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Log-Hölder regularity of stationary measures

Grigorii Monakov

TL;DR

This work establishes log-Hölder regularity for stationary measures of random dynamical systems generated by Hölder or Lipschitz homeomorphisms under a finite logarithmic moment, assuming no common invariant measure. The authors develop an energy-method framework, introducing the singular energy $oldsymbol{ extE}_{oldsymbol{ ext{α}},oldsymbol{ extvarepsilon}}$ and its $L^2$-density analogue $ ilde{oldsymbol{ extE}}_{oldsymbol{ ext{α}},oldsymbol{ extvarepsilon}}$, and prove contraction of these energies under random dynamics. This contraction yields quantitative log-Hölder bounds for stationary measures and a uniform non-stationary contraction theorem over compact families of maps, with a Lipschitz-variant treated by analogous, adjusted estimates. The results have broad implications for random matrix products and spectral problems, where log-Hölder regularity plays a key role in limit theorems and regularity of invariant-type measures, including connections to the integrated density of states and related stochastic-deterministic interactions.

Abstract

We consider Lipschitz and Hölder continuous random dynamical systems defined by a distribution with a finite logarithmic moment. We prove that under suitable non-degeneracy conditions every stationary measure must be $\log$-Hölder continuous.

Log-Hölder regularity of stationary measures

TL;DR

This work establishes log-Hölder regularity for stationary measures of random dynamical systems generated by Hölder or Lipschitz homeomorphisms under a finite logarithmic moment, assuming no common invariant measure. The authors develop an energy-method framework, introducing the singular energy and its -density analogue , and prove contraction of these energies under random dynamics. This contraction yields quantitative log-Hölder bounds for stationary measures and a uniform non-stationary contraction theorem over compact families of maps, with a Lipschitz-variant treated by analogous, adjusted estimates. The results have broad implications for random matrix products and spectral problems, where log-Hölder regularity plays a key role in limit theorems and regularity of invariant-type measures, including connections to the integrated density of states and related stochastic-deterministic interactions.

Abstract

We consider Lipschitz and Hölder continuous random dynamical systems defined by a distribution with a finite logarithmic moment. We prove that under suitable non-degeneracy conditions every stationary measure must be -Hölder continuous.
Paper Structure (16 sections, 46 theorems, 220 equations)

This paper contains 16 sections, 46 theorems, 220 equations.

Table of Contents

  1. Introduction
  2. Outline of the main results
  3. Applications
  4. Main results
  5. Hölder continuous homeomorphisms
  6. Outline of the proof
  7. Choice of the functions $U_{\alpha, \varepsilon}$ and $\varphi_{\alpha, \varepsilon}$
  8. Properties of $U_{\alpha, \varepsilon}$
  9. Definition and properties of $\mathcal{E}_{\alpha, \varepsilon}$: behaviour under images and convolutions
  10. Definition of $\widetilde{\mathcal{E}}_{\alpha, \varepsilon}$ and comparison to $\mathcal{E}_{\alpha, \varepsilon}$
  11. Properties of $\theta_{\alpha, \varepsilon} (\nu)$ and $\widetilde{\mathcal{E}}_{\alpha, \varepsilon} (\nu)$ under Hölder homeomorphisms
  12. Contraction of $\widetilde{\mathcal{E}}_{\alpha, \varepsilon}$ under random dynamics
  13. Proof of Theorem \ref{['thm:mainHol-3']}
  14. Lipschitz homeomorphisms
  15. Technical lemmata
  16. ...and 1 more sections

Key Result

Theorem 1.1 .1

Let $M$ be a closed Riemannian manifold. Suppose that $\mu$ is a probability distribution on $\mathrm{Diff}^1(M)$ such that $\int \|f\|_{\mathrm{Diff}^1}^\gamma d\mu(f) < \infty$ for some $\gamma > 0$. Suppose also that there is no probability measure $m$ on the manifold $M$ invariant under every ma

Theorems & Definitions (82)

  • Theorem 1.1 .1: Gorodetski, Kleptsyn, M., GKM
  • Theorem 1.1 .2
  • Theorem 1.1 .3
  • Theorem 1.2 .1: Guivarc’h, Gu
  • Theorem 1.2 .2: Le Page, L, Guivarc'h, Raugi, GR, Gol'dsheı̆d, Margulis, GM
  • Theorem 1.2 .3: Benuist, Quint, BQ16
  • Definition 2.1.1
  • Definition 2.1.2
  • Theorem 2.1.3
  • Theorem 2.1.4
  • ...and 72 more