论文标题
在大参数范围内采样的L-S-V地图的随机组成
Random Composition of L-S-V Maps Sampled Over Large Parameter Ranges
论文作者
论文摘要
Liverani-Saussol-Vaienti（L-S-V）地图形成了$ [0,1] $上的分段可区分动力系统系列，具体取决于一个参数$ω\ in \ Mathbb r^+$。这些地图无处不在，除了中立的固定点。众所周知，根据接近中性点的膨胀量，它们具有绝对连续的不变概率度量和相关性的多项式衰减（$ω<1 $），或者是一个独特的物理措施，在中性点（$ω> 1 $）都具有单数且集中在中性点上。在本文中，我们研究了L-S-V地图的组成，其参数是从$ \ Mathbb r^+$中的范围内随机采样的，以及这两个对比行为混合在一起。我们表明，如果参数$ω<1 $以正概率采样，那么随机系统的固定度量绝对是连续的。将退火的相关速率接近（或在某些情况下）等于采样系统的衰减速率最快。合适的伯克霍夫（Birkhoff）平均汇集以限制法律。与以前的研究相反，在[0,1] $中$ω\，我们允许$ω> 1 $在我们的采样分布中。我们还表明，当相对于一个光滑，重尾分布的家族进行采样时，可以获得$ω\ $ω\的相关率的类似衰减。
Liverani-Saussol-Vaienti (L-S-V) maps form a family of piecewise differentiable dynamical systems on $[0,1]$ depending on one parameter $ω\in\mathbb R^+$. These maps are everywhere expanding apart from a neutral fixed point. It is well known that depending on the amount of expansion close to the neutral point, they have either an absolutely continuous invariant probability measure and polynomial decay of correlations ($ω<1$), or a unique physical measure that is singular and concentrated at the neutral point ($ω>1$). In this paper, we study the composition of L-S-V maps whose parameters are randomly sampled from a range in $\mathbb R^+$, and where these two contrasting behaviours are mixed. We show that if the parameters $ω<1$ are sampled with positive probability, then the stationary measure of the random system is absolutely continuous; the annealed decay rate of correlations is close (or in some cases equal) to the fastest rate of decay among those of the sampled systems; and suitably rescaled Birkhoff averages converge to limit laws. In contrast to previous studies where $ω\in [0,1]$, we allow $ ω>1$ in our sampling distribution. We also show that one can obtain similar decay of correlation rates for $ω\in [0,\infty)$, when sampling is done with respect to a family of smooth, heavy-tailed distributions.
