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**Invariant measures and arithmetic unique ergodicity. Appendix by E. Lindenstrauss and D. Rudolph.**
*(English)*
Zbl 1104.22015

The main result of this paper is part of a sequence of ‘measure rigidity’ results of the following general form: under certain additional conditions, probability measures which are invariant under some ‘algebraic’ group action are themselves ‘algebraic’.

The main result of the paper deals with the classification of probability measures \(\mu\) on homogeneous spaces of the form \(\Gamma\setminus G\), \(G= \text{SL}(2,\mathbb{R})\times L\), where \(L\) is a finite Cartesian product of algebraic groups over \(\mathbb{R}\), \(\mathbb{C}\) or \(\mathbb{Q}_p\) and \(\Gamma\subset G\) a discrete subgroup, which are invariant under the right action of the diagonal subgroup \(A\subset\text{SL}(2,\mathbb{R})\) and which satisfy a recurrence condition under \(L\). Under an entropy condition (that all \(A\)-ergodic components of \(\mu\) have positive entropies) it is shown that any such measure is a convex combination of probability measures which are invariant under, and carried by orbits of, closed subgroups of \(G\) containing \(\text{SL}(2,\mathbb{R})\).

This result has several applications, among them ‘quantum unique ergodicity’, the statement that the sequence of absolutely continuous probability measures whose densities are joint eigenfunctions of the Laplacian and the Hecke operators on a compact surface of the form \(M=\Gamma\setminus\mathbb{H}\), where \(\Gamma\) is a lattice in \(\text{SL}(2,\mathbb{R})\), converge weakly to the normalized invariant volume on \(M\). For noncompact \(M\) it is shown that (under appropriate conditions on \(\Gamma\)) any limit of these measures is a multiple of the invariant value on \(M\).

Another application of this result assumes that \(G=\text{SL}(2,\mathbb{R})\times\text{SL}(2,\mathbb{R})\), \(H\subset G\) is isomorphic to \(\text{SL}(2,\mathbb{R})\), and \(\Gamma\subset G\) a discrete subgroup whose kernels under the two coordinate projections are finite. Then every probability measure \(\mu\) on \(\Gamma\setminus G\) which is invariant under the Cartesian product \(B\) of the two diagonal subgroups of \(\text{SL}(2,\mathbb{R})\) in \(G\) is either algebraic or has zero entropy under every one-parameter subgroup of \(B\).

The approach to measure rigidity developed in this paper, combined with earlier work by M. Einsiedler and A. Katok, has since led to significant progress on the Littlewood conjecture on simultaneous Diophantine approximation of pairs of real numbers [M. Einsiedler, A. Katok and E. Lindenstrauss, Ann. of Math. (2) 164, No. 2, 513–560 (2006; Zbl 1109.22004)].

The main result of the paper deals with the classification of probability measures \(\mu\) on homogeneous spaces of the form \(\Gamma\setminus G\), \(G= \text{SL}(2,\mathbb{R})\times L\), where \(L\) is a finite Cartesian product of algebraic groups over \(\mathbb{R}\), \(\mathbb{C}\) or \(\mathbb{Q}_p\) and \(\Gamma\subset G\) a discrete subgroup, which are invariant under the right action of the diagonal subgroup \(A\subset\text{SL}(2,\mathbb{R})\) and which satisfy a recurrence condition under \(L\). Under an entropy condition (that all \(A\)-ergodic components of \(\mu\) have positive entropies) it is shown that any such measure is a convex combination of probability measures which are invariant under, and carried by orbits of, closed subgroups of \(G\) containing \(\text{SL}(2,\mathbb{R})\).

This result has several applications, among them ‘quantum unique ergodicity’, the statement that the sequence of absolutely continuous probability measures whose densities are joint eigenfunctions of the Laplacian and the Hecke operators on a compact surface of the form \(M=\Gamma\setminus\mathbb{H}\), where \(\Gamma\) is a lattice in \(\text{SL}(2,\mathbb{R})\), converge weakly to the normalized invariant volume on \(M\). For noncompact \(M\) it is shown that (under appropriate conditions on \(\Gamma\)) any limit of these measures is a multiple of the invariant value on \(M\).

Another application of this result assumes that \(G=\text{SL}(2,\mathbb{R})\times\text{SL}(2,\mathbb{R})\), \(H\subset G\) is isomorphic to \(\text{SL}(2,\mathbb{R})\), and \(\Gamma\subset G\) a discrete subgroup whose kernels under the two coordinate projections are finite. Then every probability measure \(\mu\) on \(\Gamma\setminus G\) which is invariant under the Cartesian product \(B\) of the two diagonal subgroups of \(\text{SL}(2,\mathbb{R})\) in \(G\) is either algebraic or has zero entropy under every one-parameter subgroup of \(B\).

The approach to measure rigidity developed in this paper, combined with earlier work by M. Einsiedler and A. Katok, has since led to significant progress on the Littlewood conjecture on simultaneous Diophantine approximation of pairs of real numbers [M. Einsiedler, A. Katok and E. Lindenstrauss, Ann. of Math. (2) 164, No. 2, 513–560 (2006; Zbl 1109.22004)].

Reviewer: Harald Rindler (Wien)

### MSC:

22E40 | Discrete subgroups of Lie groups |

11F72 | Spectral theory; trace formulas (e.g., that of Selberg) |

37A45 | Relations of ergodic theory with number theory and harmonic analysis (MSC2010) |

37D40 | Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) |