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Thermodynamic formalism of countably generated self-affine sets

Antti Käenmäki, Ian D. Morris

TL;DR

The paper extends the thermodynamic formalism to affine IFS with a countable alphabet by analyzing the singular value pressure $P(\mathsf{A},s)$ and its equilibrium states, establishing existence, structure, and dependence on irreducibility. A crucial reduction to completely reducible matrix tuples allows the authors to transfer key properties to block-structured systems and connect pressure to finite subsystems, enabling approximation of the affinity dimension by the supremum of finite-subsystem dimensions. The work yields dimension-theoretic consequences for self-affine sets, including bounds and equalities for Hausdorff dimension in low dimensions and generic translations in higher dimensions, as well as a framework for dimension drop under removal of maps. Overall, the article builds a rigorous bridge between algebraic structure (via wedge powers and Zariski topology) and dynamical quantities (pressure, equilibrium states) to handle infinitely generated self-affine sets within the thermodynamic formalism, with explicit quasi-multiplicativity tools for counting arguments.

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.

Thermodynamic formalism of countably generated self-affine sets

TL;DR

The paper extends the thermodynamic formalism to affine IFS with a countable alphabet by analyzing the singular value pressure and its equilibrium states, establishing existence, structure, and dependence on irreducibility. A crucial reduction to completely reducible matrix tuples allows the authors to transfer key properties to block-structured systems and connect pressure to finite subsystems, enabling approximation of the affinity dimension by the supremum of finite-subsystem dimensions. The work yields dimension-theoretic consequences for self-affine sets, including bounds and equalities for Hausdorff dimension in low dimensions and generic translations in higher dimensions, as well as a framework for dimension drop under removal of maps. Overall, the article builds a rigorous bridge between algebraic structure (via wedge powers and Zariski topology) and dynamical quantities (pressure, equilibrium states) to handle infinitely generated self-affine sets within the thermodynamic formalism, with explicit quasi-multiplicativity tools for counting arguments.

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 , , and satisfying mild deterministic assumptions and in arbitrary dimension with generic translations.
Paper Structure (17 sections, 41 theorems, 205 equations)

This paper contains 17 sections, 41 theorems, 205 equations.

Table of Contents

  1. Introduction and statement of results
  2. Background
  3. Singular value pressure
  4. Equilibrium states
  5. Self-affine sets
  6. Existence and finiteness of pressure
  7. Preliminaries on energy and entropy
  8. Reduction to completely reducible matrices
  9. Conditional proofs of the results
  10. Behaviour of the pressure
  11. Description of the equilibrium states
  12. Dimension of infinitely generated self-affine sets
  13. Algebraic arguments: Proof of Theorem \ref{['thm:conc']}
  14. Linear algebraic groups and the Zariski topology
  15. Key results
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $\mathsf{A}=(A_i)_{i \in \mathcal{I}} \in \mathrm{GL}_d(\mathbb{R})^\mathcal{I}$, where $\mathcal{I}$ is either finite or countably infinite. Then the following four assertions hold:

Theorems & Definitions (73)

  • Theorem 1.1
  • Proposition 1.2
  • Proposition 1.3
  • Proposition 1.4
  • Theorem 1.5
  • Proposition 1.6
  • Proposition 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 63 more