论文标题
lyapunov指数的规律性，以分裂为主导
Regularity of Lyapunov Exponents for Diffeomorphisms with Dominated Splitting
论文作者
论文摘要
我们认为具有主导分裂的差异性家庭并保留了Borel概率措施，我们研究了与不变捆绑包相关的Lyapunov指数的规律性。我们获得至少是两个不变束的规律性的总和（对于$ [0,1] $中的规律性），在适当的条件下，我们为衍生物获得公式。对于流量家族而言，也获得了类似的结果，并且对于不变度量取决于地图的情况。 我们还获得了几个应用程序。接近负曲率表面的大地测量流的一张图是相对于参数，体积的度量熵是Lipschitz。在恒定负曲率的歧管上的一张地图上，拓扑熵相对于参数是不同的，我们给出了衍生物的公式。在某些规律性条件下，Lyapunov指数函数的临界点是非灯的（对于某些家庭而言，第二个衍生物是非零的）。同样，在某些规律性条件下，Lyapunov指数函数的关键性意味着地图的某种刚性，因为体积沿两个免费叶子将产品分解为产品。特别是对于保存Anosov差异性的区域，唯一的关键点是与线性图相结合的地图，对应于全球极值。
We consider families of diffeomorphisms with dominated splittings and preserving a Borel probability measure, and we study the regularity of the Lyapunov exponents associated to the invariant bundles with respect to the parameter. We obtain that the regularity is at least the sum of the regularities of the two invariant bundles (for regularities in $[0,1]$), and under suitable conditions we obtain formulas for the derivatives. Similar results are obtained for families of flows, and for the case when the invariant measure depends on the map. We also obtain several applications. Near the time one map of a geodesic flow of a surface of negative curvature the metric entropy of the volume is Lipschitz with respect to the parameter. At the time one map of a geodesic flow on a manifold of constant negative curvature the topological entropy is differentiable with respect to the parameter, and we give a formula for the derivative. Under some regularity conditions, the critical points of the Lyapunov exponent function are non-flat (the second derivative is nonzero for some families). Also, again under some regularity conditions, the criticality of the Lyapunov exponent function implies some rigidity of the map, in the sense that the volume decomposes as a product along the two complimentary foliations. In particular for area preserving Anosov diffeomorphisms, the only critical points are the maps smoothly conjugated to the linear map, corresponding to the global extrema.
