L^qdimensions and projections of random measures

Abstract

We prove preservation of L q dimensions (for 1 < q ≤ 2) under all orthogonal projections for a class of random measures on the plane, which includes (deterministic) homogeneous self-similar measures and a well-known family of measures supported on 1-variable fractals as special cases. We prove a similar result for certain convolutions, extending a result of Nazarov, Peres and Shmerkin. Recently many related results have been obtained for Hausdorff dimension, but much less is known for L q dimensions.2000 Mathematics Subject Classification. Primary 28A80, Secondary 28A78, 37H99.[8]). In general, q → D q may be strictly decreasing (this is a reflection of the multifractality of µ), but it may also be constant. For example, if µ is Ahlforsregular with exponent d (that is, if C −1 r d ≤ µ(B(x, r)) ≤ C r d for all x ∈ supp(µ)) then D q µ = dim H µ = d for all q. For many measures of dynamical origin, such as self-similar measures, the limit in the definition of D q is known to exist, see [22].The only previous result on L q dimensions of projected measures was obtained in [19]. There it is proved that if µ, ν are self-similar measures satisfying certain natural assumptions, then for any q ∈ (1, 2], D q (Π(µ × ν)) = min(D q (µ × ν), 1)

On Furstenberg's intersection conjecture, self-similar measures, and the $L^q$ norms of convolutions

“…Other than the uniformity in x, the claim (4.3) follows from [22, Theorem 3.5], which in turn is established by inspecting the proof of the subadditive ergodic theorem given by Katznelson and Weiss [31] (recall that for uniquely ergodic systems all points are generic). To deduce the uniform convergence, we recall that the ergodic averages of the continuous functions φ n,ε converge uniformly (thanks to unique ergodicity), and apply [22,Eq.(18)].…”

“…We point out that in the range q ∈ (1, 2], the above result was proved in [35] in some special cases and then, extending the same ideas, in [22, Corollary 6.2], in even greater generality. For example, in [22] no separation assumptions are made on η 1 , η 2 . However, the methods of [35,22] ultimately rely on Marstrand's projection theorem, which is known to fail in general if q > 2.…”

On normal numbers and self-similar measures

“…Our method to disintegrate µ p is based upon a technique that first appeared in [10], and was subsequently applied in [15] and [21]. In these papers the authors used this technique to express an arbitrary self-similar measure as the integral of a collection of random measures.…”

“…We will prove each item in turn. The proof of item 1 can be found in any of [10,15,21]. We include the details for completion.…”

Absolute continuity in families of parametrised non-homogeneous self-similar measures

“…In the non-homogeneous setting of Theorem 1.5, the measures µ u no longer have an infinite convolution structure, and hence the method of [8,9] is not directly applicable. A way around this problem was found in [2]: Galicer, Saglietti, Shmerkin, and Yavicoli (see [2,Lemma 6.6]) discovered a way to express non-homogeneous self-similar measures as averages over measures with an infinite convolution structure. This naturally comes at a price: the infinite convolutions are no longer self-similar measures.…”