三阶Lyness差分方程的动力学
Dynamics of the third order Lyness' difference equation
摘要：
本文研究了具有正初始条件的三阶Lyness常返式$x_{k+3}=(a+x_{k+1}+x_{k+2})/x_k,$的迭代，其中$a$也是正参数.已知对于$a=1$，由该递归生成的所有序列都是8周期的。证明了对于每个$a\ne1$，有无穷多个初始条件产生几乎所有偶数周期的周期序列，并且对于初始条件的满测度集，由递归产生的序列在$\R.$的一个或两个不相交的有界区间上稠密。最后证明了产生奇数周期的周期序列的初始条件集包含在余维1中代数簇（因此它具有零测度），并且对于$a$的值的开集，它也包含所有的奇数，除了其中的100个。
英文摘要：
This paper studies the iterates of the third order Lyness' recurrence $x_{k+3}=(a+x_{k+1}+x_{k+2})/x_k,$ with positive initial conditions, being $a$ also a positive parameter. It is known that for $a=1$ all the sequences generated by this recurrence are 8-periodic. We prove that for each $a\ne1$ there are infinitely many initial conditions giving rise to periodic sequences which have almost all the even periods and that for a full measure set of initial conditions the sequences generated by the recurrence are dense in either one or two disjoint bounded intervals of $\R.$ Finally we show that the set of initial conditions giving rise to periodic sequences of odd period is contained in a codimension one algebraic variety (so it has zero measure) and that for an open set of values of $a$ it also contains all the odd numbers, except finitely many of them.
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