Patterson-Sullivan Measures and Quasi-Conformal Deformations
Patterson-Sullivan Measures and Quasi-Conformal Deformations
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Patterson-Sullivan 测量和准共形变形
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发表时间：
2005
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通讯作者：
Edward C. Taylor
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作者：
M. Bridgeman;Edward C. Taylor
本文将Kleinian群在线元空间上的遍历作用与群在无穷远球面上的共形作用联系起来。特别地，我们证明了对于一对几何同构的凸余紧Kleinian群，线元空间上的Patterson-Sullivan测度的长度与其推进长度之比在极限集的Hausdorff维数之比下有界.我们的主要技术来自遍历理论和Patterson-Sullivan理论。设Isom+（H）n ≥ 2是H_n的保向等距空间。众所周知，这种等距空间可以由紧集上的一致收敛所诱导的拓扑给出。Kleinian群Γ是Isom+（H）的离散子群。因此，r不连续地作用在Hn上，并且因为我们假设该作用是无挠的，所以商流形N = Hn/r是一个常曲率-1的完整黎曼流形。克莱因群Γ也是球面在无穷远处Sn−1 ∞的共形自同构的离散子群;这个作用将Sn −1 ∞划分为两个不相交的集合。正则集ΩΓ是Sn−1 ∞中最大的开集，Γ在其上不连续地作用，极限集LΓ是它的补集。在LΓ包含多于2个点的情况下，极限集被刻画为Sn−1 ∞的最小闭Γ-不变子集。定义极限集LΓ的凸船体CH（LΓ）为H_n的最小凸子集，使得所有两个极限点都在LΓ内的测地线都包含在CH（LΓ）中.我们可以取CH（LΓ）与Γ的商（用C（Γ）表示）;这是
In this paper we relate the ergodic action of a Kleinian group on the space of line elements to the conformal action of the group on the sphere at infinity. In particular, we show that for a pair of geometrically isomorphic convex co-compact Kleinian groups, the ratio of the length of the Patterson-Sullivan measure on line element space to the length of its push-forward is bounded below by the ratio of the Hausdorff dimensions of the limit sets. Our primary techniques come from ergodic theory and Patterson-Sullivan theory. 1 Basics and Statement of Results Let Isom+(H) n ≥ 2 be the space of orientation-preserving isometries of Hn. As is well-known, this space of isometries can be given the topology induced by uniform convergence on compact sets. A Kleinian group Γ is a discrete subgroup of Isom+(H). As such, Γ acts discontinuously on Hn, and because we make a standing assumption that the action is torsion-free, the quotient manifoldN = Hn/Γ is a complete Riemannian manifold of constant curvature −1. A Kleinian group Γ also acts as a discrete subgroup of conformal automorphisms of the sphere at infinity Sn−1 ∞ ; this action partitions S n−1 ∞ into two disjoint sets. The regular set ΩΓ is the largest open set in Sn−1 ∞ on which Γ acts properly discontinuously, and the limit set LΓ is its complement. In the case that LΓ contains more than 2 points, the limit set is characterized as being the smallest closed Γ-invariant subset of Sn−1 ∞ . Define the convex hull CH(LΓ) of the limit set LΓ to be the smallest convex subset of Hn so that all geodesics with both limit points in LΓ are contained in CH(LΓ). We can take the quotient of CH(LΓ) by Γ (denoted by C(Γ)); this is