On natural invariant measures on generalised iterated function systems

TL;DR

This paper investigates invariant measures on generalized iterated function systems, establishing the existence of equilibrium states under natural potentials and linking ergodic measures to the Hausdorff dimension of self-affine sets.

Contribution

It proves the existence of invariant probability measures satisfying equilibrium states for generalized IFS with natural potentials, connecting ergodic measures to set dimensions.

Findings

Existence of invariant measures satisfying equilibrium states.

Ergodic invariant measures share Hausdorff dimension with the set.

Applicability to typical self-affine sets.

Abstract

We consider the limit set of generalised iterated function systems. Under the assumption of a natural potential, the so called cylinder function, we prove the existence of the invariant probability measure satisfying the equilibrium state. We motivate this approach by showing that for typical self-affine sets there exists an ergodic invariant measure having the same Hausdorff dimension as the set itself.