arXiv 1406.72012014-06math.DS

正熵曲面上随机动力学的刚度度量

Measure rigidity for random dynamics on surfaces with positive entropy

Aaron W. Brown, Federico Rodriguez Hertz

备注：: The results in this paper are superseded by those in arXiv:1506.06826; this version will be kept as a preprint

摘要：

给定曲面$M$和$M$的$C^2$ -纯同态群上的Borel概率测度$ν$，研究了$M$上的$ν$ -平稳概率测度.在熵为正的假设下，证明了随机动力学的稳定分布是非随机的，或者测度是SRB。在$ν$ -a. e.双同态保持了一个共同的光滑测度$m$，我们表明，对于任何正熵平稳测度$μ$，要么存在一个$ν$ -几乎必然不变的$μ$ -可测线场（对应于几乎每一个随机组合物的稳定分布）或测度$μ$是$ν$ -几乎必然不变的，并与$m$的遍历分量相一致。为了证明上述结果，我们引入了一个带有增子$σ$的保保测变换上的带表面纤维的斜积。给定一个不变测度$μ$的斜积，并假设$\hat F$ -可测的“过去的动态”和纤维明智的条件措施，我们证明了二分法：要么纤维明智的稳定分布是可测的相关增加子$σ$ -代数，或措施$μ$是纤维明智的SRB。

英文摘要：

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.

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