Equidistribution of polynomially bounded o-minimal curves in homogeneous spaces

We extend Ratner’s theorem on equidistribution of individual orbits of unipotent flows on finite volume homogeneous spaces of Lie groups to trajectories of noncontracting curves definable in a polynomially bounded o-minimal structure. To be precise, let φ:[0,∞)→SL(n,R) be a continuous curve whose coordinate functions are definable in a polynomially bounded o-minimal structure; for example, rational functions. Suppose that φ is noncontracting; that is, for any linearly independent vectors v1,…,vk in Rn, φ(t)⋅(v1∧⋯∧vk)↛0 as t→∞. Then there exists a unique closed connected subgroup Hφ⊆SL(n,R) of smallest dimension such that Hφ is generated by unipotent one-parameter subgroups and such that φ(t)Hφ→g0Hφ in SL(n,R)∕Hφ as t→∞, for some g0∈SL(n,R). Let G⊆SL(n,R) be a closed subgroup, and let Γ⊆G be a lattice. Suppose that φ([0,∞))⊆G. Then Hφ⊆G, and for any x∈G∕Γ, the trajectory {φ(t)x:t∈[0,T]} is equidistributed with respect to the measure g0μLx as T→∞, where L⊆G is a closed subgroup such that Hφx‾=Lx and Lx admits a unique L-invariant probability measure, denoted by μLx. A crucial new ingredient is the proof that for any finite-dimensional representation V of SL(n,R), there exist T0>0, C>0, and α>0 such that for any v∈V, the map t↦‖φ(t)v‖ is (C,α)-good on [T0,∞).