Generalized dimensions of measures on almost self-affine sets

Abstract

We establish a generic formula for the generalised q-dimensions of measures supported by almost self-affine sets, for all q > 1. These q-dimensions may exhibit phase transitions as q varies. We first consider general measures and then specialise to Bernoulli and Gibbs measures. Our method involves estimating expectations of moment expressions in terms of 'multienergy' integrals which we then bound using induction on families of trees.

Generalized Energy Inequalities and Higher Multifractal Moments

“…A further extension of the calculation establishes that ( ineq 3.3) remains valid for any real number q > 1, see Fa6, FX [9,11] for further details. In applications f (i) ≡ f s (i) typically depends on a parameter s such that…”

“…The upper bound ( dqeq1 5.12) comes from splitting ellipses of the form that occur in the intersections in ( ellipses 5.4) into appropriate pieces and summing the powers of the measures, see Fa5,Fa6 [8,9]. For the case where q ≥ 2 is an integer and µ ω a random almost self-affine set, let j 1 , .…”

Asymptotic Order of the Geometric Mean Error for Self-Affine Measures on Bedford–mcmullen Carpets

“…Self-affine sets and self-affine measures have attracted great attention of mathematicians (cf. [1,2,11,14,17,19]) in the past decades, and related problems are often rather difficult. Let G y := Proj y G and ϑ := log m log n .…”

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

“…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].…”