Exact dimensionality and projections of random self-similar measures and sets

Abstract

We study the geometric properties of random multiplicative cascade measures defined on self‐similar sets. We show that such measures and their projections and sections are almost surely exact dimensional, generalizing a result of Feng and Hu for self‐similar measures. This, together with a compact group extension argument, enables us to generalize Hochman and Shmerkin's theorems on projections of deterministic self‐similar measures to these random measures without requiring any separation conditions on the underlying sets. We give applications to self‐similar sets and fractal percolation, including new results on projections, C1‐images and distance sets.

Self-Similar Sets: Projections, Sections and Percolation

“…The open set condition is not essential here since, for all ε > 0, a Vitali argument may be used to set up a new IFS, consisting of certain compositions of the f i , that satisfies SSC, with attractor E ⊂ E such that dim H E > dim H E − ε; we can also ensure that the new IFS has dense rotations if the original one has, see [7,9,29,31].…”

“…For example, the conclusions of Theorem 1.1 hold for all projections for certain classes of self-similar sets [10,15,31,36] as well as for random subsets of certain self-similar sets [7,33,35,34,38].…”

Projections of Gibbs measures on self-conformal sets

“…Since then, their work has been followed up by many mathematicians, see the recent survey papers [6,21,16] and the references therein. In particular, Falconer and Jin [7] extended their result to random cascade measures (including self-similar measures as special cases) without requiring any separation condition. In [12] Hochman and Shmerkin also considered the projections of products of Gibbs measures on one-dimensional non-linear Cantor sets.…”

“…We also make clear how to formulate an analogue of the dense rotations condition for the minimality of the underlining dynamical system. The methods from [7] also remove the requirement of any separation condition on the underlying sets.…”

Fractal Geometry and Stochastics V

“…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.…”