Orbit-counting in non-hyperbolic dynamical systems
Orbit-counting in non-hyperbolic dynamical systems
复制标题
非双曲动力系统中的轨道计数
DOI：
10.1515/crelle.2007.056
复制
发表时间：
2005
影响因子：
0.9
通讯作者：
T. Ward
中科院分区：
文献类型：
--
作者：
G. Everest;R. Miles;S. Stevens;T. Ward
摘要 对于具有双曲行为的动力系统，素数定理和 Mertens 定理有众所周知的类似物。在这里，我们针对最简单的非双曲代数系统考虑同样的问题。轨道计数函数的渐近行为由相关紧群上的旋转控制，在简单的例子中，我们展示了轨道计数函数的无数不同的渐近增长率。默滕斯定理在这种情况下也成立，具有从非双曲本征方向的算术属性获得的显式有理主导系数。梅尔滕斯定理的动力学类比的证明使用了超越理论和狄利克雷特征。
Abstract There are well-known analogues of the prime number theorem and Mertens' Theorem for dynamical systems with hyperbolic behaviour. Here we consider the same question for the simplest non-hyperbolic algebraic systems. The asymptotic behaviour of the orbit-counting function is governed by a rotation on an associated compact group, and in simple examples we exhibit uncountably many different asymptotic growth rates for the orbit-counting function. Mertens' Theorem also holds in this setting, with an explicit rational leading coefficient obtained from arithmetic properties of the non-hyperbolic eigendirections. The proof of the dynamical analogue of Mertens' Theorem uses transcendence theory and Dirichlet characters.