关于模为素数的二次动力系统周期点的个数
On the Number of Periodic Points of Quadratic Dynamical Systems Modulo a Prime
摘要：
2004年Vasiga和Shallit研究了模素数的两个离散二次映射的周期点的个数。他们发现的渐近行为总和的数量周期点的所有素数小于一些限制，假设扩展黎曼假设。同年晚些时候，周和Shparlinski证明了这一渐近结果，而无需假设任何未经证实的假设。受此启发，我们进行实验，并发现一个惊人的模式的行为的总和的周期点的二次映射的其他两个特定的研究。从模拟看来，这种类型的所有二次映射的周期点的数目之和表现相同。最后，我们发现，数值上的周期点的数量分布似乎是瑞利。
英文摘要：
In 2004 Vasiga and Shallit studied the number of periodic points of two particular discrete quadratic maps modulo prime numbers. They found the asymptotic behaviour of the sum of the number of periodic points for all primes less than some bound, assuming the Extended Riemann Hypothesis. Later that same year Chou and Shparlinski proved this asymptotic result without assuming any unproven hypotheses. Inspired by this we perform experiments and find a striking pattern in the behaviour of the sum of the number of periodic points for quadratic maps other than the two particular ones studied previously. From simulations it appears that the sum of the number of periodic points of all quadratic maps of this type behave the same. Finally we find that numerically the distribution of the amounts of periodic points seems to be Rayleigh.
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