扩张叶状结构上的Margulis测度：构造与刚性
Margulis Measures on Expanding Foliations: Construction and Rigidity
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中文总结 AI 辅助
研究在保持特定叶状结构的微分同胚下，构造参考测度并证明最大\(u\)-熵测度相关性质，还给出对数雅可比行列式与常数上同调的结论及在阿诺索夫微分同胚等方面的应用。
中文摘要 AI 辅助
给定一个保持具有齐次指数增长的一维扩张叶状结构\(\mathcal F\)的微分同胚，我们在叶状结构的每一叶上构造一族具有可控雅可比行列式和吉布斯性质的参考测度。然后证明对于任何最大\(u\)-熵测度，其在每一叶上的条件测度必定与参考测度等价。当最大\(u\)-熵测度是吉布斯\(\mathcal F\)-态时，我们证明\(f\)的对数雅可比行列式必定通过一个可测函数与一个常数上同调。我们给出了几个应用，包括阿诺索夫微分同胚的强叶状结构和中心叶状结构、阿诺索夫微分同胚的商、以及负曲率曲面上测地线流的时间-1映射的扰动。
英文摘要
Given a diffeomorphism preserving a one-dimensional expanding foliation $\mathcal F$ with homogeneous exponential growth, we construct a family of reference measures on each leaf of the foliation with controlled Jacobian and a Gibbs property. We then prove that for any measure of maximal $u$-entropy, its conditional measures on each leaf must be equivalent to the reference measures. When the measure of maximal $u$-entropy is a Gibbs $\mathcal F$-state (i.e., when the reference measures are equivalent to the leafwise Lebesgue measure), we prove that the log-Jacobian of $f$ must be cohomologous to a constant via a measurable function. We provide several applications, including the strong and center foliations of Anosov diffeomorphisms, factor over Anosov diffeomorphisms, and perturbations of the time-one map of geodesic flows on surfaces with negative curvature.
