Hausdorff Dimension for Randomly Perturbed Self Affine Attractors

Generalized dimensions of measures on almost self-affine sets

“…, ω im , provided T i < 1 2 for all i, see [5,24]. Equality also holds with probability one for the almost self-affine sets introduced in [15], where the {ω i : i ∈ I} are independent with identical distributions of bounded density within a region D.…”

“…The main theorem of [15] states that, in this setting, the dimension of E ω equals the affinity dimension almost surely.…”

Generalized Energy Inequalities and Higher Multifractal Moments

“…Inequality ( dimineq 5.8) is obtained by a covering method. Almost sure equality for self-affine sets and random almost self-affine sets may be derived from energy estimates for measures supported on the sets, see Fa1 [5] and JPS [12] for the two settings. 2…”

On Falconer's formula for the generalised Rényi dimension of a self-affine measure

“…, σ d (A), to be the non-negative square roots of the eigenvalues of the positive semi-definite matrix A * A, listed in decreasing order. Let us define a function ϕ : (0, +∞) × M d (R) → [0, +∞) by by B. Solomyak [25], and to 1 by Jordan, Simon and Pollicott for a notion of "almost self-affine set" which incorporates additional random translations [19]. While it is well-known that the Hausdorff dimension of Z T can fail to depend continuously on the affinites T 1 , .…”

“…In the later article [12] this result was extended to a more general class of almost self-affine measures, under the weaker hypotheses max A i < 1 and q > 1 but requiring randomised translations in a similar manner to [19]. An alternative extension of this result, which allows q ∈ [1, Q] for certain values of Q > 2, was recently obtained by Barral and Feng [4,§6].…”