Horospherically invariant measures and finitely generated Kleinian groups

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ \Gamma &lt; {\rm{PSL}}_2( \mathbb{C}) $\end{document}</tex-math></inline-formula> be a Zariski dense finitely generated Kleinian group. We show all Radon measures on <inline-formula><tex-math id="M2">\begin{document}$ {\rm{PSL}}_2( \mathbb{C}) / \Gamma $\end{document}</tex-math></inline-formula> which are ergodic and invariant under the action of the horospherical subgroup are either supported on a single closed horospherical orbit or quasi-invariant with respect to the geodesic frame flow and its centralizer. We do this by applying a result of Landesberg and Lindenstrauss [<xref ref-type="bibr" rid="b18">18</xref>] together with fundamental results in the theory of 3-manifolds, most notably the Tameness Theorem by Agol [<xref ref-type="bibr" rid="b2">2</xref>] and Calegari-Gabai [<xref ref-type="bibr" rid="b10">10</xref>].</p>

Plant Layout Optimization for Chemical Industry Considering Inner Frame Structure Design

Plant layout design is a complex task requiring a wealth of engineering experience. A well-designed layout can extraordinarily reduce various costs, so layout study is of great value. To promote the research depth, plenty of considerations have been taken. However, an actual plant may have several frames and how to distribute facilities and determine the location of them in the different frames has not been well studied. In this work, frames are set as a special kind of inner structure and are added into the model to assign facilities into several blocks. A quantitative method for assigning facilities is proposed to let the number of cross-frame connections be minimized. After allocating the facilities into several blocks, each frame is optimized to obtain initial frame results. With designer decisions and cross-frame flow information, the relative locations of frames are determined and then the internal frame layouts are optimized again to reach the coupling optimization between frame and plant layout. Minimizing the total cost involving investment and operating costs is set to be the objective. In the case study, a plant with 138 facilities and 247 material connections is studied. All the facilities are assigned into four frames, and only 17 connections are left to be cross-frame ones. Through the two optimizations of each frame, the length of cross-frame connections reduces by 582.7 m, and the total cost decreases by 4.7 × 105 ¥/a. Through these steps, the idea of frame is successfully applied and the effectiveness of the proposed methodology is proved.

Local Mixing and Invariant Measures for Horospherical Subgroups on Abelian Covers

Abelian covers of hyperbolic three-manifolds are ubiquitous. We prove the local mixing theorem of the frame flow for abelian covers of closed hyperbolic three-manifolds. We obtain a classification theorem for measures invariant under the horospherical subgroup. We also describe applications to the prime geodesic theorem as well as to other counting and equidistribution problems. Our results are proved for any abelian cover of a homogeneous space Γ0∖G where G is a rank one simple Lie group and Γ0 &lt; G is a convex cocompact Zariski dense subgroup.

Equilibrium measures for certain isometric extensions of Anosov systems

We prove that for the frame flow on a negatively curved, closed manifold of odd dimension other than 7, and a Hölder continuous potential that is constant on fibers, there is a unique equilibrium measure. Brin and Gromov’s theorem on the ergodicity of frame flows follows as a corollary. Our methods also give a corresponding result for automorphisms of the Heisenberg manifold fibering over the torus.