satisfying the SSC with

O an irrational rotation, such that dimHE > 1 but L(projθE) = 0 for θ ∈ S for a dense Gδ set S ⊂ [0, π). (O is a rotation by a reciprocal of a pisot number).

Kenneth Falconer Projections of Fractals - Old and New

Percolation on a self-similar set

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

Kenneth Falconer Projections of Fractals - Old and New

Percolation on a self-similar set

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

Kenneth Falconer Projections of Fractals - Old and New

Percolation on a self-similar set

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

Kenneth Falconer Projections of Fractals - Old and New

Percolation on a self-similar set

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

Kenneth Falconer Projections of Fractals - Old and New

Percolation on a self-similar set

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

Kenneth Falconer Projections of Fractals - Old and New

Percolation on a self-similar set

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

Kenneth Falconer Projections of Fractals - Old and New

Percolation on a self-similar set

If p > 1/m then Ep = ∅ with positive probability, conditional on which dimHEp = s, where p m

i=1 rs i = 1

Kenneth Falconer Projections of Fractals - Old and New

Projection of percolation on a self-similar set

Let E ⊂ R2 be a self-similar set defined by a family {f1, . . . , fm} of similarities with dense rotations. Perform the percolation process

• n E with respect to its hierarchical construction, with each similar

component retained independently with probability p > 1/m, to

• btain a percolation set Ep ⊂ E. Let s be given by p m

i=1 rs i = 1,

so, conditional on Ep = ∅, dimH = s almost surely.

Kenneth Falconer Projections of Fractals - Old and New

Projection of percolation on a self-similar set

Let E ⊂ R2 be a self-similar set defined by a family {f1, . . . , fm} of similarities with dense rotations. Perform the percolation process

• n E with respect to its hierarchical construction, with each similar

component retained independently with probability p > 1/m, to

• btain a percolation set Ep ⊂ E. Let s be given by p m

i=1 rs i = 1,

so, conditional on Ep = ∅, dimH = s almost surely. Theorem (Jin & F, 2014) Conditional on Ep = ∅, almost surely dimHprojθEp = min{dimHEp, 1} for all θ.

Kenneth Falconer Projections of Fractals - Old and New

Projection of percolation on a self-similar set

Let E ⊂ R2 be a self-similar set defined by a family {f1, . . . , fm} of similarities with dense rotations. Perform the percolation process

• n E with respect to its hierarchical construction, with each similar

component retained independently with probability p > 1/m, to

• btain a percolation set Ep ⊂ E. Let s be given by p m

i=1 rs i = 1,

so, conditional on Ep = ∅, dimH = s almost surely. Theorem (Jin & F, 2014) Conditional on Ep = ∅, almost surely dimHprojθEp = min{dimHEp, 1} for all θ. Theorem (Jin & F, 2015) If s > 1, then conditional on Ep = ∅, almost surely L(projθEp) > 0 for all θ except for a set of θ of Hausdorff dimension 0. [These results on projections of random percolating sets have implications for sections of deterministic sets.]

Kenneth Falconer Projections of Fractals - Old and New

Self-affine sets

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

Kenneth Falconer Projections of Fractals - Old and New

Self-affine sets

The singular values α1 ≥ α2 ≥ 0 of a linear mapping T : R2 → R2 are the +ve square roots of the eigenvalues of TT ∗. The singular value function of T is φs(T) = αs

1

(0 ≤ s ≤ 1) α1αs−1

2

(1 ≤ s) The affinity dimension of the self-affine set E defined by the IFS {fi(x) = Ti(x) + ti}m

i=1 is

dimAff E =

• s : lim

k→∞ i1...ik

φs(Ti1 ◦ · · · ◦ Tik) 1/k = 1

• .

In particular, if Ti < 1

2 for all i, then dimHE = dimAff E for almost all

(t1, . . . , tm) ∈ R2m.

Kenneth Falconer Projections of Fractals - Old and New

Projections of self-affine sets

A Bernoulli measure µ on a self-affine E is of the form µ(Ti1 ◦ · · · ◦ TikE) = pi1 · · · pik where (p1, . . . , pm) is a given probability vector. Theorem (Kempton & F, 2016) If E is a self-affine set defined by strictly positive matrices and there exists a sequence (µn) of Bernoulli measures supported on E with dimHµn → dimAff E, then dimHE = dimAff E and dimHprojθ(E) = min{dimHE, 1} for all θ ∈ [0, π) unless the Ti have a common maximal eigenvector when there may be

• ne exceptional direction.

Kenneth Falconer Projections of Fractals - Old and New

Projections of self-affine sets

A Bernoulli measure µ on a self-affine E is of the form µ(Ti1 ◦ · · · ◦ TikE) = pi1 · · · pik where (p1, . . . , pm) is a given probability vector. Theorem (Kempton & F, 2016) If E is a self-affine set defined by strictly positive matrices and there exists a sequence (µn) of Bernoulli measures supported on E with dimHµn → dimAff E, then dimHE = dimAff E and dimHprojθ(E) = min{dimHE, 1} for all θ ∈ [0, π) unless the Ti have a common maximal eigenvector when there may be

• ne exceptional direction.

Corollary (Kempton & F, 2016) If Ti are positive matrices with Ti < 1

2

for all i with no common maximal eigenvector, then for almost all (t1, . . . , tm) ∈ R2m, dimHE = dimAff E and dimHprojθ(E) = min{dimHE, 1} for all θ ∈ [0, π). However, this corollary refers to ’generic’ self-affine sets, and to find specific E to which the theorem can be applied can be awkward.

Kenneth Falconer Projections of Fractals - Old and New

Projections of self-affine sets

Very recently Morris & Shmerkin showed that all projections behave well for a large class of identifiable self-affine sets E. Theorem (Morris & Shmerkin, 2017) Suppose that the IFSs satisfy SOSC, the Ti have no common eigenvector, one of the Ti is hyperbolic, and the Ti have ‘exponential separation’. If either dimAff E ≥ 3

2

• r

α1(Ti)2 ≤ α2(Ti) for all i then dimHE = dimBE = dimAff E and dimHprojθ(E) = min{dimHE, 1} for all θ ∈ [0, π). The proof depends on showing that given ǫ > 0 there exists k ∈ N and a Bernoulli measure µ with respect to {Ti1 ◦ · · · ◦ Tik} supported on the self-affine set E such that dimH µ > dimAff E − ǫ so that the previous Corollary can be applied.

Kenneth Falconer Projections of Fractals - Old and New

Specific self-affine sets arising from 50 and 2 IFS mappings with all projections of Hausdorff dimension 1

Kenneth Falconer Projections of Fractals - Old and New

Thank you!

Kenneth Falconer Projections of Fractals - Old and New