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: 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 is non-contracting; that is, for any linearly independent vectors v₁, , vₖ in Rⁿ, (t). (v₁ vₖ) 0 as t. Then, there exists a unique smallest subgroup H_ of SL (n, R) generated by unipotent one-parameter subgroups such that (t) H_ g₀H_ in SL (n, R) /H_ as t for some g₀ SL (n, R). Let G be a closed subgroup of SL (n, R) and be a lattice in G. Suppose that ([0, ) ) G. Then H_ G, and for any x G/, the trajectory \ (t) x: t [0, T\ gets equidistributed with respect to the measure g₀₋ₗ as T, where L is a closed subgroup of G such that Hx=Lx and Lx admits a unique L-invariant probability measure, denoted by ₋ₗ. 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 >0 such that for any v G, the map t \| (t) v\| is (C, ) -good on [T₀, ).