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 $\phi :[0,\infty )\to \text{SL}(n,\mathbb{R})$ be a continuous map whose coordinate functions are definable in a polynomially bounded o-minimal structure; for example, rational functions. Suppose that $\phi$ is non-contracting; that is, for any linearly independent vectors $v_1,\dots ,v_k$ in ${\mathbb{R}}^n$, $\phi (t).(v_1\wedge \cdots \wedge v_k)\nrightarrow 0$ as $t\to \infty$. Then, there exists a unique smallest subgroup $H_{\phi }$ of $\text{SL}(n,\mathbb{R})$ generated by unipotent one-parameter subgroups such that $\phi (t)H_{\phi }\to g_0{H}_{\phi }$ in $\text{SL}(n,\mathbb{R})/H_{\phi }$ 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 $\Gamma$ be a lattice in $G$. Suppose that $\phi ([0,\infty ))\subset G$. Then $H_{\phi }\subset G$, and for any $x\in G/\Gamma$, the trajectory $\{\phi (t)x:t\in [0,T]\}$ gets equidistributed with respect to the measure $g_0{\mu }_{Lx}$ 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 ${\mu }_{Lx}$. 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 $\alpha >0$ such that for any $v\in G$, the map $t\mapsto \Vert \phi (t)v\Vert$ is $(C,\alpha )$-good on $[T_0,\infty )$.