Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Analyticity of the pressure function for products of matrices

Arnaud Hautecœur

TL;DR

This work proves that the pressure function $\mathbf{P}(s)$ for products of non-invertible matrices is real-analytic on the natural interval $I_\mu=(s_-,s_+)$ under weakened irreducibility (Irr) and proximality (Cont) assumptions, extending previous results for invertible cases. The authors construct Doob-relativised Markov operators $Q_s$ to obtain a quasi-compact spectral framework for the core operator $\Gamma_s$, establishing a spectral gap and enabling Kato perturbation theory to yield analyticity of $k(s)=e^{\mathbf P(s)}$ and thus of $\mathbf P(s)$. They also develop a Polish-alphabet variational principle, define equilibrium states $\mathbb Q^s$, and prove exponential decay of correlations, connecting spectral theory to thermodynamic formalism. The paper further provides counterexamples showing the necessity of the hypothesis and discusses the dynamical-thermodynamic implications, including a general variational principle and uniqueness of the equilibrium state. Overall, the results deepen our understanding of the regularity of generalized Lyapunov exponents and the absence of phase transitions in broad non-invertible settings with irreducibility.

Abstract

The pressure function is a fundamental object in various areas of mathematics. Its regularity is studied to derive insights into phase transitions in certain physical systems or to determine the Hausdorff dimension of self-affine sets. In this paper, we prove the analyticity of the pressure function for products of non-invertible matrices satisfying an irreducibility and a contractivity assumptions. Additionally, we establish a variational principle for the pressure function, thereby generalizing previous results.

Analyticity of the pressure function for products of matrices

TL;DR

This work proves that the pressure function for products of non-invertible matrices is real-analytic on the natural interval under weakened irreducibility (Irr) and proximality (Cont) assumptions, extending previous results for invertible cases. The authors construct Doob-relativised Markov operators to obtain a quasi-compact spectral framework for the core operator , establishing a spectral gap and enabling Kato perturbation theory to yield analyticity of and thus of . They also develop a Polish-alphabet variational principle, define equilibrium states , and prove exponential decay of correlations, connecting spectral theory to thermodynamic formalism. The paper further provides counterexamples showing the necessity of the hypothesis and discusses the dynamical-thermodynamic implications, including a general variational principle and uniqueness of the equilibrium state. Overall, the results deepen our understanding of the regularity of generalized Lyapunov exponents and the absence of phase transitions in broad non-invertible settings with irreducibility.

Abstract

The pressure function is a fundamental object in various areas of mathematics. Its regularity is studied to derive insights into phase transitions in certain physical systems or to determine the Hausdorff dimension of self-affine sets. In this paper, we prove the analyticity of the pressure function for products of non-invertible matrices satisfying an irreducibility and a contractivity assumptions. Additionally, we establish a variational principle for the pressure function, thereby generalizing previous results.
Paper Structure (26 sections, 41 theorems, 247 equations)

This paper contains 26 sections, 41 theorems, 247 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Assumptions
  4. The analyticity of the function $s \mapsto \mathbf{P}(s)$
  5. Notations and Sketch of proof
  6. Notations
  7. Sketch of proof of the analyticity
  8. Construction of Markov Operators $Q_s$
  9. $k(s)$ is an eigenvalue of $\Gamma_s$.
  10. Uniqueness of the invariant measure of the operators $Q_s$.
  11. Quasi-compactness of the operators $Q_s$
  12. The ${\mathbb Q}^s$-almost sure convergence of $a_2(\widetilde{W}_n)$ towards 0
  13. Properties of the function $J_s$
  14. Radon-Nykodim derivative
  15. The properties of $h_\alpha$
  16. ...and 11 more sections

Key Result

Theorem 2.1

Let $0<s_{-}<s_+$ and $\mu$ be a measure on $\mathrm M_d({\mathbb C})$ such that: Then, the pressure function $\mathbf{P}: I_\mu \to {\mathbb R}$ of $\mu$: is analytic on $(s_-,s_+)$.

Theorems & Definitions (75)

  • Theorem 2.1
  • Theorem 3.1
  • Theorem 3.2
  • Proposition 4.1
  • Lemma 4.2: Lemma 2.7 GLP15
  • Lemma 4.3
  • proof : Proof of Lemma \ref{['Lemma:ExistenceSigma']}.
  • Remark 4.4
  • Proposition 4.5
  • proof
  • ...and 65 more