A variational principle relating self-affine measures to self-affine sets

A breakthrough result of B\'ar\'any, Hochman and Rapaport published in 2019 established that every self-affine measure on $\mathbb{R}^2$ satisfying certain mild non-degeneracy conditions has Hausdorff dimension equal to its Lyapunov dimension. In combination with a variational principle established earlier by Morris and Shmerkin this result implied as a corollary that the attractor of a planar affine iterated function system satisfying the same conditions necessarily has Hausdorff dimension equal to a value proposed by Falconer in 1988. In this article we extend the variational principle of Morris and Shmerkin from the planar context to the case of affine iterated function systems acting on $\mathbb{R}^d$. This allows a recent theorem of Rapaport on the dimensions of self-affine measures in $\mathbb{R}^3$ to be extended into a characterisation of the dimensions of the corresponding self-affine subsets of $\mathbb{R}^3$. At the core of the present work is an algebraic result concerned with finding large Zariski-dense Schottky semigroups inside a given finitely generated completely reducible semigroup of linear transformations.