Entropy and geometric measure theory · Entropy and geometric measure theory Tuomas Sahlsten...

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}.