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Measure rigidity for generalized u-Gibbs states and stationary measures via the factorization method

Aaron Brown, Alex Eskin, Simion Filip, Federico Rodriguez Hertz

TL;DR

This work establishes measure rigidity results for stationary measures arising from random walks generated by diffeomorphisms and for SL(2,\mathbb{R})-actions on smooth manifolds, introducing generalized $u$-Gibbs states via normal forms and subresonant holonomies. The authors prove a central inductive step, valid for a single map or flow, which yields extra invariance under compatible subgroup structures and a detailed classification of invariant/stationary measures in both random and Teichmüller settings. Their framework encompasses products of moduli strata, extending Eskin–Mirzakhani–Mohammadi-type rigidity to product spaces and providing a unified mechanism (the QNI condition) for obtaining invariant conditional measures along stable/unstable foliations. The results hinge on a sophisticated blend of normal forms, holonomies, Lyapunov-adapted geometry, and entropy techniques for skew extensions, culminating in a comprehensive measure classification in SL(2,R) dynamics, random dynamics, and Teichmüller dynamics with implications for moduli spaces and affine invariant submanifolds.

Abstract

We obtain measure rigidity results for stationary measures of random walks generated by diffeomorphisms, and for actions of $\operatorname{SL}(2,\mathbb{R})$ on smooth manifolds. Our main technical result, from which the rest of the theorems are derived, applies also to the case of a single diffeomorphism or $1$-parameter flow and establishes extra invariance of a class of measures that we call ``generalized u-Gibbs states''.

Measure rigidity for generalized u-Gibbs states and stationary measures via the factorization method

TL;DR

This work establishes measure rigidity results for stationary measures arising from random walks generated by diffeomorphisms and for SL(2,\mathbb{R})-actions on smooth manifolds, introducing generalized -Gibbs states via normal forms and subresonant holonomies. The authors prove a central inductive step, valid for a single map or flow, which yields extra invariance under compatible subgroup structures and a detailed classification of invariant/stationary measures in both random and Teichmüller settings. Their framework encompasses products of moduli strata, extending Eskin–Mirzakhani–Mohammadi-type rigidity to product spaces and providing a unified mechanism (the QNI condition) for obtaining invariant conditional measures along stable/unstable foliations. The results hinge on a sophisticated blend of normal forms, holonomies, Lyapunov-adapted geometry, and entropy techniques for skew extensions, culminating in a comprehensive measure classification in SL(2,R) dynamics, random dynamics, and Teichmüller dynamics with implications for moduli spaces and affine invariant submanifolds.

Abstract

We obtain measure rigidity results for stationary measures of random walks generated by diffeomorphisms, and for actions of on smooth manifolds. Our main technical result, from which the rest of the theorems are derived, applies also to the case of a single diffeomorphism or -parameter flow and establishes extra invariance of a class of measures that we call ``generalized u-Gibbs states''.

Paper Structure

This paper contains 325 sections, 181 theorems, 874 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Random Dynamics
  3. SL(2,R) actions
  4. Teichmüller dynamics
  5. Products of strata
  6. Outline and General Remarks
  7. Previous results
  8. The role of normal forms
  9. Subgroups compatible with the measure
  10. QNI condition
  11. Some notational conventions
  12. Technical preliminaries
  13. Assumptions on the total space
  14. Notation
  15. Norms on jet bundles
  16. ...and 310 more sections

Key Result

Theorem 1.1.1

Suppose $\mu$ is volume preserving and satisfies uniform expansion in dimension $d$ for all $1 \le d < \dim(Q)$. Suppose also that either $Q$ is compact, or there exists an admissible Margulis function on $Q$ (see def:random:margulis:function). Then, any $\mu$-stationary measure is $\mu$-invariant.

Figures (4)

  • Figure 4.5.1: Left: The general arrangement of points in the argument. Right: More detailed notation. Points in a transparent gray blob are exponentially close in the parameter $\ell$, for some exponent that depends only on the Lyapunov spectrum, and constants that depend on the Lusin set.
  • Figure 6.2.1: Interpolating the curves.
  • Figure 7.1.1: The initial, and halfway points.
  • Figure 11.2.1:

Theorems & Definitions (421)

  • Definition 1
  • Definition 2: Uniform expansion
  • Theorem 1.1.1
  • Definition 3: Uniform expansion gaps
  • Remark 1: On uniform expansion and gaps
  • Theorem 1.1.2
  • Definition 4: $\mu$-invariant subbundle
  • Theorem 1.1.3
  • Conjecture 1
  • Definition 5: Nonnegative Laplacian
  • ...and 411 more