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
• 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
• 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
• 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
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)
.
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)
.
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)
.
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)
.
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)
.
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)
.
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)
.
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)
.
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).
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).
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).
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).
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.
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.
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.
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.
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}.
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}.
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}.
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}.
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}.
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}.
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}.
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