逐段单调函数的高度与拓扑共轭

Height and Topological Conjugacy of Piecewise Monotonic Functions

迭代运算下,函数值可以交叉于不同的子区间,使得逐段单调函数的高度异常复杂.本文考虑一个非单调点的连续函数类.首先给出高度的充分必要条件,以此获得此类函数的一种划分.其次针对函数类的一个非空子集,给出判定拓扑共轭的充分必要条件和构造拓扑共轭的新方法.进一步地,我们阐明这样的事实：两个逐段单调函数拓扑共轭是其高度相等的充分不必要条件,最后举例说明.

Computing height of piecewise monotonic functions is difficult because the value of functions may interact each other under iteration. In this paper we consider the set of continuous functions with a single non-monotonic point. We first present a sufficient and necessary condition for heights which gives a classification of those functions. Then we provide an equivalent condition and a new construction method for topological conjugacy for a nonempty subset of the mentioned continuous functions.Furthermore, we prove that topological conjugacy is a sufficient but not necessary condition for equal heights of piecewise monotonic functions. Finally, some examples are given to illustrate our results.