In this talk, I will discuss my joint work with Michael Bersudsky and Hao Xing extending 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 a polynomially bounded o-minimal structure.
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:

∀ linearly independent v1,...,vk ∈ R^n, 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 → g_0 H_φ in SL(n, R)/H_φ as t→∞ for some g_0 ∈SL(n,R).

For any Lie subgroup G of SL(n,R) such that φ([0,∞)) ⊂ G, we have H_φ ⊂G. For any lattice Γ in G and x∈G/Γ, the trajectory { φ(t)x:t≥0} gets equidistributed with respect to the measure g_0.μ_F_x, where H_φ x = Fx for a closed connected subgroup F of G and μ_F_x is the unique F-invariant probability measure on Fx.