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 non-contracting curves definable in polynomially bounded o-minimal structures. To be precise, let φ:[0,∞)→ SL(n, R) be a continuous map whose coordinate functions are definable in a polynomially bounded o-minimal structure; for example, rational functions. Suppose that φ is non-contracting; that is, for any linearly independent vectors v₁,…,vₖ in Rⁿ, φ(t).(v₁⋯ vₖ)→0 as t→∞. Then, there exists a unique smallest subgroup H_φ of SL(n, R) generated by unipotent one-parameter subgroups such that φ(t)H_φ→ g₀H_φ in SL(n, R)/H_φ as t→∞ for some g₀∈ SL(n, R). Let G be a closed subgroup of SL(n, R) and Γ be a lattice in G. Suppose that φ([0,∞))⊂ G. Then H_φ⊂ G, and for any x∈ G/Γ, the trajectory \φ(t)x:t∈ [0,T]\ gets equidistributed with respect to the measure g₀μLx as T→∞, where L is a closed subgroup of G such that Hx̄=Lx and $Lx$ admits a unique L-invariant probability measure, denoted by μLx. A crucial new ingredient in this work is proving 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∈ G, the map t↦ \|φ(t)v\| is (C,α)-good on [T₀,∞).