Rigidity of measures—The high entropy case and non-commuting foliations

Abstract

Abstract. We consider invariant measures for partially hyperbolic, semisimple, higher rank actions on homogeneous spaces defined by products of real and p-adic Lie groups. In this paper we generalize our earlier work to establish measure rigidity in the high entropy case in that setting. We avoid any additional ergodicity-type assumptions but rely on, and extend the theory of conditional measures.

Diagonalizable flows on locally homogeneous spaces and number theory

“…This case has been treated in [17] using the methods developed by M. E. and A. Katok in [13] (to be precise, only v = ∞ is considered in [17] , but there are no difficulties in extending that treatment to Q v for any v). The proof of the more general Theorem 2.11 requires also the results in [20].…”

The work of Einsiedler, Katok and Lindenstrauss on the Littlewood conjecture

“…This is an example of the fact, noted in §1. 3, that purely analytic methods can often handle cases when the number of variables is sufficiently large relative to the degree.…”

“…We have seen (see (3)) that the Littlewood conjecture is (almost, with a constraint x 1 = 0) equivalent to the assertion that |P (x)| < ε is solvable in integers. 10 The set of H-invariant probability measures forms, clearly, a convex set in the space of all probability measures.…”

“…We saw that in the discussion preceding (10) that the failure of the Littlewood conjecture for a fixed pair (α, β) would correspond to the A + 3 -orbit of a certain L α,β ∈ L 3 being bounded. If (20) fails, indeed, there exists a set of lattices L α,β of box dimension ≥ 0.01 (say) whose A + 3 -orbits all remain within a fixed bounded set inside L 3 .…”

“…Let µ be an A 3 -invariant measure on L 3 . Then, for (k, ) = (i, j), (j, i) we have µ ij n k (t)x ∝ µ ij x , for µ k x -almost all t ∈ R and for µ-almost every x ∈ L 3 .…”