论文标题
使用Hecke操作员在一个地方使用正熵
Positive entropy using Hecke operators at a single place
论文作者
论文摘要
我们证明了以下语句：让$ x = \ text {sl} _n（\ Mathbb {z}）\ backslash \ text {sl} _n（\ Mathbb {r}）$，并考虑对角色组的标准操作$令$μ$为$ x $上的$ a $ a $ invariant概率度量，这是一个限制$$ $ =λ\ lim_i | ϕ_i | n |^2dx，其中$ ϕ_i $在某些固定位置$ p $ p $ p $和$λ> 0的$λ> 0 $ 0的$ ϕ_i $是标准化的hecke algebra functions。然后，任何常规元素$ a \ in $ ACT上的$μ$上都有几乎每个ergodic组件的正熵。我们还证明了来自$ \ mathbb {q} $的分区代数的晶格的结果类似，并为当地相关的对称空间得出了量子唯一的奇异性结果。这概括了Brooks和Lindenstrauss的结果。
We prove the following statement: Let $X=\text{SL}_n(\mathbb{Z})\backslash \text{SL}_n(\mathbb{R})$, and consider the standard action of the diagonal group $A<\text{SL}_n(\mathbb{R})$ on it. Let $μ$ be an $A$-invariant probability measure on $X$, which is a limit $$ μ=λ\lim_i|ϕ_i|^2dx, $$ where $ϕ_i$ are normalized eigenfunctions of the Hecke algebra at some fixed place $p$, and $λ>0$ is some positive constant. Then any regular element $a\in A$ acts on $μ$ with positive entropy on almost every ergodic component. We also prove a similar result for lattices coming from division algebras over $\mathbb{Q}$, and derive a quantum unique ergodicity result for the associated locally symmetric spaces. This generalizes a result of Brooks and Lindenstrauss.
