Density of periodic orbit measures for transformations on the interval with two monotonic pieces
Density of periodic orbit measures for transformations on the interval with two monotonic pieces
复制标题
两个单调片段区间变换的周期轨道密度测量
DOI：
10.4064/fm_1998_157_2-3_1_221_234
复制
发表时间：
1998
影响因子：
0.6
通讯作者：
P. Raith
中科院分区：
文献类型：
--
作者：
F. Hofbauer;P. Raith
考虑了具有两个单调部分的变换T：[0，1]→[0，1]。在T是拓扑传递且HTOP(T)>；0的假设下，证明了集中在周期轨道上的不变测度在所有不变概率测度的集合中是稠密的。导言。为了研究拓扑动力系统不变测度的一般性质，R.Bowen[2]引入了规范性质。这是一个拓扑性质，它意味着集中在周期轨道上的测度在所有不变测度的集合中是稠密的。规范性质意味着不同类型的不变度量的一般性质，例如遍历度量、非原子度量、具有零熵的度量和强混合度量(见[3])。公理A-微分同胚的基本集([2]，[3])、单调mod-变换([5])和区间上的连续映射([1])都具有规范性质。本文研究由分段单调映射生成的动力系统。如果这些映射具有不连续性，则证明周期轨道测度的密度就变得很复杂。除了不变测度的一般性质外，考虑分段单调映射T的这个问题还有两个原因：[0，1]→[0，1]。我们将在下面描述这些原因。1991年数学学科分类：58F03、58F11、54H20。
Transformations T : [0, 1] → [0, 1] with two monotonic pieces are considered. Under the assumption that T is topologically transitive and htop(T ) > 0, it is proved that the invariant measures concentrated on periodic orbits are dense in the set of all invariant probability measures. Introduction. In order to investigate generic properties of invariant measures for a topological dynamical system R. Bowen [2] introduced the specification property. This is a topological property which implies that the measures concentrated on periodic orbits are dense in the set of all invariant measures. The specification property implies generic properties for different types of invariant measures, e.g. ergodic measures, nonatomic measures, measures with zero entropy and strongly mixing measures (see [3]). It is known that the specification property holds for basic sets of axiom A-diffeomorphisms ([2], [3]), for monotonic mod one transformations ([5]) and for continuous maps on the interval ([1]). We investigate in this paper dynamical systems generated by piecewise monotonic maps. If these maps have discontinuities, it becomes complicated to prove the density of periodic orbit measures. Besides generic properties of invariant measures there are two more reasons to consider this problem for piecewise monotonic maps T : [0, 1]→ [0, 1]. We describe these reasons below. 1991 Mathematics Subject Classification: 58F03, 58F11, 54H20.