Entropy and geometric measure theory · Entropy and geometric measure theory Tuomas Sahlsten...
Transcript of Entropy and geometric measure theory · Entropy and geometric measure theory Tuomas Sahlsten...
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Entropy and geometric measure theory
Tuomas Sahlsten
Advances on Fractals and Related TopicsThe Chinese University of Hong Kong, 11.12.2012
joint work with Ville Suomala and Pablo Shmerkin
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• M. Hochman, P. Shmerkin: Local entropy averages and projectionsof fractal measures, Ann. of Math. (2), 175(3):1001–1059, 2012
• P. Shmerkin: The dimension of weakly mean porous measures: aprobabilistic approach, Int. Math. Res. Not. IMRN, (9):2010–2033,2012• T. S., P. Shmerkin, V. Suomala: Dimension, entropy and the local
distribution of measures, J. London Math. Soc., appeared online, 2012
![Page 3: Entropy and geometric measure theory · Entropy and geometric measure theory Tuomas Sahlsten AdvancesonFractalsandRelatedTopics TheChineseUniversityofHongKong,11.12.2012 jointworkwithVille](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f7598e94fc9343487520a14/html5/thumbnails/3.jpg)
• M. Hochman, P. Shmerkin: Local entropy averages and projectionsof fractal measures, Ann. of Math. (2), 175(3):1001–1059, 2012• P. Shmerkin: The dimension of weakly mean porous measures: a
probabilistic approach, Int. Math. Res. Not. IMRN, (9):2010–2033,2012
• T. S., P. Shmerkin, V. Suomala: Dimension, entropy and the localdistribution of measures, J. London Math. Soc., appeared online, 2012
![Page 4: Entropy and geometric measure theory · Entropy and geometric measure theory Tuomas Sahlsten AdvancesonFractalsandRelatedTopics TheChineseUniversityofHongKong,11.12.2012 jointworkwithVille](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f7598e94fc9343487520a14/html5/thumbnails/4.jpg)
• M. Hochman, P. Shmerkin: Local entropy averages and projectionsof fractal measures, Ann. of Math. (2), 175(3):1001–1059, 2012• P. Shmerkin: The dimension of weakly mean porous measures: a
probabilistic approach, Int. Math. Res. Not. IMRN, (9):2010–2033,2012• T. S., P. Shmerkin, V. Suomala: Dimension, entropy and the local
distribution of measures, J. London Math. Soc., appeared online, 2012
![Page 5: Entropy and geometric measure theory · Entropy and geometric measure theory Tuomas Sahlsten AdvancesonFractalsandRelatedTopics TheChineseUniversityofHongKong,11.12.2012 jointworkwithVille](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f7598e94fc9343487520a14/html5/thumbnails/5.jpg)
Local entropy averages
Let Qk,x be the dyadic cube of generation k ∈ N containing x ∈ Rd.• The 2a-adic entropy in Qk,x of a measure µ in Rd is
Ha(µ,Qk,x) =∑
Q is a generation k+a
dyadic subcube of Qk,x
− µ(Q)µ(Qk,x)
log µ(Q)µ(Qk,x)
.
![Page 6: Entropy and geometric measure theory · Entropy and geometric measure theory Tuomas Sahlsten AdvancesonFractalsandRelatedTopics TheChineseUniversityofHongKong,11.12.2012 jointworkwithVille](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f7598e94fc9343487520a14/html5/thumbnails/6.jpg)
Local entropy averages
Let Qk,x be the dyadic cube of generation k ∈ N containing x ∈ Rd.
• The 2a-adic entropy in Qk,x of a measure µ in Rd is
Ha(µ,Qk,x) =∑
Q is a generation k+a
dyadic subcube of Qk,x
− µ(Q)µ(Qk,x)
log µ(Q)µ(Qk,x)
.
![Page 7: Entropy and geometric measure theory · Entropy and geometric measure theory Tuomas Sahlsten AdvancesonFractalsandRelatedTopics TheChineseUniversityofHongKong,11.12.2012 jointworkwithVille](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f7598e94fc9343487520a14/html5/thumbnails/7.jpg)
Local entropy averages
Let Qk,x be the dyadic cube of generation k ∈ N containing x ∈ Rd.• The 2a-adic entropy in Qk,x of a measure µ in Rd is
Ha(µ,Qk,x) =∑
Q is a generation k+a
dyadic subcube of Qk,x
− µ(Q)µ(Qk,x)
log µ(Q)µ(Qk,x)
.
![Page 8: Entropy and geometric measure theory · Entropy and geometric measure theory Tuomas Sahlsten AdvancesonFractalsandRelatedTopics TheChineseUniversityofHongKong,11.12.2012 jointworkwithVille](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f7598e94fc9343487520a14/html5/thumbnails/8.jpg)
Local entropy averages
Let Qk,x be the dyadic cube of generation k ∈ N containing x ∈ Rd.• The 2a-adic entropy in Qk,x of a measure µ in Rd is
Ha(µ,Qk,x) =∑
Q is a generation k+a
dyadic subcube of Qk,x
− µ(Q)µ(Qk,x)
log µ(Q)µ(Qk,x)
.
![Page 9: Entropy and geometric measure theory · Entropy and geometric measure theory Tuomas Sahlsten AdvancesonFractalsandRelatedTopics TheChineseUniversityofHongKong,11.12.2012 jointworkwithVille](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f7598e94fc9343487520a14/html5/thumbnails/9.jpg)
Local entropy averages
Let Qk,x be the dyadic cube of generation k ∈ N containing x ∈ Rd.• The 2a-adic entropy in Qk,x of a measure µ in Rd is
Ha(µ,Qk,x) =∑
Q is a generation k+a
dyadic subcube of Qk,x
− µ(Q)µ(Qk,x)
log µ(Q)µ(Qk,x)
.
![Page 10: Entropy and geometric measure theory · Entropy and geometric measure theory Tuomas Sahlsten AdvancesonFractalsandRelatedTopics TheChineseUniversityofHongKong,11.12.2012 jointworkwithVille](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f7598e94fc9343487520a14/html5/thumbnails/10.jpg)
Local entropy averages
Let Qk,x be the dyadic cube of generation k ∈ N containing x ∈ Rd.• The 2a-adic entropy in Qk,x of a measure µ in Rd is
Ha(µ,Qk,x) =∑
Q is a generation k+a
dyadic subcube of Qk,x
− µ(Q)µ(Qk,x)
log µ(Q)µ(Qk,x)
.
![Page 11: Entropy and geometric measure theory · Entropy and geometric measure theory Tuomas Sahlsten AdvancesonFractalsandRelatedTopics TheChineseUniversityofHongKong,11.12.2012 jointworkwithVille](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f7598e94fc9343487520a14/html5/thumbnails/11.jpg)
Local entropy averages
Let Qk,x be the dyadic cube of generation k ∈ N containing x ∈ Rd.• The 2a-adic entropy in Qk,x of a measure µ in Rd is
Ha(µ,Qk,x) =∑
Q is a generation k+a
dyadic subcube of Qk,x
− µ(Q)µ(Qk,x)
log µ(Q)µ(Qk,x)
.
![Page 12: Entropy and geometric measure theory · Entropy and geometric measure theory Tuomas Sahlsten AdvancesonFractalsandRelatedTopics TheChineseUniversityofHongKong,11.12.2012 jointworkwithVille](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f7598e94fc9343487520a14/html5/thumbnails/12.jpg)
Local entropy averages
Let Qk,x be the dyadic cube of generation k ∈ N containing x ∈ Rd.• The 2a-adic entropy in Qk,x of a measure µ in Rd is
Ha(µ,Qk,x) =∑
Q is a generation k+a
dyadic subcube of Qk,x
− µ(Q)µ(Qk,x)
log µ(Q)µ(Qk,x)
.
![Page 13: Entropy and geometric measure theory · Entropy and geometric measure theory Tuomas Sahlsten AdvancesonFractalsandRelatedTopics TheChineseUniversityofHongKong,11.12.2012 jointworkwithVille](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f7598e94fc9343487520a14/html5/thumbnails/13.jpg)
Local entropy averages
Let Qk,x be the dyadic cube of generation k ∈ N containing x ∈ Rd.• The 2a-adic entropy in Qk,x of a measure µ in Rd is
Ha(µ,Qk,x) =∑
Q is a generation k+a
dyadic subcube of Qk,x
− µ(Q)µ(Qk,x)
log µ(Q)µ(Qk,x)
.
Lemma (Llorente, Nicolau; Hochman, Shmerkin; Peres)
Let µ be a measure on Rd and a ∈ N. Then at µ almost every x ∈ Rd:
lim infr↘0
logµ(B(x, r))
log r= lim inf
N→∞
1
Na log 2
N∑k=1
Ha(µ,Qk,x);
lim supr↘0
logµ(B(x, r))
log r= lim sup
N→∞
1
Na log 2
N∑k=1
Ha(µ,Qk,x).
![Page 14: Entropy and geometric measure theory · Entropy and geometric measure theory Tuomas Sahlsten AdvancesonFractalsandRelatedTopics TheChineseUniversityofHongKong,11.12.2012 jointworkwithVille](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f7598e94fc9343487520a14/html5/thumbnails/14.jpg)
Local entropy averages
Let Qk,x be the dyadic cube of generation k ∈ N containing x ∈ Rd.• The 2a-adic entropy in Qk,x of a measure µ in Rd is
Ha(µ,Qk,x) =∑
Q is a generation k+a
dyadic subcube of Qk,x
− µ(Q)µ(Qk,x)
log µ(Q)µ(Qk,x)
.
Lemma (Llorente, Nicolau; Hochman, Shmerkin; Peres)
Let µ be a measure on Rd and a ∈ N.
Then at µ almost every x ∈ Rd:
lim infr↘0
logµ(B(x, r))
log r= lim inf
N→∞
1
Na log 2
N∑k=1
Ha(µ,Qk,x);
lim supr↘0
logµ(B(x, r))
log r= lim sup
N→∞
1
Na log 2
N∑k=1
Ha(µ,Qk,x).
![Page 15: Entropy and geometric measure theory · Entropy and geometric measure theory Tuomas Sahlsten AdvancesonFractalsandRelatedTopics TheChineseUniversityofHongKong,11.12.2012 jointworkwithVille](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f7598e94fc9343487520a14/html5/thumbnails/15.jpg)
Local entropy averages
Let Qk,x be the dyadic cube of generation k ∈ N containing x ∈ Rd.• The 2a-adic entropy in Qk,x of a measure µ in Rd is
Ha(µ,Qk,x) =∑
Q is a generation k+a
dyadic subcube of Qk,x
− µ(Q)µ(Qk,x)
log µ(Q)µ(Qk,x)
.
Lemma (Llorente, Nicolau; Hochman, Shmerkin; Peres)
Let µ be a measure on Rd and a ∈ N. Then at µ almost every x ∈ Rd:
lim infr↘0
logµ(B(x, r))
log r= lim inf
N→∞
1
Na log 2
N∑k=1
Ha(µ,Qk,x);
lim supr↘0
logµ(B(x, r))
log r= lim sup
N→∞
1
Na log 2
N∑k=1
Ha(µ,Qk,x).
![Page 16: Entropy and geometric measure theory · Entropy and geometric measure theory Tuomas Sahlsten AdvancesonFractalsandRelatedTopics TheChineseUniversityofHongKong,11.12.2012 jointworkwithVille](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f7598e94fc9343487520a14/html5/thumbnails/16.jpg)
Local entropy averages
Let Qk,x be the dyadic cube of generation k ∈ N containing x ∈ Rd.• The 2a-adic entropy in Qk,x of a measure µ in Rd is
Ha(µ,Qk,x) =∑
Q is a generation k+a
dyadic subcube of Qk,x
− µ(Q)µ(Qk,x)
log µ(Q)µ(Qk,x)
.
Lemma (Llorente, Nicolau; Hochman, Shmerkin; Peres)
Let µ be a measure on Rd and a ∈ N. Then at µ almost every x ∈ Rd:
lim infr↘0
logµ(B(x, r))
log r= lim inf
N→∞
1
Na log 2
N∑k=1
Ha(µ,Qk,x);
lim supr↘0
logµ(B(x, r))
log r= lim sup
N→∞
1
Na log 2
N∑k=1
Ha(µ,Qk,x).
![Page 17: Entropy and geometric measure theory · Entropy and geometric measure theory Tuomas Sahlsten AdvancesonFractalsandRelatedTopics TheChineseUniversityofHongKong,11.12.2012 jointworkwithVille](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f7598e94fc9343487520a14/html5/thumbnails/17.jpg)
Local entropy averages
Let µ be a measure on Rd and a ∈ N. Then at µ almost every x ∈ Rd:
dimloc(µ, x) = limN→∞
1
Na log 2
N∑k=1
Ha(µ,Qk,x).
The Problem: Relating dimension to local distribution
HeuristicsIf the dimension of a measure µ is “large”, then the distribution of µ is“spread out” and “flat” at many scales.
![Page 18: Entropy and geometric measure theory · Entropy and geometric measure theory Tuomas Sahlsten AdvancesonFractalsandRelatedTopics TheChineseUniversityofHongKong,11.12.2012 jointworkwithVille](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f7598e94fc9343487520a14/html5/thumbnails/18.jpg)
Local entropy averages
Let µ be a measure on Rd and a ∈ N. Then at µ almost every x ∈ Rd:
dimloc(µ, x) = limN→∞
1
Na log 2
N∑k=1
Ha(µ,Qk,x).
The Problem: Relating dimension to local distribution
HeuristicsIf the dimension of a measure µ is “large”, then the distribution of µ is“spread out” and “flat” at many scales.
![Page 19: Entropy and geometric measure theory · Entropy and geometric measure theory Tuomas Sahlsten AdvancesonFractalsandRelatedTopics TheChineseUniversityofHongKong,11.12.2012 jointworkwithVille](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f7598e94fc9343487520a14/html5/thumbnails/19.jpg)
Local entropy averages
Let µ be a measure on Rd and a ∈ N. Then at µ almost every x ∈ Rd:
dimloc(µ, x) = limN→∞
1
Na log 2
N∑k=1
Ha(µ,Qk,x).
The Problem: Relating dimension to local distribution
Heuristics
If the dimension of a measure µ is “large”, then the distribution of µ is“spread out” and “flat” at many scales.
![Page 20: Entropy and geometric measure theory · Entropy and geometric measure theory Tuomas Sahlsten AdvancesonFractalsandRelatedTopics TheChineseUniversityofHongKong,11.12.2012 jointworkwithVille](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f7598e94fc9343487520a14/html5/thumbnails/20.jpg)
Local entropy averages
Let µ be a measure on Rd and a ∈ N. Then at µ almost every x ∈ Rd:
dimloc(µ, x) = limN→∞
1
Na log 2
N∑k=1
Ha(µ,Qk,x).
The Problem: Relating dimension to local distribution
HeuristicsIf the dimension of a measure µ is “large”, then the distribution of µ is“spread out” and “flat” at many scales.
![Page 21: Entropy and geometric measure theory · Entropy and geometric measure theory Tuomas Sahlsten AdvancesonFractalsandRelatedTopics TheChineseUniversityofHongKong,11.12.2012 jointworkwithVille](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f7598e94fc9343487520a14/html5/thumbnails/21.jpg)
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Time is running out!
As a sample, one of the results in the plane:
Theorem (S., Shmerkin, Suomala 2012)
For s ∈ (1, 2) and α ∈ (0, 1) there exist p > 0 and c > 0 such that
• if a measure µ in R2 satisfies dimH µ > s, then at µ-a.e. x ∈ R2:
lim infN→∞
∣∣∣{k = 1, . . . , N : inf`
µ(C(x,`,α,2−k))µ(B(x,2−k))
> c}∣∣∣
N> p. (1)
• if dimp µ > s, then same holds with lim inf replaced by lim sup in (1).
C(x, `, α, r) := {y ∈ B(x, r) : dist(y − x, `) < α|y − x|, (y − x) · ` > 0}.
![Page 37: Entropy and geometric measure theory · Entropy and geometric measure theory Tuomas Sahlsten AdvancesonFractalsandRelatedTopics TheChineseUniversityofHongKong,11.12.2012 jointworkwithVille](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f7598e94fc9343487520a14/html5/thumbnails/37.jpg)
Time is running out! As a sample, one of the results in the plane:
Theorem (S., Shmerkin, Suomala 2012)
For s ∈ (1, 2) and α ∈ (0, 1) there exist p > 0 and c > 0 such that
• if a measure µ in R2 satisfies dimH µ > s, then at µ-a.e. x ∈ R2:
lim infN→∞
∣∣∣{k = 1, . . . , N : inf`
µ(C(x,`,α,2−k))µ(B(x,2−k))
> c}∣∣∣
N> p. (1)
• if dimp µ > s, then same holds with lim inf replaced by lim sup in (1).
C(x, `, α, r) := {y ∈ B(x, r) : dist(y − x, `) < α|y − x|, (y − x) · ` > 0}.
![Page 38: Entropy and geometric measure theory · Entropy and geometric measure theory Tuomas Sahlsten AdvancesonFractalsandRelatedTopics TheChineseUniversityofHongKong,11.12.2012 jointworkwithVille](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f7598e94fc9343487520a14/html5/thumbnails/38.jpg)
Time is running out! As a sample, one of the results in the plane:
Theorem (S., Shmerkin, Suomala 2012)
For s ∈ (1, 2) and α ∈ (0, 1) there exist p > 0 and c > 0 such that
• if a measure µ in R2 satisfies dimH µ > s, then at µ-a.e. x ∈ R2:
lim infN→∞
∣∣∣{k = 1, . . . , N : inf`
µ(C(x,`,α,2−k))µ(B(x,2−k))
> c}∣∣∣
N> p. (1)
• if dimp µ > s, then same holds with lim inf replaced by lim sup in (1).
C(x, `, α, r) := {y ∈ B(x, r) : dist(y − x, `) < α|y − x|, (y − x) · ` > 0}.
![Page 39: Entropy and geometric measure theory · Entropy and geometric measure theory Tuomas Sahlsten AdvancesonFractalsandRelatedTopics TheChineseUniversityofHongKong,11.12.2012 jointworkwithVille](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f7598e94fc9343487520a14/html5/thumbnails/39.jpg)
Time is running out! As a sample, one of the results in the plane:
Theorem (S., Shmerkin, Suomala 2012)
For s ∈ (1, 2) and α ∈ (0, 1) there exist p > 0 and c > 0 such that
• if a measure µ in R2 satisfies dimH µ > s,
then at µ-a.e. x ∈ R2:
lim infN→∞
∣∣∣{k = 1, . . . , N : inf`
µ(C(x,`,α,2−k))µ(B(x,2−k))
> c}∣∣∣
N> p. (1)
• if dimp µ > s, then same holds with lim inf replaced by lim sup in (1).
C(x, `, α, r) := {y ∈ B(x, r) : dist(y − x, `) < α|y − x|, (y − x) · ` > 0}.
![Page 40: Entropy and geometric measure theory · Entropy and geometric measure theory Tuomas Sahlsten AdvancesonFractalsandRelatedTopics TheChineseUniversityofHongKong,11.12.2012 jointworkwithVille](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f7598e94fc9343487520a14/html5/thumbnails/40.jpg)
Time is running out! As a sample, one of the results in the plane:
Theorem (S., Shmerkin, Suomala 2012)
For s ∈ (1, 2) and α ∈ (0, 1) there exist p > 0 and c > 0 such that
• if a measure µ in R2 satisfies dimH µ > s, then at µ-a.e. x ∈ R2:
lim infN→∞
∣∣∣{k = 1, . . . , N : inf`
µ(C(x,`,α,2−k))µ(B(x,2−k))
> c}∣∣∣
N> p. (1)
• if dimp µ > s, then same holds with lim inf replaced by lim sup in (1).
C(x, `, α, r) := {y ∈ B(x, r) : dist(y − x, `) < α|y − x|, (y − x) · ` > 0}.
![Page 41: Entropy and geometric measure theory · Entropy and geometric measure theory Tuomas Sahlsten AdvancesonFractalsandRelatedTopics TheChineseUniversityofHongKong,11.12.2012 jointworkwithVille](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f7598e94fc9343487520a14/html5/thumbnails/41.jpg)
Time is running out! As a sample, one of the results in the plane:
Theorem (S., Shmerkin, Suomala 2012)
For s ∈ (1, 2) and α ∈ (0, 1) there exist p > 0 and c > 0 such that
• if a measure µ in R2 satisfies dimH µ > s, then at µ-a.e. x ∈ R2:
lim infN→∞
∣∣∣{k = 1, . . . , N : inf`
µ(C(x,`,α,2−k))µ(B(x,2−k))
> c}∣∣∣
N> p. (1)
• if dimp µ > s, then same holds with lim inf replaced by lim sup in (1).
C(x, `, α, r) := {y ∈ B(x, r) : dist(y − x, `) < α|y − x|, (y − x) · ` > 0}.
![Page 42: Entropy and geometric measure theory · Entropy and geometric measure theory Tuomas Sahlsten AdvancesonFractalsandRelatedTopics TheChineseUniversityofHongKong,11.12.2012 jointworkwithVille](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f7598e94fc9343487520a14/html5/thumbnails/42.jpg)
Time is running out! As a sample, one of the results in the plane:
Theorem (S., Shmerkin, Suomala 2012)
For s ∈ (1, 2) and α ∈ (0, 1) there exist p > 0 and c > 0 such that
• if a measure µ in R2 satisfies dimH µ > s, then at µ-a.e. x ∈ R2:
lim infN→∞
∣∣∣{k = 1, . . . , N : inf`
µ(C(x,`,α,2−k))µ(B(x,2−k))
> c}∣∣∣
N> p. (1)
• if dimp µ > s, then same holds with lim inf replaced by lim sup in (1).
C(x, `, α, r) := {y ∈ B(x, r) : dist(y − x, `) < α|y − x|, (y − x) · ` > 0}.