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Equidistribution of expanding translates of smooth curves in homogeneous spaces under the action of a product of SO(n,1)'s

Yubin Shin

TL;DR

The paper proves that expanding translates of a smooth curve segment in a product of hyperbolic groups $G={\rm SO}(n_1,1)\times\cdots\times{\rm SO}(n_k,1)$ acting on a finite-volume space $X=L/\Gamma$ become equidistributed with respect to the $L$-invariant measure $\mu_L$, provided the curve avoids a countable family of algebraic obstructions. The authors develop a shrinking-interval, Taylor-polynomial approximation framework to reduce the problem to polynomial curves, establish nondivergence and unipotent invariance of limit measures, and apply a refined linearization (Sha96 style) together with Kempf’s invariant theory to rule out concentration on lower-dimensional invariant sets. The obstructions in the closed-orbit case are shown to be Möbius-embedded products of subspheres, and in the non-closed case they arise from unstable orbit data; these obstructions form a countable set, enabling almost-everywhere avoidance. The result extends prior analytic and smooth-curve equidistribution results to the setting of a product of ${\rm SO}(n_i,1)$ factors and clarifies how algebraic obstructions govern the asymptotic distribution of expanding translates on homogeneous spaces.

Abstract

We study the limiting distributions of expanding translates of a compact segment of a smooth curve under a diagonal subgroup of $G=\mathrm{SO}(n_1,1)\times\cdots\times\mathrm{SO}(n_k,1)$, where $G$ acts on a finite volume homogeneous space $L/Γ$ as a subgroup. We show that the expanding translates of the curve become equidistributed in the orbit closure of $G$, provided that Lebesgue almost every point on the curve avoids a certain countable collection of algebraic obstructions. The proof involves Ratner's measure classification theorem, Kempf's geometric invariant theory, and the linearization technique.

Equidistribution of expanding translates of smooth curves in homogeneous spaces under the action of a product of SO(n,1)'s

TL;DR

The paper proves that expanding translates of a smooth curve segment in a product of hyperbolic groups acting on a finite-volume space become equidistributed with respect to the -invariant measure , provided the curve avoids a countable family of algebraic obstructions. The authors develop a shrinking-interval, Taylor-polynomial approximation framework to reduce the problem to polynomial curves, establish nondivergence and unipotent invariance of limit measures, and apply a refined linearization (Sha96 style) together with Kempf’s invariant theory to rule out concentration on lower-dimensional invariant sets. The obstructions in the closed-orbit case are shown to be Möbius-embedded products of subspheres, and in the non-closed case they arise from unstable orbit data; these obstructions form a countable set, enabling almost-everywhere avoidance. The result extends prior analytic and smooth-curve equidistribution results to the setting of a product of factors and clarifies how algebraic obstructions govern the asymptotic distribution of expanding translates on homogeneous spaces.

Abstract

We study the limiting distributions of expanding translates of a compact segment of a smooth curve under a diagonal subgroup of , where acts on a finite volume homogeneous space as a subgroup. We show that the expanding translates of the curve become equidistributed in the orbit closure of , provided that Lebesgue almost every point on the curve avoids a certain countable collection of algebraic obstructions. The proof involves Ratner's measure classification theorem, Kempf's geometric invariant theory, and the linearization technique.

Paper Structure

This paper contains 14 sections, 15 theorems, 156 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main result
  4. Paper Organization and Proof Outline
  5. Basic lemma for shrinking pieces of curves
  6. Nondivergence of limit measures and unipotent invariance
  7. Nondivergence of $\mu_{s_0,t}$
  8. Unipotent invariance of the limit measure
  9. Equidistribution of Limit Measures
  10. $Gp_H$ is closed.
  11. $Gp_H$ is not closed.
  12. Proof of Theorem \ref{['thm: 2. equidistribution of polyn approx']}
  13. proof of Theorem \ref{['Thm : 1. shrinking equidistribuiton at point']}
  14. Proof of Theorem \ref{['Theorem : 1. main theorem']}

Key Result

Theorem 1.1

Let $\psi=(\psi_i)_{i=1}^k : I \to G=\prod_{i=1}^k G_i$ be a curve such that $\mathcal{I}\circ \psi$ is a $C^l$-map for some $l >\frac{2\zeta_1}{\zeta_k}$ and $(\mathcal{I}_i\circ\psi_i)'(s) \neq 0$ for all $1\leq i\leq k$ and almost every $s\in I$. Suppose that Then, for every $f \in C_c(L/\Gamma)$, we have where $x_0=e\Gamma$, $\overline{Gx_0}=L/\Gamma$, and $\mu_L$ is the unique $L$-invariant

Theorems & Definitions (32)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.1
  • Theorem 1.2: Equidistribution of expanding translate of shirinking pieces of a curve
  • Theorem 2.1
  • Definition 2.1
  • Proposition 2.2
  • Remark 2.13
  • Lemma 2.3: ShahLYang, Lemma 4.1
  • Lemma 2.4: ShahPYang24, Lemma 3.5
  • ...and 22 more