tailieunhanh - Đề tài " Invariant measures and arithmetic quantum unique ergodicity "

We classify measures on the locally homogeneous space Γ\ SL(2, R) × L which are invariant and have positive entropy under the diagonal subgroup of SL(2, R) and recurrent under L. This classification can be used to show arithmetic quantum unique ergodicity for compact arithmetic surfaces, and a similar but slightly weaker result for the finite volume case. Other applications are also presented. In the appendix, joint with D. Rudolph, we present a maximal ergodic theorem, related to a theorem of Hurewicz, which is used in theproof of the main result. . | Annals of Mathematics Invariant measures and arithmetic quantum unique ergodicity By Elon Lindenstrauss Annals of Mathematics 163 2006 165 219 Invariant measures and arithmetic quantum unique ergodicity By Elon Lindenstrauss Appendix with D. Rudolph Abstract We classify measures on the locally homogeneous space r SL 2 R X L which are invariant and have positive entropy under the diagonal subgroup of SL 2 R and recurrent under L. This classification can be used to show arithmetic quantum unique ergodicity for compact arithmetic surfaces and a similar but slightly weaker result for the finite volume case. Other applications are also presented. In the appendix joint with D. Rudolph we present a maximal ergodic theorem related to a theorem of Hurewicz which is used in theproof of the main result. 1. Introduction We recall that the group L is S-algebraic if it is a finite product of algebraic groups over R C or Qp where S stands for the set of fields that appear in this product. An S-algebraic homogeneous space is the quotient of an S-algebraic group by a compact subgroup. Let L be an S-algebraic group K a compact subgroup of L G SL 2 R X L and r a discrete subgroup of G for example r can be a lattice of G and consider the quotient X r G K. The diagonal subgroup 0 A M p e_tj t e M c SL 2 R acts on X by right translation. In this paper we wish to study probablilty measures p on X invariant under this action. Without further restrictions one does not expect any meaningful classification of such measures. For example one may take L SL 2 Qp K The author acknowledges support of NSF grant DMS-0196124. 166 ELON LINDENSTRAUSS SL 2 Zp and r the diagonal embedding of SL 2 Z p in G. As is well-known r G K SL 2 Z SL 2 R . Any A-invariant measure f on r G K is identified with an A-invariant measure f on SL 2 Z SL 2 R . The A-action on SL 2 Z SL 2 R is very well understood and in particular such measures f are in finite-to-one correspondence with shift invariant measures on a .