The ergodic behaviour of piecewise monotonic transformations
The ergodic behaviour of piecewise monotonic transformations
复制标题
分段单调变换的遍历行为
DOI：
10.1007/bf00538119
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发表时间：
1979
期刊：
影响因子：
--
通讯作者：
G. Wagner
中科院分区：
文献类型：
--
作者：
G. Wagner
设T是单位区间I=[0，1）到自身上的分段单调变换，它满足一些附加条件，这些条件将在后面给出.然后T分裂成一个有限数目的遍历组件在以下意义上：有一个分解的I成100多个不相交的可测集A1. A s和B，使得（a）A p是不变的（TA p = A p mod 0），限制TA a = rp关于Lebesgue测度m是遍历的，（B）在每个A o上存在一个不变测度# p，它等价于m对A p的限制，（c）集合B满足关系T-1 B c B和lim m（T-n B）= 0. n~ co（d）对应于每个分量Ap，对应于变换Tp的幂Tp*= T~ p）和不相交分解Ap= Aplu…wAp，..，（p）使得T*（Ap-）= Ao-mod 0（a= 1，.，re（p）），T* 是Ap~上的正合自同态.变换T o循环地置换Ap，~。
Let T be a piecewise monotonic transformation of the unit interval I=[0, 1) onto itself satisfying some additional conditions to be specified later. Then T splits into a finite number of ergodic components in the following sense: There is a decomposition of I into finitely many disjoint measurable sets A 1..... A s and B such that (a) the Ap's are invariant (TAp= Ap mod 0) and the restrictions TIA= rp are ergodic with respect to Lebesgue measure m,(b) on each A o there exists an invariant measure# p which is equivalent to the restriction of m to Ap,(c) the set B satisfies the relations T-1B c B and lim m (T-nB)= O. n~ co (d) to each component Ap there corresponds a power Tp*= T~ p) of the transformation Tp and a disjoint decomposition Ap= Aplu... wAp,,,(p) such that T*(Ap~)= Ao~ mod 0 (a= 1,..., re (p)) and T* is an exact endomorphism on each Ap~. The transformation T o permutes the Ap,~ cyclically.