On Radon Measures Invariant Under Horospherical Flows on Geometrically Infinite Quotients

Invariant Radon Measures for Horocycle Flows on Abelian Covers

We classify the ergodic invariant Radon measures for horocycle flows on Zd–covers of compact Riemannian surfaces of negative curvature, thus proving a conjecture of M. Babillot and F. Ledrappier. An important tool is a result in the ergodic theory of equivalence relations concerning the reduction of the range of a cocycle by the addition of a coboundary.

Measures Invariant under Horospherical Subgroups in Positive Characteristic

Measure rigidity for the action of maximal horospherical subgroups on homogeneous spaces over a field of positive characteristic is proved. In the case when the lattice is uniform we prove the action of any horospherical subgroup is uniquely ergodic.

On Measures Invariant under Tori on Quotients of Semi-simple Groups

We classify invariant and ergodic probability measures on arithmetic homogeneous quotients of semisimple S-algebraic groups invariant under a maximal split torus in at least one simple local factor, and show that the algebraic support of such a measure splits into the product of four homogeneous spaces: a torus, a homogeneous space on which the measure is (up to finite index) the Haar measure, ...

On Quasi-Invariant Transverse Measures for the Horospherical Foliation of a Negatively Curved Manifold

Nonexistence of Quasi-invariant Measures on Infinite-dimensional Linear Spaces1

It was shown by V. N. Sudakov [7] that a locally convex topological linear space with a nontrivial quasi-invariant <r-finite measure on its weakly measurable sets must be finite-dimensional. Sudakov's ingenious proof uses a theorem of Ulam [6, Footnote 3]. In some unpublished notes [8], Y. Umemara attempted to prove the same theorem, by utilizing Weil's "converse to the existence of Haar measur...