Effective equidistribution of twisted horocycle flows and horocycle maps
Effective equidistribution of twisted horocycle flows and horocycle maps
复制标题
扭曲的周环流和周环图的有效均匀分布
DOI：
10.1007/s00039-016-0385-4
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发表时间：
2015
影响因子：
2.2
通讯作者：
James Tanis
中科院分区：
文献类型：
--
作者：
L. Flaminio;G. Forni;James Tanis
我们证明了双曲曲面的horocycle流的扭曲遍历平均的界，无论是在紧的和非紧的有限区域的情况下。从这些界限，我们得到有效的均匀分布的结果，单圈地图。作为我们的主要定理在紧情形中的应用，我们进一步改进了Venkatesh的结果，最近已经由Tanis和Vishe改进了，关于Shah和Margulis提出的经典horocycle流的稀疏等分布问题，并且在一般的非紧，有限区域情形中，我们证明了尖点形式的Fourier系数的界，其与全纯情形中最好的已知界相当，和伯恩斯坦和Reznikov在马斯（非全纯）的情况下。我们的方法是基于Sobolev估计的上同调方程的解决方案和扭曲horocycle流量的不变分布的缩放。
We prove bounds for twisted ergodic averages for horocycle flows of hyperbolic surfaces, both in the compact and in the non-compact finite area case. From these bounds we derive effective equidistribution results for horocycle maps. As an application of our main theorems in the compact case we further improve on a result of Venkatesh, recently already improved by Tanis and Vishe, on a sparse equidistribution problem for classical horocycle flows proposed by Shah and Margulis, and in the general non-compact, finite area case we prove bounds on Fourier coefficients of cusp forms which are comparable to the best known bounds of Good in the holomorphic case, and of Bernstein and Reznikov in the Maass (non-holomorphic) case. Our approach is based on Sobolev estimates for solutions of the cohomological equation and on scaling of invariant distributions for twisted horocycle flows.