SLIDE 1 Distortion of dimension by Sobolev and quasiconformal mappings
17 June 2014
- I. Introduction and overview, quasiconformal maps of Rn and their effect on
Hausdorff dimension
- II. Global quasiconformal dimension in Rn
- III. Conformal dimension of metric spaces
- IV. Sobolev dimension distortion in Rn and in metric spaces
- V. QC and Sobolev dimension distortion in the sub-Riemannian Heisenberg group
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SLIDE 2 In this lecture we discuss Pansu’s conformal dimension. Definition (Pansu, 1989) Let X be a metric space. The conformal dimension of X is Cdim X = inf{dim Y : Y a metric space, Y
qs
∼ X}. Recall: f : X → Y is η-quasisymmetric (qs) if |f (x) − f (a)| ≤ η(t)|f (x) − f (b)| whenever x, a, b ∈ X satisfy |x − a| ≤ t|x − b| and t > 0.
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SLIDE 3 In this lecture we discuss Pansu’s conformal dimension. Definition (Pansu, 1989) Let X be a metric space. The conformal dimension of X is Cdim X = inf{dim Y : Y a metric space, Y
qs
∼ X}. Recall: f : X → Y is η-quasisymmetric (qs) if |f (x) − f (a)| ≤ η(t)|f (x) − f (b)| whenever x, a, b ∈ X satisfy |x − a| ≤ t|x − b| and t > 0. Goal for today: estimates from below for Cdim X. In particular, when is Cdim X = dim X?
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SLIDE 4 In this lecture we discuss Pansu’s conformal dimension. Definition (Pansu, 1989) Let X be a metric space. The conformal dimension of X is Cdim X = inf{dim Y : Y a metric space, Y
qs
∼ X}. Recall: f : X → Y is η-quasisymmetric (qs) if |f (x) − f (a)| ≤ η(t)|f (x) − f (b)| whenever x, a, b ∈ X satisfy |x − a| ≤ t|x − b| and t > 0. Goal for today: estimates from below for Cdim X. In particular, when is Cdim X = dim X? From last time: Cdim X = 0 if X ⊂ Rn is a self-similar Cantor set or has dim < 1.
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SLIDE 5 Lower bounds for conformal dimension
The general philosophy is: lower bounds on Cdim X arise from “well distributed” families of curves inside X. The model case is the foliation of Rn by lines parallel to a fixed direction.
V V V
a T
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SLIDE 6 Proposition A (after M. Bourdon, P. Pansu) Let (X, d, µ) be a doubling metric measure space and let 1 < p < ∞. Let Γ be a family of curves in X equipped with a probability measure ν s.t. (i) the support of Γ is bounded, (ii) the elements of Γ have diameters uniformly bounded away from zero, and (iii) ∃ C > 0 s.t. ν{γ ∈ Γ : γ ∩ B(x, r) = ∅} ≤ Cµ(B(x, r))1/p ∀ B(x, r). Then Cdim X ≥ p′, where p′ =
p p−1.
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SLIDE 7 Proposition A (after M. Bourdon, P. Pansu) Let (X, d, µ) be a doubling metric measure space and let 1 < p < ∞. Let Γ be a family of curves in X equipped with a probability measure ν s.t. (i) the support of Γ is bounded, (ii) the elements of Γ have diameters uniformly bounded away from zero, and (iii) ∃ C > 0 s.t. ν{γ ∈ Γ : γ ∩ B(x, r) = ∅} ≤ Cµ(B(x, r))1/p ∀ B(x, r). Then Cdim X ≥ p′, where p′ =
p p−1.
Remark If µ is assumed s-regular, then (iii) can be replaced by (iii’) there exists C > 0 s.t. ν{γ ∈ Γ : γ ∩ B(x, r) = ∅} ≤ Cr s/p ∀ B(x, r). If p =
s s−1 then the conclusion is Cdim X = dim X = s. In this case s p = s − 1.
If we can foliate a piece of X (of positive measure) by a family of curves which is “uniformly 1-codimensional”, then X is minimal for conformal dimension.
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SLIDE 8 Examples
- 1. X = Rn (s = n), Γ foliation by parallel lines Va = V + a, ν Lebesgue measure
- n V ⊥, p =
n n−1. Conclusion: Cdim Rn = dim Rn = n.
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SLIDE 9 Examples
- 1. X = Rn (s = n), Γ foliation by parallel lines Va = V + a, ν Lebesgue measure
- n V ⊥, p =
n n−1. Conclusion: Cdim Rn = dim Rn = n.
- 2. X = Hn Heisenberg group with left invariant Carnot–Carath´
eodory metric dcc (s = 2n + 2), Γ foliation by integral curves of a horiz vector field V , Γ can be equipped with a measure ν s.t. the previous condition holds with p =
s s−1.
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SLIDE 10 Examples
- 1. X = Rn (s = n), Γ foliation by parallel lines Va = V + a, ν Lebesgue measure
- n V ⊥, p =
n n−1. Conclusion: Cdim Rn = dim Rn = n.
- 2. X = Hn Heisenberg group with left invariant Carnot–Carath´
eodory metric dcc (s = 2n + 2), Γ foliation by integral curves of a horiz vector field V , Γ can be equipped with a measure ν s.t. the previous condition holds with p =
s s−1.
V ∈ span{X1, Y1, . . . , Xn, Yn} γ ∈ Γ satisfies γ′(s) = V (γ(s)) for all s ν satisfies |A| =
- Γ length(γ ∩ A) dν(γ) for all A ⊂ Hn
Conclusion: Cdim(Hn, dcc) = dim(Hn, dcc) = s. (due to Pansu)
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SLIDE 11 Examples
nski carpet SC ⊂ R2 s = log 8
log 3 ≈ 1.893 . . .
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SLIDE 12 Examples
nski carpet SC ⊂ R2 s = log 8
log 3 ≈ 1.893 . . .
Γ family of horiz lines (parameterized by 1
3 Cantor set C along y-axis)
ν = Hlog 2/ log 3 C, p = log 8/ log 3
log 2/ log 3 = 3.
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SLIDE 13 Examples
nski carpet SC ⊂ R2 s = log 8
log 3 ≈ 1.893 . . .
Γ family of horiz lines (parameterized by 1
3 Cantor set C along y-axis)
ν = Hlog 2/ log 3 C, p = log 8/ log 3
log 2/ log 3 = 3. Conclusion: Cdim SC ≥ p′ = 3 2 > 1.
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SLIDE 14 The estimate can be improved. Consider X1 = C × [0, 1] ⊂ SC. The measure µ1 = Ht × L1 X1 ≃ Ht+1 X1 t = log 2 log 3 is Ahlfors regular on X1, and ν({γ ∈ Γ : γ ∩ B(x, r) = ∅}) ≤ Cµ1(B(x, r))1/p with p = 1+t
t .
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SLIDE 15 The estimate can be improved. Consider X1 = C × [0, 1] ⊂ SC. The measure µ1 = Ht × L1 X1 ≃ Ht+1 X1 t = log 2 log 3 is Ahlfors regular on X1, and ν({γ ∈ Γ : γ ∩ B(x, r) = ∅}) ≤ Cµ1(B(x, r))1/p with p = 1+t
t . Hence Cdim X1 ≥ p′ = 1 + t and
Cdim SC ≥ Cdim X1 = 1 + log 2
log 3 > 3 2.
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SLIDE 16 Lower bounds for conformal dimension can also be obtained using moduli of curve families. Definition Let Γ be a family of curves in a metric measure space (X, d, µ) and let p ≥ 1. The p-modulus of Γ is Modp(Γ) = inf
ρp dµ where the infimum is taken over all admissible Borel functions ρ : X → [0, ∞], i.e.,
- γ ρ ds ≥ 1 for all locally rectifiable γ ∈ Γ.
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SLIDE 17 Lower bounds for conformal dimension can also be obtained using moduli of curve families. Definition Let Γ be a family of curves in a metric measure space (X, d, µ) and let p ≥ 1. The p-modulus of Γ is Modp(Γ) = inf
ρp dµ where the infimum is taken over all admissible Borel functions ρ : X → [0, ∞], i.e.,
- γ ρ ds ≥ 1 for all locally rectifiable γ ∈ Γ.
Example Let (Z, d, ν) be any compact mms and X = Z × [0, h] with the usual product metric and measure µ = ν ⊗ L1. Let Γ = {γz : z ∈ Z}, γz : [0, h] → X, γz(s) = (z, s). Modp(Γ) = ν(Y ) hp−1 . “≤”: ρ(z, s) = 1
h is admissible
“≥”: apply Fubini’s theorem and H¨
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SLIDE 18
Proposition B Let (X, d, µ) be a doubling metric measure space satisfying the upper mass bound µ(B(x, r)) ≤ r s for all x ∈ X and r > 0. Assume that there exists a curve family Γ in X s.t. Modp(Γ) > 0 for some 1 < p ≤ s. Then Cdim X ≥ s.
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SLIDE 19
Proposition B Let (X, d, µ) be a doubling metric measure space satisfying the upper mass bound µ(B(x, r)) ≤ r s for all x ∈ X and r > 0. Assume that there exists a curve family Γ in X s.t. Modp(Γ) > 0 for some 1 < p ≤ s. Then Cdim X ≥ s. Corollary Assume that X is Ahlfors s-regular and supports a curve family Γ in X s.t. Mods(Γ) > 0. Then Cdim X = dim X = s.
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SLIDE 20
Minimal sets for global qc dimension
For any t ∈ (0, n − 1) choose a compact t-regular set Z ⊂ Rn−1 (e.g., Z a suitable self-similar Cantor set). Let X = Z × [0, 1] ⊂ Rn equipped with product metric and measure µ = Ht × L1 ≃ Hs, s = t + 1.
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SLIDE 21 Minimal sets for global qc dimension
For any t ∈ (0, n − 1) choose a compact t-regular set Z ⊂ Rn−1 (e.g., Z a suitable self-similar Cantor set). Let X = Z × [0, 1] ⊂ Rn equipped with product metric and measure µ = Ht × L1 ≃ Hs, s = t + 1. Then X is s-regular and supports a curve family Γ with Modp(Γ) > 0 for any p. Alternatively, the criteria of Proposition A hold with ν = Ht Z and p =
s s−1.
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SLIDE 22 Minimal sets for global qc dimension
For any t ∈ (0, n − 1) choose a compact t-regular set Z ⊂ Rn−1 (e.g., Z a suitable self-similar Cantor set). Let X = Z × [0, 1] ⊂ Rn equipped with product metric and measure µ = Ht × L1 ≃ Hs, s = t + 1. Then X is s-regular and supports a curve family Γ with Modp(Γ) > 0 for any p. Alternatively, the criteria of Proposition A hold with ν = Ht Z and p =
s s−1.
Hence Cdim X = dim X = s. In particular, GQCdimRn X = s.
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SLIDE 23 Proposition A Let (X, d, µ) be a doubling metric measure space, 1 < p < ∞, Γ a family of curves equipped with a prob meas ν s.t. (i) Γ has bounded support, (ii) the elements of Γ have diameters ≥ c > 0, and (iii) ∃ C > 0 s.t. ν{γ ∈ Γ : γ ∩ B(x, r) = ∅} ≤ Cµ(B(x, r))1/p ∀ B(x, r). Then Cdim X ≥ p′. Suppose f : X → Y is η-qs and dim Y < p′. Uniform continuity of f implies all elements of f (Γ) have diameters ≥ c′ > 0. Cover Y with balls {B′
i } s.t. { 1 5B′ i } are disjoint and i(diam B′ i )p′ < ǫ.
Preimages of the balls B′
i under f are roughly balls; choose Bi ⊂ X s.t.
Bi ⊂ f −1( 1
5B′ i ) ⊂ f −1(B′ i ) ⊂ HBi where H = η(5). Note: {Bi} are disjoint.
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SLIDE 24 For each γ,
i =∅
diam(B′
i ) ≥ c′ > 0.
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SLIDE 25 For each γ,
i =∅
diam(B′
i ) ≥ c′ > 0.
Integrate over Γ w.r.t. ν: c′ ≤
(diam B′
i )ν{γ : f (γ) ∩ B′ i = ∅}
≤
(diam B′
i )ν{γ : γ ∩ HBi = ∅} ≤ C
(diam B′
i )µ(HBi)1/p.
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SLIDE 26 For each γ,
i =∅
diam(B′
i ) ≥ c′ > 0.
Integrate over Γ w.r.t. ν: c′ ≤
(diam B′
i )ν{γ : f (γ) ∩ B′ i = ∅}
≤
(diam B′
i )ν{γ : γ ∩ HBi = ∅} ≤ C
(diam B′
i )µ(HBi)1/p.
H¨
- lder’s inequality and the doubling property of µ give
0 < C ≤
(diam B′
i )p′
1/p′
i
µ(Bi) 1/p ≤ Cǫ1/p′ µ(A)1/p where A is a suitable neighborhood of the support of Γ. This leads to a contradiction if ǫ is sufficiently small.
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SLIDE 27
Proposition B Let (X, d, µ) be a doubling metric measure space satisfying the upper mass bound µ(B(x, r)) ≤ r s for all x ∈ X and r > 0. Assume that there exists a curve family Γ in X s.t. Modp(Γ) > 0 for some 1 < p ≤ s. Then Cdim X ≥ s.
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SLIDE 28 Proposition B Let (X, d, µ) be a doubling metric measure space satisfying the upper mass bound µ(B(x, r)) ≤ r s for all x ∈ X and r > 0. Assume that there exists a curve family Γ in X s.t. Modp(Γ) > 0 for some 1 < p ≤ s. Then Cdim X ≥ s. The proof is similar. For simplicity let p = s. Where previously we integrated over Γ w.r.t. ν, we now construct an admissible density ρ for the p-modulus of Γ. Choosing {B′
i } and {Bi} as before, set
ρ(x) = C
diam B′
i
ri
χ2HBi(x) for a suitable (large) constant C.
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SLIDE 29 Proposition B Let (X, d, µ) be a doubling metric measure space satisfying the upper mass bound µ(B(x, r)) ≤ r s for all x ∈ X and r > 0. Assume that there exists a curve family Γ in X s.t. Modp(Γ) > 0 for some 1 < p ≤ s. Then Cdim X ≥ s. The proof is similar. For simplicity let p = s. Where previously we integrated over Γ w.r.t. ν, we now construct an admissible density ρ for the p-modulus of Γ. Choosing {B′
i } and {Bi} as before, set
ρ(x) = C
diam B′
i
ri
χ2HBi(x) for a suitable (large) constant C. Admissibility of ρ follows from the uniform lower bound on diam f (γ). Then 0 < Mods(Γ) ≤ C
diam B′
i
ri
χ2HBi s dµ ≤ C
(diam B′
i )s µ(Bi) r s
i
→ 0.
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SLIDE 30
Annular linear connectivity and lower bounds for Cdim
Definition (X, d) is annularly linearly connected (ALC) if ∃ L ≥ 2 s.t. any two points x, y ∈ B(x0, 2r) \ B(x0, r/2) can be joined by an arc in B(x0, Lr) \ B(x0, r/L). Intuitively, X is annularly linearly connected if points in any annulus can be joined without going too far away from or too close to the center of the annulus. For instance, Rn is ALC if n ≥ 2. Heisenberg group (Hn, dcc) is ALC for any n. Sierpi´ nski carpet SC is ALC.
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SLIDE 31
Annular linear connectivity and lower bounds for Cdim
Definition (X, d) is annularly linearly connected (ALC) if ∃ L ≥ 2 s.t. any two points x, y ∈ B(x0, 2r) \ B(x0, r/2) can be joined by an arc in B(x0, Lr) \ B(x0, r/L). Intuitively, X is annularly linearly connected if points in any annulus can be joined without going too far away from or too close to the center of the annulus. For instance, Rn is ALC if n ≥ 2. Heisenberg group (Hn, dcc) is ALC for any n. Sierpi´ nski carpet SC is ALC. Theorem (Mackay) Let (X, d) be complete, doubling and annularly linearly connected. Then Cdim X ≥ 1 + ǫ > 1. The constant ǫ can be chosen to depend only on the doubling and annular linear connectivity constants of X.
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SLIDE 32
X complete, doubling, ALC ⇒ Cdim X ≥ 1 + ǫ > 1
Sketch of the proof: Start with a single arc γ in X. Choose an annulus A centered on γ. Pick two points x, y on γ ∩ A on opposite sides of the center.
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SLIDE 33
X complete, doubling, ALC ⇒ Cdim X ≥ 1 + ǫ > 1
Sketch of the proof: Start with a single arc γ in X. Choose an annulus A centered on γ. Pick two points x, y on γ ∩ A on opposite sides of the center. ALC ⇒ can choose a second arc joining x to y.
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SLIDE 34
X complete, doubling, ALC ⇒ Cdim X ≥ 1 + ǫ > 1
Sketch of the proof: Start with a single arc γ in X. Choose an annulus A centered on γ. Pick two points x, y on γ ∩ A on opposite sides of the center. ALC ⇒ can choose a second arc joining x to y. Repeat with both the old and the new arcs to get 4 arcs, 8 arcs, etc. Completeness ⇒ ∃ a Cantor set’s worth of arcs. Hausdorff distance on this family of arcs Γ is bi-Lipschitz equivalent to a standard self-similar metric on the Cantor set C s.t. dimension = t > 0. The standard self-similar measure on C pushes forward to a measure ν on Γ s.t. ν({γ ∈ Γ : γ ∩ B(x, r) = ∅}) ≤ Cr t. By Proposition A, Cdim X ≥ 1 + c(t) > 1. All steps are quantitative.
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SLIDE 35 Additional remarks
- 1. As for global quasiconformal dimension, conformal dimension takes on no
values strictly between zero and one. In fact this follows from Kovalev’s theorem, since nonseparable metric spaces have infinite conformal dimension and every separable metric space embeds isometrically into a real separable Banach space.
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SLIDE 36 Additional remarks
- 1. As for global quasiconformal dimension, conformal dimension takes on no
values strictly between zero and one. In fact this follows from Kovalev’s theorem, since nonseparable metric spaces have infinite conformal dimension and every separable metric space embeds isometrically into a real separable Banach space.
- 2. We saw that under mild extra assumptions, the existence of a curve family of
positive modulus suffices for minimality for conformal dimension. It is a much deeper fact that this criterion is also necessary.
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SLIDE 37
Theorem (Keith–Laakso) Let (X, d, µ) be a compact metric measure space, Ahlfors s-regular for some s > 1. If CdimA X = s then there exists some weak tangent X∞ of X which carries a family Γ of curves with diameters ≥ c > 0 and s.t. Mods(Γ) > 0.
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SLIDE 38 Theorem (Keith–Laakso) Let (X, d, µ) be a compact metric measure space, Ahlfors s-regular for some s > 1. If CdimA X = s then there exists some weak tangent X∞ of X which carries a family Γ of curves with diameters ≥ c > 0 and s.t. Mods(Γ) > 0. “Weak tangent”: Gromov–Hausdorff limit of rescaled spaces (X, r −1
j
d, sjµ).
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SLIDE 39 Theorem (Keith–Laakso) Let (X, d, µ) be a compact metric measure space, Ahlfors s-regular for some s > 1. If CdimA X = s then there exists some weak tangent X∞ of X which carries a family Γ of curves with diameters ≥ c > 0 and s.t. Mods(Γ) > 0. “Weak tangent”: Gromov–Hausdorff limit of rescaled spaces (X, r −1
j
d, sjµ). Assouad dimension interacts better with weak tangents than does Hausdorff dimension. Easy fact: If X∞ is a weak tangent of X, then dimA X∞ ≤ dimA X Quasisymmetric maps pass to weak tangents ⇒ CdimA X∞ ≤ CdimA X.
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SLIDE 40 Theorem (Keith–Laakso) Let (X, d, µ) be a compact metric measure space, Ahlfors s-regular for some s > 1. If CdimA X = s then there exists some weak tangent X∞ of X which carries a family Γ of curves with diameters ≥ c > 0 and s.t. Mods(Γ) > 0. “Weak tangent”: Gromov–Hausdorff limit of rescaled spaces (X, r −1
j
d, sjµ). Assouad dimension interacts better with weak tangents than does Hausdorff dimension. Easy fact: If X∞ is a weak tangent of X, then dimA X∞ ≤ dimA X Quasisymmetric maps pass to weak tangents ⇒ CdimA X∞ ≤ CdimA X. Conclusion: If X is s-regular for some s > 1 and has a weak tangent space X∞ which contains a curve family of positive p-modulus for some 1 < p ≤ s, then CdimA X = s. s = CdimA X∞ ≤ CdimA X ≤ dimA X = s
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SLIDE 41
- 3. Conformal dimension of the Sierpi´
nski carpet SC contains no curve families of positive p-modulus (for any p). Since SC is self-similar, neither does any weak tangent of SC. By Keith–Laakso, it follows that Cdim SC ≤ CdimA SC< dim SC = log 8
log 3.
1 < 1 + log 2 log 3 ≤ Cdim SC < dim SC = log 8 log 3 Question: Cdim SC =?
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