2024/07/06 by Bersudsky, Michael, Shah, Nimish A., Xing, Hao · 1 citation
#03C64 #37A17 #Dynamical Systems (math.DS) #FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.2407.04935
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,\mathbb 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 \mathbb Rn, φ(t).(v1\wedge⋯\wedge vk)\not→0 as t→∞. Then, there exists a unique smallest subgroup Hφ of SL(n,\mathbb R) generated by unipotent one-parameter subgroups such that φ(t)Hφ→ g0Hφ in SL(n,\mathbb R)/Hφ as t→∞ for some g0∈ SL(n,\mathbb R). Let G be a closed subgroup of SL(n,\mathbb R) and Γ be a lattice in G. Suppose that φ([0,∞))⊂ G. Then Hφ⊂ 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 Hx=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,\mathbb 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,∞).