Thermodynamic formalism of countably generated self-affine sets

TL;DR

This paper extends thermodynamic formalism to countably generated self-affine sets, establishing equilibrium states and relating affinity and Hausdorff dimensions in infinite systems.

Contribution

It generalizes the thermodynamic formalism for affine iterated function systems from finite to countably infinite transformations, providing new dimension results.

Findings

Equilibrium states characterized for countably infinite affine systems

Affinity dimension equals supremum of finite subsystem dimensions

Hausdorff dimension results for self-affine sets in various dimensions

Abstract

In this article, we further develop the thermodynamic formalism of affine iterated function systems with countably many transformations by showing the existence and extending earlier characterisations of the equilibrium states of finite affine iterated function systems to the countably infinite case. As an application, under mild conditions, we prove that the affinity dimension of a countable affine iterated function system is equal to the supremum of the affinity dimensions of its finite subsystems. We deduce corollaries concerning the Hausdorff dimension of countably generated self-affine sets in dimensions $1$, $2$, and $3$ satisfying mild deterministic assumptions and in arbitrary dimension with generic translations.