Paper of Ian Morris on fractal analysis appears in GAFA

The paper “Equilibrium states of generalised singular value potentials and applications to affine iterated function systems“, co-authored by Ian Morris and Jairo Bochi, has been accepted to Geometric and Functional Analysis (GAFA). It is the result of his visit to the Pontifical Catholic University of Chile, where Jairo works, in January 2017.  A link the arXiv version is here.

Self-similar fractals such as the classical Sierpiński gasket and Menger sponge have the property of being composed of exact rescaled copies of themselves, and their dimension theory has been understood for many decades.  Self-affine fractals have the property of being composed of linearly-distorted copies of themselves and have been investigated since the 1980s by notable mathematicians such as Kenneth Falconer and Curtis McMullen. Self-affine fractals can be thought of as models for repelling invariant sets in non-conformal dynamical systems. In 1988 Kenneth Falconer gave a formula for the Hausdorff dimension of a self-affine set which is known to be an upper bound, and is believed to be sharp except in certain highly degenerate cases.  To obtain a lower bound, one must construct a measure on the fractal with large Hausdorff dimension. In the early 2000s Antti Käenmäki gave a variational formula which these measures must satisfy. In the present paper Jairo and Ian completely describe the solutions to this variational formula. The paper extends the use of algebraic geometry which was introduced to this area by Antti Käenmäki and Ian in their recent PLMS paper Structure of equilibrium states on self-affine sets and strict monotonicity of affinity dimension.