[Paper Review] Pointwise equidistribution for one parameter diagonalizable group action on homogeneous space

Ronggang Shi|arXiv (Cornell University)|May 8, 2014

Mathematical Dynamics and Fractals25 references3 citations

TL;DR

This paper establishes pointwise equidistribution for one-parameter $<em>\mathrm{Ad}$-diagonalizable group actions on homogeneous spaces, proving that for almost every $u$ in a certain unstable horospherical subgroup $U$, the trajectory $\{g_t u x\}_{0 \leq t \leq T}$ becomes equidistributed with respect to the invariant probability measure $\mu_{\overline{Gx}}$ as $T \to \infty$. The key result shows that the Haar measure on the unstable horospherical subgroup $G^+$ is $(g_t, \mu_{\overline{Gx}})$-generic at every point $x \in L/\Gamma$, extending equidistribution results beyond unipotent flows.

ABSTRACT

Let $Γ$ be a lattice of a semisimple Lie group $L$. Suppose that one parameter Ad-diagonalizable subgroup $\{g_t\}$ of $L$ acts ergodically on $L/Γ$ with respect to the probability Haar measure $μ$. For certain proper subgroup $U$ of the unstable horospherical subgroup of $\{g_t\}$ we show that given $x\in L/Γ$ for almost every $u\in U$ the trajectory $\{g_tux: 0\le t\le T\}$ is uniformly distributed with respect to $μ$ as $T o \infty$.

Motivation & Objective

• To investigate pointwise equidistribution for one-parameter $\mathrm{Ad}$-diagonalizable group actions on homogeneous spaces, particularly when the invariant measure is singular to Haar measure.
• To determine whether the Haar measure on the unstable horospherical subgroup $G^+$ is $(g_t, \mu_{\overline{Gx}})$-generic at every point $x \in L/\Gamma$.
• To extend equidistribution results from unipotent flows to $\mathrm{Ad}$-diagonalizable flows in the setting of homogeneous dynamics.
• To establish genericity of trajectories under $g_t$-action for measures supported on $G^+$, even when singular to the Haar measure.

Proposed method

• Utilizes the $g_1$-expanding property of the unstable horospherical subgroup $G^+$, ensuring that the action of $g_t$ expands vectors in the Lie algebra $\mathfrak{g}^+$.
• Applies representation theory of $\mathfrak{sl}_2$-triples to construct a semisimple subalgebra $\mathfrak{g}_1 \subset \mathfrak{g}$ containing the generator $z$ of the one-parameter group $\{g_t\}$.
• Constructs an abelian subalgebra $\mathfrak{u} \subset \mathfrak{g}_1$ from root spaces corresponding to positive roots $\beta_i$ with $\beta_i + \beta_j \notin \Phi^+$, ensuring $\mathfrak{u}$ is abelian and $g_1$-expanding.
• Uses the fact that $\rho(z)v = av$ with $a > 0$ for nontrivial representations $\rho$, proving $V^{\mathfrak{u}} \subset V^+$, which confirms the $g_1$-expanding property of $U_a \leq G^+$.
• Applies the Birkhoff ergodic theorem in the context of homogeneous dynamics, showing that for $f \in C_c(L/\Gamma)$, the time average of $f(g_t u x)$ converges to $\int f \, d\mu_{\overline{Gx}}$ for $\mu_{G^+}$-a.e. $u$.
• Relies on the structure of real semisimple Lie groups, Cartan decomposition, and root space decomposition to analyze the dynamics of $g_t$-action on $L/\Gamma$.

Experimental results

Research questions

• RQ1 Is the Haar measure on the unstable horospherical subgroup $G^+$ $(g_t, \mu_{\overline{Gx}})$-generic at every point $x \in L/\Gamma$ for $\mathrm{Ad}$-diagonalizable one-parameter flows?
• RQ2 Can pointwise equidistribution be established for $\mathrm{Ad}$-diagonalizable flows when the invariant measure is not the full Haar measure?
• RQ3 Under what conditions does the trajectory $\{g_t u x\}_{0 \leq t \leq T}$ become equidistributed with respect to $\mu_{\overline{Gx}}$ as $T \to \infty$?
• RQ4 What structural properties of the Lie algebra and root system ensure that a subgroup $U \leq G^+$ is $g_1$-expanding?

Key findings

• The Haar measure on the unstable horospherical subgroup $G^+$ is $(g_t, \mu_{\overline{Gx}})$-generic at every point $x \in L/\Gamma$ under the assumptions of Theorem 1.1.
• For almost every $u \in G^+$, the trajectory $\{g_t u x\}_{0 \leq t \leq T}$ becomes equidistributed with respect to $\mu_{\overline{Gx}}$ as $T \to \infty$.
• The construction of a $g_1$-expanding abelian subgroup $U_a \leq G^+$ via $\mathfrak{sl}_2$-triples and root space decomposition ensures the necessary expansion for equidistribution.
• The key technical step is showing that $V^{\mathfrak{u}} \subset V^+$ via representation theory, which implies $U_a$ is $g_1$-expanding.
• The result holds under the condition that the projection of $g_1$ to each simple factor of $G$ is nontrivial, ensuring non-degenerate expansion.
• The method applies to the case $G = \mathrm{SL}_2(\mathbb{R}) \times \{1\} \subset \mathrm{SL}_3(\mathbb{R})$, $\Gamma = \mathrm{SL}_3(\mathbb{Z})$, where the result is new and nontrivial.