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}.