论文标题
bowen $ \ unicode {x2013} $系列编码和Zeta功能的零
The Bowen$\unicode{x2013}$Series coding and zeros of zeta functions
论文作者
论文摘要
我们讨论了经典的Bowen $ \ unicode {x2013} $系列编码，尤其是将其应用于与Geodesic Flow及其零相关的Zeta功能的应用。在恒定负曲率$ -1 $的紧凑表面的情况下，Selberg zeta函数向整个复杂平面的分析扩展是经典的，并且可以使用Selberg Trace公式来实现。但是，另一种动力学方法是使用bowen $ \ unicode {x2013} $串联编码在Infinity的边界上进行编码，以获得分段分析扩展的映射，可以从中使用相关传输操作员的属性从中获得Zeta函数的扩展。后一种方法的优点是，如果没有尖头，则它在无限面积表面的情况下也适用。对于这样的例子，零的位置更加神秘。但是，在特别简单的示例中，当我们采用适当的限制时，零有一个惊人的结构。我们将尝试对这一现象有所了解。 除了研究一对裤子外，较新的版本还包括对与对称的一口烟托里相关的Zeta功能的启发式分析；特别是，在其小零的位置上有许多纯数值结果。
We give a discussion of the classical Bowen$\unicode{x2013}$Series coding and, in particular, its application to the study of zeta functions associated to geodesic flows and their zeros. In the case of compact surfaces of constant negative curvature $-1$ the analytic extension of the Selberg zeta function to the entire complex plane is classical, and can be achieved using the Selberg trace formula. However, an alternative dynamical approach is to use the Bowen$\unicode{x2013}$Series coding on the boundary at infinity to obtain a piecewise analytic expanding map from which the extension of the zeta function can be obtained using properties of the associated transfer operator. This latter method has the advantage that it also applies in the case of infinite area surfaces provided they do not have cusps. For such examples the location of the zeros is somewhat more mysterious. However, in particularly simple examples there is a striking structure to the zeros when we take appropriate limits. We will try to give some insight into this phenomenon. The newer version, in addition to the study of pair of pants, also includes heuristic analysis of the zeta function associated to the geodesic flow on symmetric one-funneled tori; in particular, there is a number of pure numerical results on location of its small zeros.
