Download or read book Nontrivial Time-changes of Unipotent Flows on Quotients of Lorentz Groups written by Siyuan Tang (Mathematician) and published by . This book was released on 2022 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: In a series of papers, Ratner proved a deep theorem asserting that the orbits of any one-parameter unipotent flows are well behaved: they are either closed, or are dense in some homogeneous space of finite volume. Moreover, the probability measure which is invariant and ergodic under the unipotent flow must be the Haar measure on some homogeneous space. In particular, we can conclude from Ratner's theorem that the measurable conjugacy between two unipotent flows is affine. A natural problem is, whether we can similarly obtain the rigidity of perturbations of unipotent flows. Perhaps the simplest perturbations are time-changes, that is the flows that move points along the same orbits, but with different speeds. The problem was first studied by Ratner, who focused on horocycle flows and realized that it is related to the cohomological equations. In this thesis, we show that there are certain cocompact lattices so that there are time-changes of unipotent flows on the quotient of Lorentz groups SO(n,1) that are not measurably conjugate to the unperturbed ones. Moreover, we show that these time-changes are disjoint from the unperturbed flows in the sense of Furstenberg. More precisely, in Chapter 2, we extend the works of Ratner to the case SO(n,1) via a detailed study on the centralizer of unipotent flows. Then in Chapter 3, combining a stronger version of the branching of the complementary and Flaminio-Forni argument, we are able to provide certain cocompact lattices and time-changes on the corresponding homogeneous spaces to reach our purpose. The method also helps us obtain an extension of the work of Dong, Kanigowski and Wei.