Generalised dimensions of measures on almost self-affine sets

TL;DR

This paper derives a universal formula for the generalized q-dimensions of measures on almost self-affine sets, revealing phase transitions and applying to various measure types through a novel multienergy integral approach.

Contribution

It introduces a generic formula for q-dimensions of measures on almost self-affine sets, including phase transition analysis and specialized cases for Bernoulli and Gibbs measures.

Findings

Derived a universal formula for q-dimensions.

Identified phase transitions as q varies.

Applied method to Bernoulli and Gibbs measures.

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.