S-integer dynamical systems: periodic points.
S-integer dynamical systems: periodic points.
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S-整数动力系统：周期点。
DOI：
10.1515/crll.1997.489.99
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发表时间：
1997
期刊：
影响因子：
--
通讯作者：
V. Chothi
中科院分区：
文献类型：
--
作者：
T. Ward;G. Everest;V. Chothi
我们通过对偶性将一个动力系统与每一对（RS,x）联系起来，其中RS是a域k中的s整数环，x是RS\{0}中的一个元素。这些动力系统包括圆加倍映射、某些螺线形和整体自同态、素数幂字母上的完全单边和双边移位，以及某些代数元胞自动机。在算术情况下，我们证明了S有限系统具有接近双曲系统的性质：周期点的增长率存在，周期点在哈尔测度上均匀分布。然而，动态ζ函数通常是非理性的。对于S∞，系统表现出广泛的行为范围。利用Heath-Brown关于Artin猜想的工作，我们展示了S是无限的，但周期点的上增长率为正的例子。
We associate via duality a dynamical system to each pair (RS,x), where RS is the ring of S-integers in an A-field k, and x is an element of RS\{0}. These dynamical systems include the circle doubling map, certain solenoidal and toral endomorphisms, full one- and two-sided shifts on prime power alphabets, and certain algebraic cellular automata. In the arithmetic case, we show that for S finite the systems have properties close to hyperbolic systems: the growth rate of periodic points exists and the periodic points are uniformly distributed with respect to Haar measure. The dynamical zeta function is in general irrational however. For S infinite the systems exhibit a wide range of behaviour. Using Heath-Brown's work on the Artin conjecture, we exhibit examples in which S is infinite but the upper growth rate of periodic points is positive.