论文标题
在均匀空间上可对角流动的尺寸下降
Dimension drop for diagonalizable flows on homogeneous spaces
论文作者
论文摘要
令$ x = g/γ$，其中$ g $是一个谎言组，$γ$是$ g $的晶格，让$ o $为$ x $的开放子集，让$ f = \ {g_t：t \ ge 0 \} $为$ g $ $ g $的单参数subsmigroop。考虑$ x $中的一组积分，其$ f $ -orbit会错过$ o $;如果流量为ergodic，它的测量为零。已经猜想这套集合的尺寸严格小于$ x $的尺寸。当$ x $紧凑或$ g $是一个真正的排名$ 1 $的谎言时，或者最近，对于晶格空间上的某些特殊流动时，就证明了这一猜想。在本文中，我们证明了对任意$ \ operatotorname {ad} $的猜想 - 在半圣母谎言组的不可减至的商上可对角线瓦的流动。该证明将流量的指数混合与$ g/γ$的高度功能的积分不等式的方法一起使用。我们还推导了一个线性形式的共同迪基莱特 - 不良系统的应用。
Let $X = G/Γ$, where $G$ is a Lie group and $Γ$ is a lattice in $G$, let $O$ be an open subset of $X$, and let $F = \{g_t: t\ge 0\}$ be a one-parameter subsemigroup of $G$. Consider the set of points in $X$ whose $F$-orbit misses $O$; it has measure zero if the flow is ergodic. It has been conjectured that this set has Hausdorff dimension strictly smaller than the dimension of $X$. This conjecture is proved when $X$ is compact or when $G$ is a simple Lie group of real rank $1$, or, most recently, for certain special flows on the space of lattices. In this paper we prove this conjecture for arbitrary $\operatorname{Ad}$-diagonalizable flows on irreducible quotients of semisimple Lie groups. The proof uses exponential mixing of the flow together with the method of integral inequalities for height functions on $G/Γ$. We also derive an application to jointly Dirichlet-Improvable systems of linear forms.
