Equidistribution of polynomially bounded o-minimal curves in homogeneous spaces
Michael Bersudsky, Nimish A. Shah, Hao Xing
TL;DR
The paper extends Ratner-type equidistribution to trajectories of non-contracting curves definable in polynomially bounded o-minimal structures, by introducing the hull $H_ heta$ generated by unipotent one-parameter subgroups and a bounded correcting curve $eta$ so that $eta(t) heta(t)$ remains in $H_ heta$. It proves that for any lattice $ riangleleft$ and $x otin$ singular set, the averaging measures $ rac{1}{T} extstyle loat{0}^{T} ( heta(t)x) ext{dt}$ converge to the homogeneous probability on the orbit $Lx$, where $Lx=ar{H}_ heta x$ and $ar{H}_ heta$ is the Zariski-closure-related normalizer; the limit is described by the $L$-invariant probability on $Lx$. A key technical innovation is a polynomially bounded $(C,oldsymbol{ healpha})$-goodness result for the growth of $ heta(t)igr ext{·}v$ in any rational representation $V$, which yields non-escape of mass and enables linearization arguments. The work combines o-minimal geometry (including Peterzil–Steinhorn groups and definable curves), unipotent dynamics, and Ratner-type measure classification to obtain a broad equidistribution theorem, with explicit constructions where the hull is as large as possible. This broadens Ratner’s framework to a wide class of definable dynamical systems on homogeneous spaces, offering new tools for understanding asymptotic distribution of o-minimal trajectories in Lie groups and their homogeneous quotients.
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 $\varphi:[0,\infty)\to \text{SL}(n,\mathbb R)$ 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 $Γ$ be a lattice in $G$. Suppose that $\varphi([0,\infty))\subset G$. Then $H_\varphi\subset G$, and for any $x\in G/Γ$, the trajectory $\{\varphi(t)x:t\in [0,T]\}$ 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 $t\mapsto \|\varphi(t)v\|$ is $(C,α)$-good on $[T_0,\infty)$.