While on the one hand, chaotic dynamical systems can be predicted for all time given exact knowledge of an initial state, they are also in many cases rapidly mixing, meaning that smooth probabilistic information (quantified by measures) on the system’s state has negligible value for predicting the long-term future. However, an understanding of the long-term predictive value of intermediate kinds of probabilistic information is necessary in various physical problems, and largely remains lacking. Of particular interest in data assimilation and linear response theory are the conditional measures of the Sinai–Ruelle–Bowen (SRB) measure on zero sets of general smooth functions of the phase space. In this paper we give rigorous and numerical evidence that such measures generically converge back under the dynamics to the full SRB measures, exponentially quickly. We call this property conditional mixing. While conditional mixing typically cannot be proven from standard transfer operator theory, we will prove that conditional mixing holds in a class of generalized baker’s maps, and demonstrate it numerically in some non-Markovian piecewise hyperbolic maps. Conditional mixing provides a natural limit on the effectiveness of long-term forecasting of chaotic systems via partial observations, and appears key to proving the existence of linear response outside the setting of smooth uniform hyperbolicity.

中文翻译：

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确定性混沌中的条件混合

虽然一方面，在给定初始状态的准确知识的情况下，可以随时预测混沌动力系统，但它们在许多情况下也会快速混合，这意味着系统状态的平滑概率信息（通过测量量化）对于系统状态的价值可以忽略不计。预测长期的未来。然而，在各种物理问题中，对中间类型概率信息的长期预测价值的理解是必要的，但在很大程度上仍然缺乏。数据同化和线性响应理论特别感兴趣的是相空间一般平滑函数零集的 Sinai-Ruelle-Bowen (SRB) 测度的条件测度。在本文中，我们提供了严格的数字证据，表明这些措施在动态下通常会收敛到完整的 SRB 措施，呈指数级快速增长。我们将此属性称为条件混合。虽然条件混合通常无法从标准传递算子理论中得到证明，但我们将证明条件混合在一类广义面包师映射中成立，并在一些非马尔可夫分段双曲映射中以数值方式证明它。条件混合为通过部分观测对混沌系统进行长期预测的有效性提供了自然限制，并且似乎是证明平滑均匀双曲性设置之外线性响应存在的关键。并在一些非马尔可夫分段双曲图中用数值证明了它。条件混合为通过部分观测对混沌系统进行长期预测的有效性提供了自然限制，并且似乎是证明平滑均匀双曲性设置之外线性响应存在的关键。并在一些非马尔可夫分段双曲图中用数值证明了它。条件混合为通过部分观测对混沌系统进行长期预测的有效性提供了自然限制，并且似乎是证明平滑均匀双曲性设置之外线性响应存在的关键。