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Absolute continuity of stationary measures for random surface dynamics

Aaron Brown, Homin Lee, Davi Obata, Yuping Ruan

TL;DR

This work studies when stationary measures for random surface dynamics driven by diffeomorphisms are absolutely continuous with respect to Lebesgue. The authors introduce uniform expansion on average in the future and past (UEF/UEP) and develop a nonperturbative Lasota–Yorke framework using a ρ-norm on measures and admissible curve-supported components. They prove that, under UEF/UEP and bounded C^2 control, ergodic μ-stationary measures are either atomic or AC with Lebesgue density, and there is a finite (often unique) absolutely continuous stationary measure, along with orbit classification and equidistribution results. The approach blends curvature/density estimates, distortion bounds, good/convolution machinery, and transversality to obtain a spectral-gap-type control that converts SRB-type regularity into actual absolute continuity. The results extend known hyperbolic perturbative findings to dissipative settings and apply to non-linear perturbations of volume-preserving systems, with implications for rigidity and ergodic classification of random surface dynamics.

Abstract

We find conditions for stationary measures of random dynamical systems on surfaces having dissipative diffeomorphisms to be absolutely continuous. These conditions involve a uniformly expanding on average property in the future (UEF) and past (UEP). Our results can cover random dynamical systems generated by "very dissipative" diffeomorphisms and perturbations of volume preserving surface diffeomorphisms. For example, we can consider a random dynamical system on $\mathbb{T}^2$ generated by perturbations of a pair of non-commuting infinite order toral automorphisms with any arbitrary single diffeomorphism. In this case, we conclude that stationary measures are either atomic or absolutely continuous. We also obtain an orbit classification and equidistribution result.

Absolute continuity of stationary measures for random surface dynamics

TL;DR

This work studies when stationary measures for random surface dynamics driven by diffeomorphisms are absolutely continuous with respect to Lebesgue. The authors introduce uniform expansion on average in the future and past (UEF/UEP) and develop a nonperturbative Lasota–Yorke framework using a ρ-norm on measures and admissible curve-supported components. They prove that, under UEF/UEP and bounded C^2 control, ergodic μ-stationary measures are either atomic or AC with Lebesgue density, and there is a finite (often unique) absolutely continuous stationary measure, along with orbit classification and equidistribution results. The approach blends curvature/density estimates, distortion bounds, good/convolution machinery, and transversality to obtain a spectral-gap-type control that converts SRB-type regularity into actual absolute continuity. The results extend known hyperbolic perturbative findings to dissipative settings and apply to non-linear perturbations of volume-preserving systems, with implications for rigidity and ergodic classification of random surface dynamics.

Abstract

We find conditions for stationary measures of random dynamical systems on surfaces having dissipative diffeomorphisms to be absolutely continuous. These conditions involve a uniformly expanding on average property in the future (UEF) and past (UEP). Our results can cover random dynamical systems generated by "very dissipative" diffeomorphisms and perturbations of volume preserving surface diffeomorphisms. For example, we can consider a random dynamical system on generated by perturbations of a pair of non-commuting infinite order toral automorphisms with any arbitrary single diffeomorphism. In this case, we conclude that stationary measures are either atomic or absolutely continuous. We also obtain an orbit classification and equidistribution result.

Paper Structure

This paper contains 35 sections, 59 theorems, 356 equations.

Table of Contents

  1. Introduction
  2. Motivation and main results
  3. The uniformly hyperbolic case
  4. Examples
  5. Relation with other works
  6. Organization of this paper
  7. Preliminaries on random dynamical systems
  8. ${\rm Diff}^2 (S)$ and Borel probability measures on ${\rm Diff}^2 (S)$
  9. Skew extension and stationary measure
  10. Lyapunov exponents, stable manifolds and unstable manifolds
  11. Random SRB measures
  12. Technical assumptions, basic estimates and statement of \ref{['thm:maintechnicaltheorem']}
  13. Setting and the statement of the main technical theorem
  14. Proofs of Lemmas \ref{['lem.DK.trick']}, \ref{['lem:VAP']} and Corollary \ref{['cor.mostly expand/contract']}
  15. Some consequences of UEF and UEP
  16. ...and 20 more sections

Key Result

Theorem 1

For $A,B$ and ${\mu}$ as above, every ergodic $\mu$-stationary measure $\nu$ on ${\mathbb T}^2$ is either Haar (Lebesgue) measure or finitely supported.

Theorems & Definitions (138)

  • Theorem : MR2831114MR2340439
  • Corollary 1.1
  • Theorem A
  • Theorem B
  • Theorem C
  • Remark 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Example 1.7
  • ...and 128 more