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Frostman dimension of Furstenberg measure for $\mathrm{SL}(2,\mathbb{R})$ random matrix products

Tom Rush

TL;DR

This work advances the understanding of the Furstenberg measure for random matrix products in $\mathrm{SL}(2,\mathbb{R})$ by deriving an explicit formula for its Frostman dimension under the SIP hypothesis and by delineating a left neighborhood of $0$ where Le Page-type transfer operators $P_t$ exhibit a spectral gap on Hölder spaces. The authors develop a comprehensive transfer-operator framework, introducing precise rate-quantities $I_{\gamma}(\alpha)$ and pressure functions $k(t),k^+(t),k^-(t)$, and then prove quasi-compactness and a spectral decomposition for $P_t$ in a regime $t\in(t_c',0]$, with a leading eigenfunction expressed via a dual eigenmeasure. The Frostman dimension is linked to Hölder regularity through Riesz potentials, yielding $\dim_F \nu_F=\zeta$ where $\zeta$ solves $\sup_{\alpha} \{ {}^*I_{2}(\alpha)+2\zeta\alpha\}=0$, and demonstrating that $\dim_F{}^{*}\nu_t=-t+\zeta_t$ with $\zeta_t$ aligning with $\zeta_t'$. The results unify dimension theory, spectral gap, and thermodynamic-formalism ideas in the SL$(2,\mathbb{R})$ SIP setting and pave the way for explicit large-deviation and Fourier-decay consequences for Furstenberg measures.

Abstract

For compactly supported $μ\in \mathcal{P}(\mathrm{SL}(2,\mathbb{R}))$ satisfying strong irreducibility and proximality, we obtain a formula for the Frostman dimension of the associated Furstenberg measure. We also describe the left neighbourhood of 0 for which the classical transfer operators defined by Le Page have a spectral gap on Hölder spaces in this setting.

Frostman dimension of Furstenberg measure for $\mathrm{SL}(2,\mathbb{R})$ random matrix products

TL;DR

This work advances the understanding of the Furstenberg measure for random matrix products in by deriving an explicit formula for its Frostman dimension under the SIP hypothesis and by delineating a left neighborhood of where Le Page-type transfer operators exhibit a spectral gap on Hölder spaces. The authors develop a comprehensive transfer-operator framework, introducing precise rate-quantities and pressure functions , and then prove quasi-compactness and a spectral decomposition for in a regime , with a leading eigenfunction expressed via a dual eigenmeasure. The Frostman dimension is linked to Hölder regularity through Riesz potentials, yielding where solves , and demonstrating that with aligning with . The results unify dimension theory, spectral gap, and thermodynamic-formalism ideas in the SL SIP setting and pave the way for explicit large-deviation and Fourier-decay consequences for Furstenberg measures.

Abstract

For compactly supported satisfying strong irreducibility and proximality, we obtain a formula for the Frostman dimension of the associated Furstenberg measure. We also describe the left neighbourhood of 0 for which the classical transfer operators defined by Le Page have a spectral gap on Hölder spaces in this setting.
Paper Structure (18 sections, 72 theorems, 353 equations)

This paper contains 18 sections, 72 theorems, 353 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Sketch of the proof
  4. Preliminaries
  5. The quantities $I_{\gamma}(\alpha)$
  6. Expansion bounds
  7. The functions $k(t)$, $k^+(t)$, and $k^-(t)$
  8. The spectral gap
  9. Introducing $t_c'$ and $\zeta_t'$
  10. Quasi-compactness
  11. Proof of the spectral gap
  12. The leading eigenfunction
  13. Concluding the proof of the main theorems
  14. Estimates for $I_{\gamma}(\alpha)$
  15. Analyticity of $k(t)$ on $(t_c,0)$
  16. ...and 3 more sections

Key Result

Theorem 1.5

The map $t \mapsto k(t)$ is analytic on $(t_c,0)$. Moreover, for all $t_c<t \leq 0$, there exists a unique probability measure $\nu_t \in \mathcal{P}(\mathbb{P})$ such that and a unique Hölder continuous function $h_t:\mathbb{P} \rightarrow \mathbb{R}$ satisfying $\nu_t(h_t)=1$ and For all $0<\zeta<\zeta_t$, $P_t:C^{\zeta}(\mathbb{P}) \rightarrow C^{\zeta}(\mathbb{P})$ has a spectral gap: where

Theorems & Definitions (136)

  • Conjecture 1.2
  • Conjecture 1.3
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 126 more