Some ideas in the proof of Ratner ’ s Theorem

Unipotent flows are very well-behaved dynamical systems. In particular, Marina Ratner has shown that every invariant measure for such a flow is of a nice algebraic form. This talk will present some consequences of this important theorem, and explain a few of the ideas of the proof. In general, algebraic technicalities will be pushed to the background as much as possible. Eg. Let X = torus T = R/Z. Any v ∈ R defines a flow on T: φt(x) = x + tv If the slope of v is irrational, it is classical that Lebesgue is the only invariant measure. Example. v = (a, b, 0) defines a flow φt on T = R/Z and T × {0} is invariant. Lebesgue on T × {0} is an invariant measure. For every v, any ergodic invariant probability measure for φt on T is the Lebesgue measure on some invariant subtorus T (0 ≤ k ≤ 3). Note that R is a Lie group. I.e., it is a group (under vector addition) and a manifold, and the group operations are smooth. The subgroup Z is closed and discrete. The quotient space Z\R = T is a manifold. Let G be any Lie group, and let Γ be a closed, discrete subgroup. Then Γ\G is a manifold, a homogeneous space. It is best if Γ\G is compact. (More generally, we can allow Γ\G to have finite volume. We say Γ is a lattice.) Eg. G = SL2(R) = { 2 × 2 real mat’s of det 1 }. Let Γ = SL2(Z). Then X = Γ\G has finite vol. Other choices of Γ can make Γ\G cpct. Define u, a:R → SL2(R) by u = ( 1 t 0 1 ) and a = ( e 0 0 e−t ) . Each of these is a homomorphism. u is a unipotent one-parameter subgroup. Define φt(x) = xu. u = “horocycle flow” a = “geodesic flow” The volume on Γ\G is a finite, invariant measure.