Infinite Non-Conformal Iterated Function Systems

TL;DR

This paper extends the theory of self-affine iterated function systems to infinite, non-conformal cases, establishing a formula for Hausdorff dimension and analyzing multifractal properties.

Contribution

It introduces a framework for infinite non-conformal IFS, deriving the Hausdorff dimension as a supremum over ergodic measures and providing a multifractal analysis with a variational principle.

Findings

Hausdorff dimension equals the supremum of ergodic measure dimensions

Established a conditional variational principle for multifractal level sets

Extended self-affine IFS theory to countably infinite, non-conformal systems

Abstract

We consider a generalisation of the self-affine iterated function systems of Lalley and Gatzouras by allowing for a countable infinity of non-conformal contractions. It is shown that the Hausdorff dimension of the limit set is equal to the supremum of the dimensions of compactly supported ergodic measures. In addition we consider the multifractal analysis for countable families of potentials. We obtain a conditional variational principle for the level sets.