  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 $\varphi:[0,\infty)\to \text{SL}(n,\mathbb R)$ be a
continuous map whose coordinate functions are definable in a polynomially
bounded o-minimal structure; for example, rational functions. Suppose that
$\varphi$ is non-contracting; that is, for any linearly independent vectors
$v_1,\ldots,v_k$ in $\mathbb R^n$, $\varphi(t).(v_1\wedge\cdots\wedge
v_k)\not\to0$ as $t\to\infty$. Then, there exists a unique smallest subgroup
$H_\varphi$ of $\text{SL}(n,\mathbb R)$ generated by unipotent one-parameter
subgroups such that $\varphi(t)H_\varphi\to g_0H_\varphi$ in
$\text{SL}(n,\mathbb R)/H_\varphi$ 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 $\varphi([0,\infty))\subset G$. Then
$H_\varphi\subset G$, and for any $x\in G/\Gamma$, the trajectory
$\{\varphi(t)x:t\in [0,T]\}$ 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 $t\mapsto
\|\varphi(t)v\|$ is $(C,\alpha)$-good on $[T_0,\infty)$.
