Local entropy averages and projections of fractal measures

Abstract

We show that for families of measures on Euclidean space which satisfy an ergodic-theoretic form of "self-similarity" under the operation of re-scaling, the dimension of linear images of the measure behaves in a semi-continuous way. We apply this to prove the following conjecture of Furstenberg: if X, Y ⊆ [0, 1] are closed and invariant, respectively, under ×m mod 1 and ×n mod 1, where m, n are not powers of the same integer, then, for any t = 0,A similar result holds for invariant measures and gives a simple proof of the Rudolph-Johnson theorem. Our methods also apply to many other classes of conformal fractals and measures. As another application, we extend and unify results of Peres, Shmerkin and Nazarov, and of Moreira, concerning projections of products of self-similar measures and Gibbs measures on regular Cantor sets. We show that under natural irreducibility assumptions on the maps in the IFS, the image measure has the maximal possible dimension under any linear projection other than the coordinate projections. We also present applications to Bernoulli convolutions and to the images of fractal measures under differentiable maps.

A family of smooth Cantor sets

“…Nazarov et al [NPS09] determined that the correlation dimension of µ r * µ s is min d r + d s , 1 whenever log r/ log s is irrational. Shmerkin informed us that in a joint work with Hochman [HS09] they generalize the work on sums of Cantor sets [PS09] and their methods imply that the dimension of the convolution H p * H p is min dim C p + dim C p , 1 whenever p/p is irrational. This in turns implies that for these parameters formula (1) holds.…”

Dynamics of the scenery flow and geometry of measures

“…Intuitively, the local dimensions of a measure should not be affected by the geometry of the measure on a density zero set of scales. This can be formalized using local entropy averages (see for example [26]). Thus heuristically one could expect that tangent distributions, defined as time averages, should encode all information on dimensions.…”

Fractal Geometry and Stochastics V

“…Then dim H proj θ (E × F ) = min{dim H (E × F ), 1} for all θ except possibly θ = 0 and θ = 1 2 π. Projection properties of self-affine measures underpin this work and there are measure analogues of these theorems, see [29,30,35].…”

“…The random percolation set is E = ∩ ∞ k=1 D k , see Figure 3. When the underlying IFS has dense rotations, Falconer and Jin [21] extended the ergodic theoretic methods of [35] to random cascade measures to obtain a random analogue of Theorem 8.2.…”

“…Then, for almost all every subspace V ∈ G(n, m) and µ-almost all x ∈ R n , dim µ (proj V x) = min{dim µ (x), m} and dim µ (proj V x) = min{dim µ (x), m}, see [24,35,39,40,85]. The (lower) Hausdorff dimension of a measure µ is defined as dim H µ = inf{dim H A : µ(A) > 0}.…”