Properties of invariant measures in dynamical systems with the shadowing property
Properties of invariant measures in dynamical systems with the shadowing property
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具有阴影特性的动力系统中不变测度的特性
DOI：
10.1017/etds.2016.125
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发表时间：
--
影响因子：
0.9
通讯作者：
OPROCHA PIOTR
中科院分区：
文献类型：
--
作者：
Jian Li;OPROCHA PIOTR
对于具有跟踪性质的动力系统，我们给出了一种用里程计支持的遍历测度及其几乎一一对应的扩展来逼近不变测度的方法。对于具有跟踪性质的拓扑可迁系统，证明了里程表支持的遍历测度在不变测度空间中是稠密的，从而遍历测度在不变测度空间中是泛类的.我们还表明，对于每个$c\geq 0$和$\unicode[STIX]{x1 D700}>0$，熵在$c$和$c+\unicode[STIX]{x1 D700}$之间的遍历测度集合（支持里程表的几乎一对一扩展）在熵至少为$c$的不变测度空间中是稠密的。此外，如果熵函数是上半连续的，则对每个c\geq 0$，熵为c的遍历测度在熵至少为c的不变测度空间中是泛类的。
For dynamical systems with the shadowing property, we provide a method of approximation of invariant measures by ergodic measures supported on odometers and their almost one-to-one extensions. For a topologically transitive system with the shadowing property, we show that ergodic measures supported on odometers are dense in the space of invariant measures, and then ergodic measures are generic in the space of invariant measures. We also show that for every $c\geq 0$ and $\unicode[STIX]{x1D700}>0$ the collection of ergodic measures (supported on almost one-to-one extensions of odometers) with entropy between $c$ and $c+\unicode[STIX]{x1D700}$ is dense in the space of invariant measures with entropy at least $c$ . Moreover, if in addition the entropy function is upper semi-continuous, then, for every $c\geq 0$ , ergodic measures with entropy $c$ are generic in the space of invariant measures with entropy at least $c$ .