反演极限与Lauwerier吸引子(Ⅱ)

Inverse Limit and Lauwerier Attractor (Ⅱ)

对适当的参数,二次映射有一条吸引的周期轨道,并且其吸引集在单位闭区间上是稠密的.根据此性质,文中定义了Lauwerier映射的一个上半连续分解.在此分解上存在一个可分商空间,通过投影将二维的Lauwerier映射降为一维的二次映射,运用二次映射反演极限空间上的移位映射来研究Lauwerier映射的动力学性质.首先对二次映射进行几乎Markov分割,然后将每个分割区间扩张成相应的小矩形区域,再对Lauwerier映射进行几乎Markov分割后,从而证明了当参数小于4时,Lauwerier映射与二次映射反演极限空间上的移位映射是拓扑半共轭的.

The quadratic mapping had an attracting periodic orbit of which the attraction set was dense in a unit closed interval for an appropriate parameter.According to that property,an upper semi-continuous decomposition of the Lauwerier mapping was defined,with respect to which there existed a separable quotient space.The 2D Lauwerier mapping was reduced to a 1D quadratic mapping through projection.The dynamic properties of the Lauwerier mapping was studied with the shift map on the inverse limit space of the quadratic mapping.First,the quadratic mapping was nearly Markov partitioned,then each partition interval was expanded to a corresponding small rectangular region,in turn the Lauwerier mapping was nearly Markov partitioned again.It is proved that the Lauwerier mapping is topologically semi-conjugate to the shift map on the inverse limit space of the quadratic mapping when the parameter is under 4.