2668. Michael Bersudsky, Nimish A. Shah, Hao Xing Equidistribution of polynomially bounded o-minimal curves in homogeneous
spaces
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Submission date: 6 July 2024
Abstract:
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
v_1,…,v_k in ℝ^n, φ(t).(v_1 ∧ … ∧
v_k) ⍆ 0 as t → ∞. Then, there exists a unique smallest subgroup
H_∞ of SL(n, ℝ) generated by unipotent one-parameter
subgroups such that φ(t)H_φ → g_0H_φ in
SL(n, ℝ)/H_φ as t → ∞ for some g_0 ∈
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
g_0 μ_{Lx} 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 μ_{Lx}.
A crucial new ingredient in this work is proving that for any
finite-dimensional representation V of SL(n, ℝ), there exist
T_0 > 0, C > 0, and α > 0 such that for any v ∈ G, the map t →
∥φ(t)v∥ is (C,α)-good on [T_0,∞).