[Paper Review] Measure rigidity for random dynamics on surfaces with positive entropy

Aaron Brown, Federico Rodriguez Hertz|arXiv (Cornell University)|Jun 27, 2014

Mathematical Dynamics and Fractals8 references3 citations

TL;DR

This paper establishes a measure rigidity dichotomy for stationary measures in random dynamical systems on surfaces with positive entropy: either the stable distribution is non-random, or the measure is Sinai-Ruelle-Bowen (SRB). Under additional invariance assumptions, if the stationary measure has positive entropy and the stable distribution is non-random, then the measure coincides with an ergodic component of a smooth invariant measure. The results extend rigidity principles from deterministic to random settings using skew product dynamics and fiber-wise absolute continuity.

ABSTRACT

Given a surface $M$ and a Borel probability measure $ν$ on the group of $C^2$-diffeomorphisms of $M$, we study $ν$-stationary probability measures on $M$. Assuming the positivity of a certain entropy, the following dichotomy is proved: either the stable distributions for the random dynamics is non-random, or the measure is SRB. In the case that $ν$-a.e. diffeomorphism preserves a common smooth measure $m$, we show that for any positive-entropy stationary measure $μ$, either there exists a $ν$-almost surely invariant $μ$-measurable line field (corresponding do the stable distributions for almost every random composition) or the measure $μ$ is $ν$-almost surely invariant and coincides with an ergodic component of $m$. To prove the above result, we introduce a skew product with surface fibers over a measure preserving transformation equipped with an increasing sub-$σ$-algebra $\hat F$. Given an invariant measure $μ$ for the skew product, and assuming the $\hat F$-measurability of the `past dynamics' and the fiber-wise conditional measures, we prove a dichotomy: either the fiber-wise stable distributions are measurable with respect to a related increasing sub-$σ$-algebra, or the measure $μ$ is fiber-wise SRB.

Motivation & Objective

• To establish a dichotomy for stationary measures in random dynamical systems on surfaces with positive metric entropy.
• To determine conditions under which such measures are either SRB or correspond to non-random stable distributions.
• To extend measure rigidity results from deterministic to random settings, particularly when a smooth invariant measure exists.
• To analyze the structure of stationary measures via skew product dynamics and fiber-wise conditional measures.
• To prove that under positivity of entropy and non-random stable distributions, the stationary measure must be an ergodic component of a smooth invariant measure.

Proposed method

• Construct a skew product system over a measure-preserving transformation with an increasing sub-σ-algebra, modeling the random dynamics on surface fibers.
• Define an invariant measure μ for the skew product and analyze the measurability of fiber-wise stable distributions with respect to the increasing sub-σ-algebra.
• Use a modified exponential drift argument and tools from non-uniform hyperbolicity to control the growth of Lyapunov exponents.
• Apply transverse absolute continuity of holonomy maps along stable manifolds to relate the ergodic basin to Lebesgue measure on fibers.
• Leverage the pointwise ergodic theorem and separability of continuous functions to characterize the ergodic basin of the stationary measure.
• Show that if the stable distribution is non-random and entropy is positive, then the stationary measure must be SRB, and under invariance, it equals an ergodic component of the smooth invariant measure.

Experimental results

Research questions

• RQ1 Under what conditions is a stationary measure with positive entropy in a random surface dynamical system necessarily SRB or associated with a non-random stable distribution?
• RQ2 How does the existence of a common smooth invariant measure affect the structure of positive-entropy stationary measures?
• RQ3 What role does the measurability of stable distributions with respect to an increasing sub-σ-algebra play in determining the nature of the stationary measure?
• RQ4 Can the ergodic basin of a stationary measure be used to reconstruct the measure as an ergodic component of a smooth invariant measure?
• RQ5 To what extent do fiber-wise absolute continuity and holonomy maps in skew products imply measure rigidity in random dynamical systems?

Key findings

• If the stable distribution is non-random and the stationary measure has positive entropy, then the measure is SRB.
• When ν-a.e. diffeomorphism preserves a smooth measure m, and the stationary measure μ has positive entropy, then either the stable distribution is non-random (implying μ is SRB), or μ is ν-a.s. invariant and coincides with an ergodic component of m.
• The ergodic basin of the stationary measure has positive Lebesgue measure on typical fibers, implying the measure is supported on a full measure set in the fibered structure.
• The stationary measure μ is shown to be equal to an ergodic component m₀ of the smooth invariant measure m, under the assumption of non-random stable distributions and positive entropy.
• The proof relies on the absolute continuity of holonomy maps along stable manifolds in the skew product setting, which allows transfer of measure-theoretic properties from the base to the fibers.
• The final result establishes that μ = m₀, confirming that the only possible positive-entropy stationary measures under the given conditions are ergodic components of the smooth invariant measure.