Equidistribution of polynomially bounded o-minimal curves in homogeneous spaces

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 $φ:[0,\infty)\to \text{SL}(n,\mathbb R)$ 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 $v_1,\ldots,v_k$ in $\mathbb R^n$, $φ(t).(v_1\wedge\cdots\wedge v_k)\not\to0$ as $t\to\infty$. Then, there exists a unique smallest subgroup $H_φ$ of $\text{SL}(n,\mathbb R)$ generated by unipotent one-parameter subgroups such that $φ(t)H_φ\to g_0H_φ$ in $\text{SL}(n,\mathbb R)/H_φ$ as $t\to\infty$ for some $g_0\in \text{SL}(n,\mathbb R)$. Let $G$ be a closed subgroup of $\text{SL}(n,\mathbb R)$ and $Γ$ be a lattice in $G$. Suppose that $φ([0,\infty))\subset G$. Then $H_φ\subset G$, and for any $x\in G/Γ$, the trajectory $\{φ(t)x:t\in [0,T]\}$ gets equidistributed with respect to the measure $g_0μ_{Lx}$ as $T\to\infty$, where $L$ is a closed subgroup of $G$ such that $\overline{Hx}=Lx$ 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 $\text{SL}(n,\mathbb R)$, there exist $T_0>0$, $C>0$, and $α>0$ such that for any $v\in G$, the map $t\mapsto \|φ(t)v\|$ is $(C,α)$-good on $[T_0,\infty)$.